The Hardness of Approximating Spanner Problems - CiteSeerX

The Hardness of Approximating Spanner Problems Michael Elkin

∗

David Peleg

†

Abstract This paper examines a number of variants of the sparse k-spanner problem, and presents hardness results concerning their approximability. Previously, it was known that most k-spanner problems are weakly inapproximable, namely, are NP-hard to approximate with ratio O(log n), for every k ≥ 2, and that the unit-length k-spanner problem for constant stretch requirement k ≥ 5 is strongly 1−ǫ inapproximable, namely, is NP-hard to approximate with ratio O(2log n ) [27]. The results of this paper significantly expand the ranges of hardness for k-spanner problems. In general, strong hardness is shown for a number of k-spanner problems, for certain ranges of the stretch requirement k depending on the particular variant at hand. The problems studied differ by the types of edge weights and lengths used, and include also directed, augmentation and client-server variants of the problem. The paper also considers k-spanner problems in which the stretch requirement k is relaxed (e.g., k = Ω(log n)). For these cases, no inapproximability results were known at all (even for a constant approximation ratio) for any spanner problem. Moreover, some versions of the k-spanner problem are known to enjoy the ratio degradation property, namely, their complexity decreases exponentially with the inverse of the stretch requirement. So far, no hardness result existed precluding any k-spanner problem from enjoying this property. This paper establishes strong inapproximability results for the case of relaxed stretch requirement (up to k = O(n1−δ ), for any 0 < δ < 1), for a large variety of k-spanner problems. It is also shown that these problems do not enjoy the ratio degradation property.

Classification: Approximation algorithms, Hardness of approximation

∗

Department of Computer Science, the Ben-Gurion University of the Negev, 84105. This work was done in the Weizmann Institute of Science, Rehovot, Israel. E-mail: [email protected] † Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100 Israel. E-mail: [email protected] Supported in part by grants from the Israel Science Foundation and the Israel Ministry of Science and Art.

1 1.1

Introduction The Sparse Spanner Problem

The concept of graph spanners has been studied in several recent papers, in the context of communication networks, distributed computing, robotics and computational geometry [3, 9, 14, 17, 15, 29, 32, 33]. Consider a connected simple graph G = (V, E, ω, l), with |V | = n vertices, where ω : E → R+ is a weight function on the edge set of the graph, and l : E → R+ is a length function on the edge set of the graph. For every pair of vertices u, v ∈ V , let P (u, v, G) be the set of all simple paths from u to v in G. We P define the distance between u and v in G to be dist(u, v, G) = minP ∈P (u,v,G) e∈P l(e). A subgraph ′

) G′ = (V, E ′ ) of G is a k − spanner if for every u, v ∈ V , dist(u,v,G dist(u,v,G) ≤ k. We refer to k as the stretch factor of G′ . Spanners for general graphs were first introduced in [33], and used to construct a new type of synchronizer for an asynchronous network. Since then, spanners were intensively used in distributed computing for synchronization [33, 5, 7], routing [34, 6, 36], approximate distance computation [19, 20] and online load balancing [4]. In all of these applications, the spanner can be thought of as a sparse backbone of the network. The ability to compute such a backbone efficiently is, consequently, crucial for applications in distributed computing and communication networks. For most applications, it is desirable that the spanner be as sparse or as light as possible, namely, has few edges or small total weight. This leads to the following problem. The cost of a subgraph G′ is its P weight, ω(G′ ) = e∈E ′ ω(e). The goal of the k-spanner problem is to find a k-spanner G′ = (V, E ′ ) with the smallest cost ω(G′ ). A number of variants of the sparse spanner problem have been considered in the literature. The general k-spanner problem allows arbitrary edge weights and lengths. However, the most basic variant of the sparse spanner problem deals with the simple unweighted uniform case, where ω(e) = l(e) = 1 for every edge e ∈ E [32, 33]. We call this variant the unweighted (or basic) k-spanner problem. In-between, one may consider a number of intermediate variants. The first is the unit-length k-spanner problem studied in [27]. In this case, the weight function ω may be arbitrary, but the length function l assigns l(e) = 1 to every edge e ∈ E. An important special case of the unit-length k-spanner problem is when the weight function ω may assign only 0 and 1 values. This problem is called the light-edges (LE) k-spanner problem, and it is equivalent to the k-spanner augmentation problem studied in [22]. Intuitively, such a Boolean function ω : E → {0, 1} captures the situation where in addition to the target graph G, we are given also an initial partially constructed subnetwork H ′ , whose edges are assumed to be given in advance for free, and it is required to augment the subnetwork H ′ into a k-spanner H for G, where edges not in H ′ must be “paid for” in order to be included in the spanner. We denote the set of zero-weight edges by L. The aim is to minimize the number of new edges needed in order to obtain a k-spanner for the given graph. A second variant, which can be thought of as the dual of the unit-length k-spanner problem, is the unit-weight k-spanner problem, studied in [2, 13]. In this case, the length function l may be arbitrary, but the weight function ω assigns ω(e) = 1 to every edge e ∈ E. Finally, a third variant considered in the literature is the uniform k-spanner [3, 2, 13, 35]. In this case, the weight and length functions coincide, i.e. ω(e) = l(e) for every edge e ∈ E, but that function ω may be arbitrary. Any one of the above versions may be also generalized to the client-server (C-S) k-spanner problem, see [22]. This is a generalization of the k-spanner problem which distinguishes between the two different roles of edges in the problem, i.e., the input specifies also a subset C of client edges which have to be spanned, and a subset S of server edges which may be used for spanning the client edges. We also distinguish three subcases of the C-S k-spanner problem. The first is the disjoint C-S k-spanner (hereafter, DJ k-spanner)

1

problem. In this variant the client and server sets are disjoint. The second is the all-client C-S k-spanner (hereafter, AC k-spanner) problem, in which the server set is a subset of the client set. Finally, the last variant is the all-server C-S k-spanner (hereafter, AS k-spanner) problem, in which the client set is a subset of the server set.

1.2

Constructions and Approximation Algorithms

It is shown in [32] that the problem of determining, for a given unweighted graph G = (V, E) and an integer m, whether there exists a 2-spanner with m or fewer edges is NP-complete. This indicates that it is unlikely to find an exact solution for the sparsest k−spanner problem even in the case k = 2. Consequently, two possible remaining courses of action for investigating the problem are establishing global bounds on the number of edges required for an unweighted k-spanner of various graph classes and devising approximation algorithms for the problem. In [32] it is shown that every unweighted n−vertex graph G has a polynomial time constructible (4k + 1)−spanner with at most O(n1+1/k ) edges. Hence in particular, every graph G has an O(log n)−spanner with O(n) edges. These results are close to the best possible in general, as implied by the lower bound given in [32]. The results of [32] were improved and generalized in [3, 2] to the uniform case, in which the edges weights and lengths coincide. Specifically, it is shown in [3] that given an n−vertex graph and an integer 1 k ≥ 1, there is a polynomially constructible (2k + 1)−spanner G′ such that |E(G′ )| < n · ⌈n k ⌉. Again, this result is shown to be asymptotically the best possible. In [13] it was shown that the weight of the 2+ǫ uniform k-spanner obtained by the construction of [3] is bounded by ω(G′ ) = O(n k−1 · ω(M ST )). They also show how the construction can be used to provide uniform log2 n-spanners with weight bounded by ω(G′ ) = O(ω(M ST )). The algorithms of [3, 32, 2] provide us with global upper bounds for sparse k−spanners, i.e., general bounds that hold for every graph. However, it may be that for specific graphs, considerably sparser spanners exist. Furthermore, the upper bounds on sparsity given by these algorithms are small (i.e., close to n) only for large values of k. It is therefore interesting to look for approximation algorithms, that yield near-optimal bounds for the specific graph at hand. |E| In [28], a log |V | approximation algorithm was presented for the unweighted 2-spanner problem. In [27] the result was extended to an O(log n)-approximation algorithm for the unit-length 2-spanner problem. A log |V|C| (C)| -approximation algorithm for the unit-length C-S 2-spanner problem was presented in [22]. Approximation algorithms with ratio log |V|C| (C)| are given also in [22] for a number of other variants of the problem, such as the unit-length 2-spanner augmentation problem and directed unit-length 2-spanner (augmentation) problems. Also, since any k-spanner for an n-vertex graph requires at least n − 1 edges, the results of [3, 2, 32] cited above can be interpreted as providing an O(n1/k )-ratio approximation algorithm for the (unweighted or weighted) uniform k-spanner problem. This implies that once the required stretch guarantee is relaxed, i.e., k is allowed to be large, the problem becomes easier to approximate. In particular, the unweighted k-spanner problem admits O(1) approximation once the stretch requirement becomes k = Ω(log n), and the uniform k-spanner problem admits an O(1) approximation ratio once the stretch requirement becomes k = Ω(log2 n). We call this property ratio degradation.

1.3

Hardness of Approximation

In this paper we consider the hardness of approximating spanner problems. Previously known hardness results for spanner problems were of two types. First, it is shown in [27] that it is NP-hard to approx-

2

The Type of k-spanner problem

The Range of the Strong Hardness Proven in the Paper

Uniform Unit-weight Directed DJ AC C-S Augmentation Unit-weight DJ Unit-weight AS Unit-length

1 0 will cause classes III and IV of [26] to collapse.

2

The MIN-REP Problem

Following [27], we define the MIN-REP problem as follows. We are given a bipartite graph G(V1 , V2 , E), S S where V1 and V2 are each split into a disjoint union of r sets; V1 = ri=1 Ai and V2 = ri=1 Bi . The sets Ai , Bi all have size N . An instance of the problem consists of the 5-tuple (V1 , V2 , E, {Ai }, {Bi }). The bipartite graph and the partition of V1 and V2 induce a supergraph H, whose vertices are the sets Ai and Bj , where i, j ∈ {1, .., r}. Two sets Ai and Bj are adjacent in H iff there exists some ai ∈ Ai and bj ∈ Bj which are adjacent in G. We assume that H is regular and (but its degree, d = deg(H), need not be O(1)). A set of vertices C is a REP-cover for H if for each super-edge (Ai , Bj ) there is a pair ai ∈ Ai and bj ∈ Bj , both belonging to C, such that (ai , bj ) ∈ E. It is required to select a minimal REP-cover C for H. Note that it is easy to test whether a MIN-REP instance admits a REP-cover, just by checking whether the all vertex set V1 ∪ V2 REP-covers all the superedges. Thus we can assume without loss of generality, that the given instance admits a REP-cover. Consider also the maximization version of this problem, called MAX-REP. In this version the REPcover C may contain at most one vertex from each supernode. It is shown in [27] that the MIN-REP problem is strongly inapproximable. Another close problem is the Label-Cover problem [26]. In this problem a superedge (Ai , Bj ) is covered if for every vertex ai ∈ Ai ∩ C there is a vertex bj ∈ Bj ∩ C such that (ai , bj ) ∈ E. It also has the 5

minimization and maximization versions called Label − CoverM IN and Label − CoverM AX respectively. Note that the Label − CoverM AX problem is equivalent to the MAX-REP problem. As recently shown in [16], for every 0 < ǫ < 1 it is NP-hard to approximate the Label − CoverM IN 1−ǫ problem with O(2log n ) approximation ratio. It was previously known that it is hard to approximate the Label-Cover problem with this ratio unless N P ⊆ DT IM E(npolylog n ) [25]. On the positive side, we show several results concerning the MIN-REP problem. 2

Theorem 2.1 The MIN-REP problem with girth(H) ≥ t admits an O(n t )-approximation algorithm. Proof: We note that by [8, 31], girth(H) ≥ t implies 2

|E(H)| ≤ r 1+ t−2 + r . We first show the claim for the MIN-REP problem. Thus if for every superedge in H we pick an arbitrary edge that REP-covers it and insert the two endpoints of the edge into the REP-cover, we 2 2 obtain a REP-cover of size O(r 1+ t−2 + r). Since |C| ≥ 2r, it is an O(r t−2 ) approximation for the n problem. Also by taking all the n = 2N r nodes to the cover we obtain an N = 2r approximation. 2/t These two approximations provide us a total O(n ) approximation ratio. Indeed, by setting r = nα , 2 the two approximation algorithms mentioned above yield approximation ratios of O(nα· t−2 ) and O(n1−α ) respectively, hence combining the two algorithms yields an algorithm with approximation ratio nβ for 2α β 1−α = n2/t , yielding , 1 − α}. This value is maximized for α = t−2 β = β(α) = min{ t−2 t , for which n = n the desired approximation ratio. Let’s now show the claim for the Label − CoverM IN . The problem is that here taking the two endpoints for each superedge may not satisfy the second requirement of the Label − CoverM IN problem. So we use a slightly more sophisticated argument. We call a cover C that satisfies both conditions of the Label − CoverM IN problem an LCMIN-cover. We assume without loss of generality that there exists an LCMIN-cover to the instance. This can be checked in polynomial time at the beginning. We pass through all the left supernodes A1 , . . . , Ar oneby-one. For each Ai we consider the set of the superedges adjacent to it. There exists a node ali ∈ Ai such that for every supernode Bj adjacent to Ai there exists a node brj ∈ Bj such that (ali , brj ) ∈ E. Indeed, if there is no such ali for some Ai , then there is no LCMIN-cover for the instance, contradicting the assumption. For each of the edges (ali , brj ) as above we take to the cover C both endpoints (ali is common to all of them for the same i). The choice is an LCMIN-cover, since we have exhausted this way S all the superedges and for every ali ∈ C ∩ ( Ai ) there is a brj ∈ C ∩ Bj for every Bj adjacent to Ai . Also 2

|C| ≤ 2|E(H)|, because for every superedge we put into C upto 2 new vertices. Thus we obtain O(r t−2 ) approximation. Also we have O(N ) = O(n/r) approximation by putting into the cover all the right representatives and for each left supernode Ai putting into the cover one representative ali as above. This representatives will have preimages in all the neighboring right supernodes in the cover. 2

Corollary 2.2 1. Label − CoverM IN with girth(H) ≥ t admits an O(n t )-approximation algorithm. 2. MIN-REP and Label − CoverM IN with girth(H) = O(log n) admit an O(1) approximation ratio. 3, For any 0 < ǫ < 1 there exists 0 < ǫ′ < 1 such that MIN-REP and Label − CoverM IN with girth(H) = O(log1−ǫ n) admit an O(2log

1−ǫ′

n)

approximation algorithm.

Also, since girth(H) ≥ 4 for any MIN-REP instance (in fact for any bipartite graph H), we conclude √ Corollary 2.3 The MIN-REP and Label − CoverM IN problems admit a n-approximation algorithm. 6

3

Low Stretch Inapproximability

In this section we start by extending the lower bound established in [27] for the unit-length k-spanner problem for every k ≥ 3. This extension has recently been established independently by Dodis and Khanna [18], by a different reduction. Our reduction for this problem is a natural starting point for presenting all our later results. We then establish a similar lower bound on the hardness of the k-spanner augmentation problem for k ≥ 4.

3.1

The Unit-Length k-Spanner Problem

In the unit-length k-spanner problem we are given a weight assignment to the edges of the graph and the lengths of all the edges are 1. The objective is to find the lightest (or minimum-weight) k-spanner, where the weight of the k-spanner is the sum of the weights of the edges participating in the spanner. The reduction from MIN-REP to the unit-length 3-spanner problem is as follows. Let (V1 , V2 , E, {Ai }, {Bi }) be an instance of MIN-REP problem. Then we construct an instance of the unit-length 3-spanner prob¯ = (V¯ , E) ¯ , where lem, defined by the following graph G V¯

=

r [

i=1

Ai ∪

r [

i=1

(i,j)

Bi ∪ {si , ti }ri=1 ∪ {d1

(i,j)

, d2

(i,j)

, c1

(i,j)

| 1 ≤ i ≤ r, 1 ≤ j ≤ N } ,

, c2

¯ = E ∪ E ′ ∪ EsA ∪ EtB ∪ EH , E

with

E′ = EsA = EtB =

r [ N [

i=1 j=1 r [

i=1 r [

i=1

(i,j)

{(si , d1

(i,j)

), (d1

(i,j)

, d2

(i,j)

), (d2

, aji )} ∪

r [ N [

i=1 j=1

(i,j)

{(ti , c1

(i,j)

), (c1

(i,j)

, c2

(i,j)

), (c2

, bji )} ,

{(si , ali ) | ali ∈ Ai },

{(bli , ti ) | bli ∈ Bi },

EH = {(si , tj ) | (Ai , Bj ) ∈ H} . The weight assignment on the edges is as follows (see Figure 1).    0,

e ∈ E ∪ E′ ω(e) = 1, e ∈ EsA ∪ EtB   ∞, e ∈ E H

Let us briefly provide some intuition for the way the reduction operates. The vertices si and tj represent the supernodes Ai and Bj , respectively. The edges of EH , between si and tj , correspond exactly to the edges of the supergraph H. We assign these edges the weight ∞ in order to prevent the optimal spanner from using them. On the other hand, the spanner will use intensively the edges of E, from the original graph. In the MIN-REP problem we pay only for the vertices of the original graph which are taken into the REP-cover. For this reason, we assign the edges of E zero weight. Finally, taking any edge of EsA , connecting si and some vertex of Ai (or any edge of EtB , connecting tj and some vertex of Bj ) into the spanner represents taking this vertex from Ai (or Bj ) into the REP-cover. Since we are interested in minimizing the number of vertices taken into the REP-cover, we assign these edges a unit weight. Hence, the weight of any finite-weight spanner equals the number of EsA ∪ EtB edges it uses. 7

E G~ S

R

L U1 E sU

T

W1 E tW EM

Figure 1: Solid lines represent unit weight edges, dotted lines represent zero weight edges, dashed lines represent infinity weight edges.

¯ has a 3-spanner of weight K iff the given MIN-REP instance has a REP-cover of size Lemma 3.1 G K. ¯ of weight K < ∞. Then we build a REP-cover by the Proof: First, suppose we have a spanner H for G following rule: For every vertex v ∈ V1 ∪ V2 , if there exists an edge e ∈ H adjacent to v then add v to the REP-cover C. ¯ Since H 3-spans G, ¯ there is To see that C is a REP-cover for H, let (Ai , Bj ) ∈ H. Then (si , tj ) ∈ E. a 3-long path in H spanning (si , tj ). This path has to include (ai , bj ) ∈ E for some ai ∈ Ai and bj ∈ Bj . Therefore ai , bj ∈ C, and so C REP-covers (Ai , Bj ). It remains to show that |C| = K. Assuming ω(H) < ∞, the only edges of weight 1 in H are edges (si , ali ) for some 1 ≤ i ≤ r and ali ∈ Ai and (ti , bli ) for some 1 ≤ i ≤ r and bli ∈ Bi . But each such edge causes exactly one vertex (ali in the first case, bli in the second) to enter C. Two different edges of these S types cannot intersect at a vertex from ri=1 (Ai ∪ Bi ), and hence |C| = K. Conversely, suppose there is a REP-cover C of size |C|. We build a spanner H of weight |C| in the following way: 1. Take all zero-weight edges into H. 2. For every ali ∈ C, take (ali , si ) into H. 3. For every bli ∈ C, take (bli , ti ) into H.

¯ All zero-weight edges (including, in particular, E ′ edges) are We argue that H is a 3-spanner for G. in the spanner, hence are spanned. All unit-weight edges are spanned by E ′ edges. ¯ hence (Ai , Bj ) ∈ H. Since C is a REPConsider some ∞-weight edge (si , tj ). It belongs to E, l m m l cover, there exist some ai in Ai and bj in Bj for which (ali , bm j ) ∈ E and ai , bj ∈ C. It follows l m l m that (ali , si ), (bm j , tj ) ∈ H. Also observe that ω(ai , bj ) = 0, and so (ai , bj ) ∈ H. Hence the edges m (si , ali ), (ali , bm j ), (bj , tj ) are all in H, and they 3-span (si , tj ). Finally, we have to show that ω(H) = |C| . Since H consists of only zero- and unit-weight edges, it remains to argue that the number of unit-weight edges in H is exactly |C|. Indeed, for every vertex v in C we take into H exactly one edge ((v, si ) for v = ali and (v, tj ) for v = blj ). The lemma follows. 8

Theorem 3.2 For any ǫ > 0 and constant integer k ≥ 3, it is quasi-NP-hard to approximate the unit1−ǫ length k-spanner problem with ratio 2log n .

3.2

The k-Spanner Augmentation Problem 1−ǫ

Now, we prove a similar lower bound of 2log n -inapproximability for the unit-length k-spanner augmentation problem for k ≥ 4. ¯ is similar to the previous one, but instead of edges of EH , connecting The construction of the graph G pairs (si , tj ) with weight ∞, we now have an edge (si , pij ) of weight zero and an edge (pij , tj ) of weight 1, where pij is a new node. Let EsP denote the set of all edges (si , pij ). The set of zero-weight edges L is, as previously, E ∪ E ′ . Also, for each edge (si , ali ) ∈ EsA and (brj , tj ) ∈ EtB we now add a new length-4 path of zero-weight edges that spans it. Suppose this graph has 4-spanner H of size K. Then we construct a REP-cover of size K or less in the following way. We start with finding another 4-spanner H ′ of the same or smaller size K ′ ≤ K, which contains no edge of type (pij , tj ). (The existence of such H ′ is guaranteed by Lemma 3.3.) Then we use S this spanner H ′ to build a REP-cover by the following rule: For every v ∈ ri=1 (Ai ∪ Bi ), take v into the REP-cover if and only if there exists some edge e ∈ H incident to v. ¯ has a 4-spanner H ′ with no edge of type (pij , tj ), that satisfies ω(H ′ ) ≤ 3ω(H ∗ ), Lemma 3.3 The graph G where H ∗ is an optimal spanner. ¯ If H does not contain edges of the form (pij , tj ) Proof: Let H be any minimum weight 4-spanner for G. then we are done. Otherwise, for every e = (pij , tj ) ∈ H, consider all the length-4 paths P ′ in H that ¯ =E ¯ and contain e. Such paths may be of one of the following four forms: 4-span some edge e′ in E(G) 1. P ′ spans some e′ = (ali , brj ) ∈ E. Specifically, P ′ = ((ali , si ), (si , pij ), (pij , tj ), (tj , brj )). 2. P ′ spans some e′ = (si , pij ) ∈ EsP . Specifically, P ′ = ((si , ali ), (ali , brj ), (brj , tj ), (tj , pij )). 3. P ′ spans some e′ = (si , ali ) ∈ EsA . Specifically, P ′ = ((ali , brj ), (brj , tj ), (tj , pij ), (pij , si )). 4. P ′ spans some e′ = (tj , brj ) ∈ EtB . Specifically, P ′ = ((brj , ali ), (ali , si ), (si , pij ), (pij , tj )). We modify H into the desired H ′ by arbitrarily picking one of the paths P ′ in which e participates and making the following replacement. Insert into H ′ the edge e′ previously spanned by P ′ and delete from H ′ the edge (pij , tj ). As a result, the weight of the spanner is either reduced by 1 or remains unchanged, and each of the five edges of the cycle C ′ = P ′ ∪ {e′ } is either self-spanned or 4-spanned by the other four. The only difficulty that might arise is if (pij , tj ) participates in another path for some other pair ′ ′ ′ ′ ail , brj , where ali may be equal to ali and brj may be equal to brj , but not both. For each such other path, there will be an unspanned edge e′ in H ′ . But in cases (1) and (2) this edge is zero-weight and therefore is in H ′ . In cases (3) and (4) the unspanned edges have weight 1, but by the construction they have an alternative 4-spanning path of zero-weight, and thus are 4-spanned by H ′ . Note also that it is possible that an edge e of this type does not participate in any spanning path of length 4 or less, but it self-spans the edge e itself. On the other hand, since the original MIN-REP ¯ contains a spanning path of length 4 for e that does not use instance admits a REP-cover, the graph G the edges of type (pij , tj ). Thus for such e, we just pick an arbitrary path as above and insert its edges into the spanner (if they were not there). Also we remove the edge e from the spanner. Since the weight of the path is 3, we increase the total weight of the spanner by at most a constant factor of 3. The validity of the reduction can now be proved in the same way as Lemma 3.1. 9

Theorem 3.4 For any ǫ > 0 and constant integer k ≥ 4, it is quasi-NP-hard to approximate the unit1−ǫ length k-spanner augmentation problem with ratio 2log n .

4

Unweighted Spanner Problems

In this section we show that the unweighted client-server k-spanner problem for k ≥ 3, also admits 1−ǫ no 2log n -ratio approximation, for any ǫ > 0. In view of the O(log |V|C| (C)| )-ratio approximation algorithm devised in [22] for k = 2, the behavior of this problem, like the unit-length k-spanner problem, becomes almost fully understood (the only unsettled issue for these problems concerns the existence of nα 1−ǫ approximation algorithms for k ≥ 3 with small α values). In fact, we show this 2log n -inapproximability result for two subproblems of the unweighted C-S k-spanner problem, the unweighted DJ k-spanner problem and the unweighted AC k-spanner problem. Each of these results implies the same for the unweighted C-S k-spanner problem.

4.1

A Preliminary Lemma

For the proofs, we need the following construction. For any REP-cover C, the set E(C) is defined as follows. l m ˜ ˜ E(C) = {E(C) ⊆ E(C) | ∀(Ai , Bj ) ∈ H, ∃(ali , bm j ) ∈ E(C) s.t. ai ∈ Ai and bj ∈ Bj } .

˜ Lemma 4.1 For every REP-cover C there is a polynomial time constructible edge set E(C) ∈ E(C) of d d−1 ˜ size |E(C)| = O(r ), where the degree of the regular supergraph H is deg(H) = r , for some constant 1 < d < 2. Proof: We go over all supernodes Ai , and for each of them we consider the superedges (Ai , Bj1 ), .., (Ai , Bjrd−1 ) it participates in. Since C is a REP-cover, for each such superedge (Ai , Bjl ) there exist nodes api , btjl in the cover C, such that (api , btjl ) belongs to E(C). ˜ We take into E(C) for each l = 1, 2, .., r d−1 one of such edges (api , btjl ). Thus overall we take at most r d edges, hence for every supernode Ai we take r d−1 edges and there are r supernodes Ai . Note also that ˜ ˜ the edge set E(C) is in E(C), because for every superedge (Ai , Bj ) ∈ H, we have inserted into E(C) an l m l m edge (ai , bj ), for ai ∈ Ai and bj ∈ Bj .

4.2

The Unweighted C-S k-Spanner Problems

The hardness of the unweighted DJ C-S k-spanner problem is proven by an extension of the reduction from the MIN-REP problem given in Section 3.1. Given a MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }) we build ¯ as previously, but without vertices c(i,j) and d(i,j) and without the edges E ′ . Also previously the graph G k k each node of a supernode Ai was connected to a node si , whereas now, each node of a supernode Ai is connected to x nodes s1i , ..., sxi and similarly, each node of a supernode Bj is connected to x nodes t1j , ..., txj , for x = r d−1 . Also we extend the definition of EH to be EH = {(spi , tpj ) | (ai , Bj ) ∈ H, 1 ≤ p ≤ x} . Let the set of clients C be EH and let the set of servers S contain all the remaining edges. Let us start with the following two lemmas. Lemma 4.2 Given a 3-spanner H for C there is a polynomial time constructible REP-cover C for (V1 , V2 , E, {Ai }, {Bj }) of size |C| ≤ |H| x . 10

Proof: Let H be a 3-spanner for C. For p = 1, 2, .., x define the sets H1 , .., Hx by l m Hp = {(spi , ali ), (tpj , bm j ) ∈ H | ai ∈ Ai , bj ∈ Bj } .

Also let HE be H ∩ E. Then H =

p [

i=1

Hp ∪ HE .

Note that Hp ∩ Hq = φ for 1 ≤ p 6= q ≤ x. Let 1 ≤ p0 ≤ x be an index such that |Hp0 | = min {|Hp |} . 1≤p≤x

Now we define the REP-cover C by p0 l p0 m C = {ali , bm j | (si , ai ), (tj , bj ) ∈ Hp0 } .

(1)

To prove that C is a REP-cover, we consider some superedge (Ai , Bj ) ∈ H. By the construction of the ¯ there is an edge e = (sp0 , tp0 ) in C. Since H is a 3-spanner for C, there is a spanning path for e graph G i j in H. By the construction, the only possible form of a spanning path for e of length not greater than 3 m l m p0 that uses only server edges is (spi 0 , ali ), (ali , bm j ), (bj , tj ), for some ai ∈ Ai and bj ∈ Bj . Thus the edges p0 p0 l m p0 (spi 0 , ali ), (bm j , tj ) are in the spanner H, and thus by definition of Hp0 , (si , ai ), (bj , tj ) ∈ Hp0 . Hence, ali , bm j ∈ C REP-cover the superedge (Ai , Bj ). For the bound on the size of the REP-cover C we note that |C| = |Hp0 | ≤

[ 1 1 ·| · |H| , Hp | ≤ x 1≤p≤x x

completing the proof. For the opposite direction we prove the following lemma. Lemma 4.3 Given a REP-cover C for the MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }), there is a polynomial time constructible 3-spanner H of C of size |H| ≤ 2x|C|. Proof: Let C be a REP-cover for the MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }). We build a 3-spanner H by l m ˜ H = {(spi , ali ), (tqj , bm j ) | ai , bj ∈ C, 1 ≤ p, q ≤ x} ∪ E(C) . It is easy to verify that H is a 3-spanner for C. Also, since any vertex in the cover C induce x edges of the spanner H, |H| = x|C| + r d . Recall that |C| ≥ r. Recall that x = r d−1 . This implies

|H| ≤ x|C| + r d ≤ 2x|C| , as required. Now we are ready to prove Theorem 4.4 itself. Theorem 4.4 For any ǫ > 0 and constant integer k ≥ 3, it is quasi-NP-hard to approximate the un1−ǫ weighted DJ k-spanner problem with ratio 2log n .

11

Proof: Suppose, towards contradiction, that there is an algorithm A that approximates the unweighted 1−ǫ DJ k-spanner problem within a ratio 2log n , for some 0 < ǫ < 1. Then given a MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }), we build an instance of the DJ 3-spanner problem as above. Then we invoke the algorithm and obtain a 3-spanner H. Using the spanner we build a REP-cover C as in Lemma 4.2. The REP-cover satisfies 1−ǫ 1 1 |C| ≤ |H| ≤ · 2log n¯ |H ∗ | , x x ∗ ¯ where n ¯ is the number of vertices in G and H is an optimal DJ 3-spanner for C. Note that n ¯ = nx ≤ nd ′ and thus for any 0 < ǫ < ǫ < 1 and sufficiently large n 2log

1−ǫ

n ¯

< 2log

1−ǫ′

n

,

(2)

implying

1 log1−ǫ′ n ∗ ·2 |H | . (3) x Let C˜ be an optimal REP-cover. As we have shown in Lemma 4.3 there is a polynomial time constructible ˜ such that 3-spanner H ˜ ≤ 2x|C| ˜ . |H| (4) |C| ≤

Since H ∗ is an optimal 3-spanner and by (3) and (4) it follows that |C| ≤

′ 1−ǫ′ 1−ǫ′ 1 1 1 log1−ǫ′ n ∗ n ˜ n ˜ = O(2log1−ǫ n )|C| ˜ , ·2 |H | ≤ · 2log |H| ≤ · 2log · 2x|C| x x x

contradicting the fact that the MIN-REP problem is NP-hard to approximate with O(2log any 0 < ǫ < 1.

1−ǫ

n)

ratio for

Next we prove the same hardness result for the unweighted AC k-spanner problem. This is done by a reduction similar to that of the previous section, with two changes. First, we add clique edges between the nodes ali for the same i, and between the nodes bm j for the same j. Denote the set of edges by EQ . ¯ (it has to be so by definition Secondly, the client edge set becomes now the set of all edges of the graph G of the AC problem), and the set of server edges consists of all the edges except those of EH . d Note that the EQ edges do not help to 3-span any edge of EH . We set x = 2N 2 + nr . Hence the following lemma is proved exactly as Lemma 4.2. Lemma 4.5 Given a 3-spanner H for C there is a polynomial time constructible REP-cover C for (V1 , V2 , E, {Ai }, {Bj }) of size |C| ≤ |H| x . In the opposite direction, we have the following lemma. Lemma 4.6 Given a REP-cover C for the MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }), there is a polynomial time constructible 3-spanner H of C of size |H| ≤ 2x|C|. Proof: Here the construction is slightly changed. Specifically. l m H = {(spi , ali ), (tqj , bm j ) | ai , bj ∈ C, 1 ≤ p, q ≤ x} ∪ E(G) ∪ EQ .

¯ Since |C| ≥ r, the spanner’s size is thus bounded by Obviously, H is a 3-spanner for the graph G. |H| ≤ x|C| + nd + 2rN 2 ≤ x(|C| + r) ≤ 2x|C| .

Theorem 4.7 For any ǫ > 0 and constant integer k ≥ 3, it is quasi-NP-hard to approximate the un1−ǫ weighted AC k-spanner problem with ratio 2log n . Proof: We note that now n ¯ ≤ nx ≤ n3 , as d < 2 and N < n, so (2) still holds for any 0 < ǫ′ < ǫ < 1, and for sufficiently large n. The theorem follows. 12

4.3

The Unweighted Directed k-Spanner Problem

Now we consider the unweighted directed k-spanner problem. This problem was presented in [32], where it was shown that there exist infinitely many graphs on which any k-spanner requires Ω(n2 /k2 ) edges. This proved that no global upper bound of nα for some 0 < α < 1 can be provided for the directed unweighted k-spanner problem. However, this says nothing about the ability to find a good approximation algorithms for the problem, since for the graphs in which the k-spanner required a large number of edges, the optimal spanner contains all the edges of the graph and thus can be found easily. The only hardness result known for the problem is log n (weak) inapproximability [27] for k ≥ 2. We significantly improve the threshold 1−ǫ for k > 2 by showing that the problem admits no 2log n approximation ratio, for any 0 < ǫ < 1. On |E| the other hand, for k = 2 a log |V | -approximation algorithm for the problem was provided in [22]. For the proof of our result we use a construction that is very close to the one used for the unweighted ¯ remain the same, with the only AC k-spanner problem. The vertex set and the edge set of the graph G change that all the edges are now directed from left to right with respect to Figure 1, except the EQ edges which are bidirected. Also all edges now act as both client and server edges, as the problem at hand does not allow other types of edges. Note that the arc directions are chosen in such a way that EH edges cannot be used to 3-span any edge, except themselves. Hence by proof arguments similar to those of Lemma 3.3 and Theorem 4.7 we show the following. ¯ has a 3-spanner H ′ with no edges of EH , that satisfies ω(H ′ ) ≤ 3ω(H ∗ ), Lemma 4.8 The graph G ∗ where H is an optimal spanner. Theorem 4.9 For any ǫ > 0 and constant integer k ≥ 3, it is quasi-NP-hard to approximate the un1−ǫ weighted directed k-spanner problem with ratio 2log n .

5

Basic κ-Spanner Problem

We next present a reduction from the M IN REPk+1 problem to the basic k-spanner problem. Let M = (V1 , V2 , E, {Ai }, {Bi }) be a M IN REPk+1 instance. It can be checked in a polynomial time whether there exists a REP-cover C for M, by checking whether V1 ∪ V2 REP-covers all the superedges. Thus, k−3 without loss of generality, we assume that there exists a REP-cover for G. We set kl = ⌊ k−3 2 ⌋, kr = ⌈ 2 ⌉, 2 ¯ as follows (see Fig. 1). Define new vertex sets and x = n /2r, and build the graph G x r k[ l +1 [ [

S =

{spii′ }

and

T =

{(spii′ , spi(i′ +1) )} ∪

p=1 i=1 i′ =1

EsU =

r [ x [

i=1 p=1

EH =

x [

{(spi1 , ui ) | ui ∈ Ui } and

p=1

(5)

¯ = E ∪ EsU ∪ EtW ∪ EH ∪ EM , where and E kl x [ r [ [

EM =

{tpjj ′ } ,

j=1 j ′ =1 p=1

i=1 i′ =1 p=1

and set V¯ = V1 ∪ V2 ∪ S ∪ T

x r k[ r +1 [ [

p EH ,

kr [ x [ r [

{(tpjj ′ , tpj(j ′ +1) } ,

EtW =

r [ x [

i=1 p=1

{(wj , tpj1 ) | wj ∈ Wj },

p where EH = {(spi(kl +1) , tpj(kr +1) ) | (Ui , Wj ) ∈ H} .

13

(6)

p=1 j ′ =1 j=1

(7)

(8)

i,j Denote also EH = {(spi(kl +1) , tpj(kr +1) ) | (Ui , Wj ) ∈ H, 1 ≤ p ≤ x}. Observe that for k = 3, kl = kr = 0 and thus EM = ∅. p Note also that each set of edges EH is an isomorphic copy of the supergraph H. The graph is built in such a way that all the edges except EH can be easily spanned, and the proof is based on establishing a connection between the size of the REP-cover and the number of edges required for spanning the edges of EH . The following two observations are immediate.

Observation 5.1 1. No path k-spanning an EH edge can pass through some other EH edge and an E edge. 2. No path using only EH edges can k-span another EH edge. The first observation holds because otherwise the length of the path would be longer than k. The second follows because girth(H) > k + 1. p cannot help each other in spanning their edges. The next lemma claims, intuitively, that the copies EH p q Lemma 5.2 No path P such that P ∩ EH 6= ∅ can be used to k-span an edge from EH , for p 6= q.

Proof: Suppose, for contradiction, that there exists a path P which k-spans some edge (sqi′ (kl +1) , tqj′ (kr +1) ) ∈ q EH while using an edge (spi(kl +1) , tpj(kr +1) ) ∈ P , for p 6= q. Then the path P must “cross” over from copy q to copy p of the supergraph and back. Hence one of the following three possibilities holds: 1. There exist 1 ≤ i′′1 , i′′2 ≤ r such that P passes through spi′′ (kl +1) and sqi′′ (kl +1) and through spi′′ (kl +1) 2 1 1 and sqi′′ (kl +1) . 2

2. There exist 1 ≤ i′′ ≤ r and 1 ≤ j ′′ ≤ r such that P passes through spi′′ (kl +1) and sqi′′ (kl +1) and through tpj′′ (kr +1) and tqj′′ (kr +1) . 3. There exist 1 ≤ j1′′ , j2′′ ≤ r such that P passes through tpj′′ (kr +1) and tqj′′ (kr +1) and through tpj′′ (kr +1) 2 1 1 and tqj′′ (kr +1) . 2

It is easy to see that in any of these three cases, the subpaths between the mentioned nodes only (in the first case, the subpaths from spi′′ (kl +1) to sqi′′ (kl +1) and from spi′′ (kl +1) to sqi′′ (kl +1) ) contain together at 1

2

1

2

least 2(kl + 2) ≥ 2⌊ k−3 2 ⌋ + 4 ≥ k edges that are not in EH . We also assumed that |P ∩ EH | ≥ 1, thus |P | ≥ k + 1, contradicting the assumption that it k-spans the edge (sqi′ (kl +1) , tqj′ (kr +1) ). i,j i,j edge e such that e 6∈ P . 6= ∅ can k-span any EH Corollary 5.3 No path P such that P ∩ EH i,j Proof: Suppose, for contradiction, that there is a path P such that e1 = (spi(kl +1) , tpj(kr +1) ) ∈ P ∩ EH

p i,j and is k-spanned by P and e2 6∈ P . Thus p 6= q. Hence e1 ∈ P ∩ EH and e2 = (sqi(kl +1) , tqj(kr +1) ) ∈ EH q e2 ∈ P ∩ EH , contradicting Lemma 5.2.

Lemma 5.4 Let P be a simple path such that P ∩ EH 6= ∅. Let the edge e satisfy e ∈ EH \ P . Then P does not k-span the edge e. Proof: Suppose, for contradiction, that there exists a simple path P such that e1 = (spi(kl +1) , tpj(kr +1) ) ∈ P ∩ EH and P k-spans some e2 = (sqi′ (kl +1) , tqj′ (kr +1) ) ∈ EH \ P .

14

We claim that P ⊆ EH . Indeed, otherwise P uses some EM edge. But by an argument similar to the proof argument of Lemma 5.2, if P uses some EH edge and some EM edge to span some EH edge then the fact that P is a simple path implies that |P | > k, contradiction. p p′ p 6= ∅ are vertex-disjoint for p 6= p′ , the fact that P ⊆ EH and P ∩ EH and EH But as the sets EH p p p implies that P ⊆ EH . Thus P ∪ {e2 } is a cycle of length smaller or equal to k + 1 in EH . But EH is isomorphic to the supergraph H and girth(H) > k + 1, contradiction. ¯ namely, either to self-span them, It follows that there are only two ways to k-span the EH edges in G, or to span them by a direct path via the original edgeset E, namely, a path of the type p p m′ p (spi(kl +1) , spikl , . . . , spi1 , um i , wj , tj1 , tj2 , . . . , tj(kr +1) ). Note that the length of such a path is exactly kl + kr + 3 = k. ¯ is called a proper spanner if it does not use any edge of EH , i.e., A k-spanner H for the graph G H ∩ EH = ∅. Corollary 5.5 A proper k-spanner H k-spans all the edges of EH by direct paths. A main observation made next is that forbidding the use of EH edges does not degrade the spanner quality by much. ¯ can be converted in polynomial time to a proper k-spanner H ′ of Lemma 5.6 Any k-spanner H for G ′ size |H | ≤ 8|H|. Proof: The polynomial time procedure that creates a proper k-spanner H ′ from some k-spanner H is as follows. ˆ=E ˆ1 ∪ E ˆ2 by setting 1. Create the set E ˆ1 = E ˆ2 = E

r x [ [

{(u1i , spi1 )} ∪

p=1 i=1 r [ [

x [ r [

p=1 j=1

{(ui , s1i1 )} ∪

ui ∈Ui i=1

{wj1 , tpj1 )} ,

[

r [

{(wj , t1j1 )} ,

wj ∈Wj j=1

ˆ . 2. Set H1 = H ∪ E ∪ EM ∪ E

3. Recall that there exists a REP-cover C for H. For every e = (spi(kl +1) , tpj(kr +1) ) ∈ H ∩ EH = H1 ∩ EH , let u(e) ∈ Ui ∩ C, w(e) ∈ Wj ∩ C such that the edge (u(e), w(e)) is in E. For every e = (spi(kl +1) , tpj(kr +1) ) ∈ H ∩ EH , replace e in H1 by the edges (spi1 , u(e)), (w(e), tpj1 ). I.e., set H ′ = H1 ∪ {(spi1 , u(e)), (w(e), tpj1 ) | e ∈ EH } \ EH . Note that after this change H ′ is still a k-spanner (later we show it formally). Note that the step 3 of the above procedure the size of the spanner grows at most by a factor of 2,

i.e., |H ′ | ≤ 2|H1 | . (9) ˆ1 | = n, since there is exactly one edge for every Thus it remains to bound the size of H1 . Note that |E ˆ1 . Also |E ˆ2 | = 2rx, since there are exactly x edges for each supernode in E ˆ2 . Thus node u ∈ V1 ∪ V2 in E ˆ + |EM | |H1 | ≤ |H| + |E| + |E|

≤ |H| + n2 + (n + rx + rx) + (rxkl + rxkr )

= |H| + n + rx(kl + 2) + rx(kr + 2) , 15

(10)

since |EM | = rxkl + rxkr . Also |V¯ | = n + rx(kl + 1) + rx(kr + 1) .

(11)

Note that kl + 2 ≤ 2(kl + 1), kr + 2 ≤ 2(kr + 1). Assume without loss of generality that |V | ≥ 3 and thus |H| ≥ 2. Since n2 = 2rx (by the choice of x), it follows that |H1 | ≤ |H| + 2|V¯ | ≤ 3|H| + 2 ≤ 4|H| .

(12)

Using (9) we conclude |H ′ | ≤ 8|H|, as required. ¯ The edges of E ∪ EM are self-spanned by It remains to argue that H ′ is a k-spanner for the graph G. ˆ ⊆ H ′ . Now consider EH edges. As discussed earlier, H ′ , and the edges of EsU ∪ EtW are k-spanned by E (1) all the edges of EH are either self-spanned or spanned by a direct path in H. Denote by EH = EH ∩ H (2) the set of edges that are self-spanned by H, and by EH those that are spanned by a direct path in H and (1) (2) are not self-spanned. Thus EH = EH ∪ EH . For any edge e = (spi(kl +1) , tpj(kr +1) ) ∈ H ∩ EH = H1 ∩ EH p m we have inserted into H ′ on step (3) of the above procedure the edges (spi1 , um i ), (wj , tj1 ) such that ′ ′ m′ (um i , wj ) ∈ E ⊆ H . Thus there is k-long spanning path for e in H of the form ′

′

′

p p p p p m m m ((spi(kl +1) , spikl ), . . . , (spi2 , spi1 ), (spi1 , um i ), (ui , wj ), (wj , tj1 ), (tj1 , tj2 ), . . . , (tjkr , tj((kr +1)) )) .

(13)

(2)

Concerning the edges e ∈ EH , by definition, every one of them has a spanning path of the form (13) in H. Note that the path of this type contains no EH edges and so its edges could not be removed from (2) H while creating H ′ . Thus the spanning path is present in H ′ too. Hence H ′ k-spans the edges of EH too. ¯ there is a polynomial time constructible REP-cover C for Lemma 5.7 Given a k-spanner H for G 8|H| M = (G, H) of size |C| ≤ x . ¯ let H ′ be a proper k-spanner for G ¯ as guaranteed in Lemma 5.6. For Proof: Given a k-spanner H for G, p = 1, 2, . . . , x define the sets ′

′

p m ′ m m Hp′ = {(spi1 , um i ), (tj1 , wj ) ∈ H | ui ∈ Ui , wj ∈ Wj } . ′ = H ′ ∩ E . Then Also let HE′ be H ′ ∩ E and HM M ′ H ′ = HE′ ∪ HM ∪

p [

Hp′ .

i=1

Note that Hp′ ∩ Hq′ = ∅ for 1 ≤ p 6= q ≤ x. Let 1 ≤ p0 ≤ x be an index such that |Hp′ 0 | = min

1≤p≤x

n

o

|Hp′ |

.

Now we define the REP-cover C by ′

′

p0 m p0 ′ m m ′ C = {um i | (si , ui ) ∈ Hp0 } ∪ {wj | (tj , wj ) ∈ Hp0 } .

To prove that C is a REP-cover, we consider some superedge (Ui , Wj ) ∈ H. By the construction of the p0 ′ ¯ there is an edge e = (sp0 ¯ ¯ graph G i(kl +1) , tj(kr +1) ) in E. Since H is a proper k-spanner for G, there is a 16

spanning path for e in H ′ , and by Corollary 5.5, the only possible form of a spanning path for e of length at most k that uses only H ′ edges is ′

′

p0 p0 p0 p0 p0 m m m 0 ((spi(k , spik0l ), . . . , (spi20 , spi10 ), (spi10 , um i ), (ui , wj ), (wj , tj1 ), (tj1 , tj2 ), . . . , (tjkr , tj((kr +1)) )) , l +1) p0 p0 ′ m m m for some um i ∈ Ui and wj ∈ Wj . Thus the edges (si , ui ), (wj , tj ) are in the spanner H , and thus by ′ m m′ m′ p0 definition of Hp′ 0 , (spi 0 , um i ), (wj , tj ) ∈ Hp0 . Hence, ui , wj ∈ C, and these two nodes REP-cover the superedge (Ui , Wj ). For the bound on the size of the REP-cover C, we note that ′

′

[ 8 1 1 ·| · |H ′ | ≤ · |H| , Hp′ | ≤ x 1≤p≤x x x

|C| = |Hp′ 0 | ≤

where the last inequality follows from Lemma 5.6. This completes the proof. For the opposite direction we prove the following lemma. Lemma 5.8 Given a REP-cover C for the M IN REP instance M, there is a polynomial time con¯ of size |H| ≤ (k + 1)x|C|. structible k-spanner H of G Proof: Let C be a REP-cover for the M IN REP instance M = (V1 , V2 , E, V1 , V2 ). We build a k-spanner ¯ by setting H for G ˆ . H = {(spi , ui ), (tqj , wj ) | ui , wj ∈ C, 1 ≤ p, q ≤ x} ∪ E ∪ EM ∪ E ¯ Also, since any vertex in the cover C induces x edges of It is easy to verify that H is a k-spanner for G. the spanner H, then in the way similar to (10) we conclude that ˆ ≤ x|C| + n2 + (n + 2rx) + rxkl + rxkr |H| = x|C| + |E| + |EM | + |E|

≤ x|C| + n2 + (n + 2rx) + 2rx(k − 3) ≤ x|C| + (n + (k − 31)2rx) .

Recall that |C| ≥ 2r and n ≤ n2 = 2rx. This implies that |H| ≤ (k + 1)x|C| , as required. Now we are ready to prove the following theorem.

Theorem 5.9 For ǫ > 0, κ ≥ 3 there exists ǫ′ > 0 such that if there exists an O(2log algorithm for the basic k-spanner problem then there exists an O(2log the M IN REPk+1 problem.

1−ǫ′

1−ǫ

n )-approximation

n )-approximation

algorithm for

Proof: Suppose, towards contradiction, that there is an algorithm A that approximates the basic k1−ǫ spanner problem within a ratio 2log n , for some 0 < ǫ < 1. Then given a M IN REP instance (G, H), we build an instance of the k-spanner problem as above. Then we invoke the algorithm and obtain a k-spanner H. Using the spanner we build a REP-cover C as in Lemma 5.7. The REP-cover satisfies |C| ≤

1−ǫ 8 8 |H| ≤ · 2log n¯ |H ∗ | , x x

¯ and H ∗ is an optimal k-spanner for G. ¯ Note that n where n ¯ is the number of vertices in G ¯ = n+2rx(k−1) ′ (by (11)), and thus for any 0 < ǫ < ǫ < 1 and sufficiently large n 2log

1−ǫ

n ¯

< 2log

17

1−ǫ′

n

,

implying |C| ≤

8 log1−ǫ′ n ∗ ·2 |H | . x

(14)

Let C˜ be an optimal REP-cover. As we have shown in Lemma 5.8 there is a polynomial time constructible ˜ such that k-spanner H ˜ ≤ (k + 1)x|C| ˜ . |H| (15) Since H ∗ is an optimal k-spanner and by (14) and (15) it follows that |C| ≤

′ 1−ǫ′ 1−ǫ′ 8 8 8 log1−ǫ′ n ∗ n ˜ n ˜ = O(2log1−ǫ n )|C| ˜ . ·2 |H | ≤ · 2log |H| ≤ · 2log · (k + 1)x|C| x x x

Thus the existence of an approximation algorithm A as above implies existence of an approximation 1−ǫ′ n ) for some 0 < ǫ′ < 1. algorithm for the M IN REPk+1 problem with a ratio of O(2log We remark that although it is known that the MINREP problem is strongly inapproximable [27], the M IN REPt problem for t > 4 is not known to be strongly inapproximable. The proof of this claim that appeared in [23] is, unfortunately, not valid.

6

Strong Inapproximability with a Relaxed Stretch Requirement

In this section we show that the k-spanner augmentation problem (and thus the unit-length k-spanner problem) remains as hard even if the stretch requirement is relaxed from some small constant to any function k = f (n) = O(n1−δ ), for any constant 0 < δ < 1. This result should be contrasted with the situation for the unweighted k-spanner problem, which admits O(1) approximation once the stretch requirement becomes k = Ω(log n), and in contrast to the behavior of the uniform k-spanner problem which admits an O(1) approximation ratio once the stretch requirement becomes k = Ω(log2 n). We then show that the k-spanner augmentation problem and the unit-length k-spanner problem do not enjoy ratio degradation property, i.e., they cannot have an algorithm similar to the ones of [3, 32] for the (unweighted and weighted) uniform k-spanner problem, whose approximation ratio decreases exponentially with the inverse of the stretch requirement k. In fact, we show a much stronger result, specifically, that these problems cannot have an algorithm whose approximation ratio decreases exponentially with the inverse of any positive power of the logarithm of the stretch requirement. Finally, at the end of the section we show that these results apply also to the unweighted DJ k-spanner problem, the unweighted AC k-spanner problem (and thus to the unweighted C-S k-spanner problem) and to the unweighted directed k-spanner problem.

6.1

Hardness with Relaxed Stretch

Let us start with the unit-length k-spanner problem. Let w∗ be the weight of the optimal k-spanner. Then the following theorem holds. Theorem 6.1 For any 0 < ǫ, δ < 1 and 4 ≤ k = O(n1−δ ), it is quasi-NP-hard to approximate the 1−ǫ ∗ unit-length k-spanner augmentation problem with ratio 2log w . Proof: The proof is also by a reduction from the MIN-REP problem. Suppose for contradiction that there is an approximating algorithm for the f (n)-spanner problem, for some function k = f (n) = θ(n1−δ ), for some constant 0 < δ < 1. Given a MIN-REP instance (V1 , V2 , E, {Ai }, {Bj }), we construct the graph ¯ = (V¯ , E) ¯ as follows. Recall that N = |Ai | = |Bj |. Let d denote the degree of the d-regular supergraph G 18

H, d = deg(H). Hence d ≤ r. Let a = 2N r + rd and b = 2N − 3rd. Calculate the smallest value of k that satisfies the equation k = f (ak + b) . (16) Note that the value of k which solves the equation is at most polynomial in n, as the function f is of the form f (x) = x1−δ . ¯ in the same way as in Section 3.1, except that the edges of EH , connecting si and Now construct G tj , are now replaced by paths of length θ(|V¯ |1−δ ). Observe that |V | = 2rN and n ¯ = |V¯ | = 2rN + 2(k − 1)rN + 2N + (k − 3)rd = k(2N r + rd) + (2N − 3rd) = ak + b. Note that the construction is polynomial, since k = f (¯ n) = O(¯ n) is at most polynomial in n. ¯ with ratio 2log1−ǫ w∗ , we can now By approximating the f (n)-spanner augmentation problem on G 1−ǫ approximate MIN-REP on G with ratio 2log n . This is because w∗ ≤ n in the specific instance of the f (n)-spanner augmentation problem constructed through the reduction. Indeed, there is a one-to-one correspondence between the unit-weight edges in the construction and the vertices of the original graph G. Let us remark that the proof argument does not work for αn-spanners for constant 0 < α < 1. Furthermore, the proof argument fails for n/logα n-spanner for constant α > 0. Theorem 6.2 For any 0 < ǫ, δ < 1 and 4 ≤ k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ unit-length k-spanner augmentation problem with ratio 2log n . Proof: Suppose for contradiction, that there exists a polynomial time algorithm for approximating the 1−ǫ k-spanner augmentation problem with ratio 2log n . Observe that the number of nodes in the reduction of Theorem 6.1 grows at most quadratically multiplied by k. It follows that krN ≤ kr 2 N 2 and krd ≤ kr 2 ≤ kr 2 N 2 .

¯ is Thus the number of nodes in G

|V¯ | = O(kr(N + d)) = O(k(rN )2 ) = O(k|V |2 ) . 1−ǫ

Having a 2log n -approximation algorithm for the O(n1−δ )-spanner augmentation problem implies 2 1−ǫ having a 2log (k|V | ) -approximation algorithm for the MIN-REP problem. But for some 0 < α < 1 2log

1−ǫ

(k|V |2 )

≤ 2log

1−ǫ

(|V |2+α )

1−ǫ ·log1−ǫ

≤ 23

We now claim that for any ǫ′ > 0 there exists 0 < ǫ < 1 such that 1−ǫ ·log1−ǫ |V |

23

≤ 2log

or

1−ǫ′

|V |

′

,

31−ǫ · log1−ǫ |V | ≤ log1−ǫ |V | . Indeed, this is satisfied by taking ǫ ≥ ǫ′ +

log 3 , log log |V | 19

|V |

.

as for this ǫ we have

′

3log1−ǫ |V | ≤ log1−ǫ |V | . 1−ǫ′

n approximation ratio for the k-spanner augmentation Hence for every 0 < ǫ′ < 1, if we have 2log problem, then there exists some n0 such that the MIN-REP cover problem on graphs with n ≥ n0 vertices 1−ǫ admits approximation ratio 2log n , for some 0 < ǫ < 1. This contradicts the hardness of approximating the MIN-REP cover problem. The theorem follows.

Analogous considerations yield the following theorem. Corollary 6.3 For any 0 < ǫ, δ < 1 and 3 ≤ k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ 1−ǫ ∗ unit-length k-spanner problem with ratio 2log n or 2log w . Turning to client-server variants, we start with proving the hardness result for the unweighted DJ k-spanner problem. Theorem 6.4 For any 0 < ǫ, δ < 1 and 3 ≤ k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ unweighted DJ k-spanner problem with ratio 2log n . Proof: We generalize the construction of Section 4, by adding a ⌈k/2⌉-long path between each spi and ali and a ⌊k/2⌋-long path between each tqj and bm j . It is easy to verify that the analysis of Section 4 still applies. The maximum stretch requirement k for which the reduction is still polynomial, is n1−δ for any ¯ Indeed, for k = n constant 0 < δ < 1. Denote by n ¯ the number of vertices in the graph G. ¯ 1−δ , n ¯ = O(nxk) = O(n2 k) = O(n2 · n ¯ 1−δ ) , thus n ¯ = O(n2/δ ), i.e., polynomial in n. Now we proceed with proving a similar hardness result for the unweighted AC k-spanner problem. Note that in this case we cannot add edges between spi and ali , because those edges will need to be spanned. Theorem 6.5 For any 0 < ǫ, δ < 1 and 3 ≤ k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ unweighted AC k-spanner problem with ratio 2log n . Proof: We generalize the construction of Section 4.2, by replacing each spi with a path of length ⌈k/2⌉ p,⌈k/2⌉ q,⌊k/2⌋ p,1 q,1 (sp,0 ). We also replace each tqj with a path of length ⌊k/2⌋ (tq,0 ). All the i , si , . . . , si j , t j , . . . , tj p,⌈k/2⌉

and ali ; all the edges between spi and tqj are edges between spi and ali are replaced by edges between si q m replaced by edges between sp,0 and tq,0 i j , and, finally, the edges between tj and bj are replaced by edges q,⌊k/2⌋

between tj and bm j . Now, the proof is analogous to that of Theorem 6.5. Next, we state such hardness result for the unweighted directed k-spanner problem. The analysis is based on the construction of Section 4 and very similar to the one of Theorem 6.5. Theorem 6.6 For any 0 < ǫ, δ < 1 and 3 ≤ k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ unweighted directed k-spanner problem with ratio 2log n .

20

6.2

Lack of Ratio Degradation

Finally, let us contrast the behavior of the problems studied in this section with that of the uniform kspanner problem and unit-weight k-spanner problem, which enjoy ratio-degradation property, i.e., admit an approximation algorithm with variable ratio depending on the stretch requirement k, i.e., n1/k . (See [32, 3, 13, 35].) We show that such performance cannot be achieved for the unit-length augmentation k-spanner problem. In fact, we show that not only an O(n1/k ) ratio, but even an O(n1/ log k ) ratio is unattainable. Theorem 6.7 For any α, β > 0 and 0 < ǫ1 , ǫ2 < 1 there is no algorithm A(G, k) that approximates the unit-length k-spanner augmentation problem on every n-vertex graph G and for every sufficiently large k 1 1 ǫ with ratio O(kα · n kβ ) or O(2log 1 k · n logǫ2 k ). 1

Proof: We first prove the nonexistence of a O(kα · n kβ )-ratio algorithm by contradiction. Suppose that there exists an algorithm A(G, k) guaranteeing such an approximation ratio for every graph G and stretch 1

1

requirement k. Set k = log β n. Then O(kα · n kβ ) = O(logα/β n) = O(polylog n). Hence we have an 1 O(polylog n)-approximation ratio for the log β n-spanner augmentation problem, contradicting Theorem 6.2. 1 ǫ To prove the nonexistence of a O(2log 1 k · n logǫ2 k )-ratio algorithm, again by contradiction, assume 1−ǫ an algorithm A(G, k) as promised, and set k = n1−δ , for some 0 < δ < 1. Then we obtain 2log n approximation ratio, where ǫ = max{ǫ1 , 1 − ǫ2 }, hence 0 < ǫ < 1. This again contradicts Theorem 6.2. It follows that the unit-length k-spanner problem does not enjoy the ratio degradation property. Corollary 6.8 For any α, β > 0 and 0 < ǫ1 , ǫ2 < 1 there is no algorithm A(G, k) that approximates the unit-length k-spanner problem on every n-vertex graph G and for every sufficiently large k with ratio 1 1 ǫ O(kα · n kβ ) or O(2log 1 k · n logǫ2 k ). Theorem 6.9 For any α, β > 0 and 0 < ǫ1 , ǫ2 < 1 there is no algorithm A(G, k) that approximates the unit-length k-spanner problem or the unit-length k-spanner augmentation problem on every n-vertex 1 1 ǫ graph G and for every sufficiently large k with ratio O(kα · w∗ kβ ) or O(2log 1 k · w∗ logǫ2 k ). Proof: Let us start with the plain unit-length k-spanner problem. We prove a stronger claim, specifically, that the conclusion of the theorem is correct even for the unit-length k-spanner problem restricted to the case when n−1 max{ω(e)} ≤ , (17) e∈E m where |V | = n and |E| = m. 1 Observe that in this case we can substitute k = log β w∗ , because by (17) w∗ ≤ n−1. By substituting 1

such k to O(kα · w∗ kβ ) we obtain an O(polylog w∗ ) approximation ratio for the restricted unit-length version. Note that the unit-length k-spanner problem reduces to the restricted case by normalization (namely, n−1 1 multiplying the weights by maxe∈E ω(e) · m ). Obviously, this trivial reduction preserves approximation 1

ratio. Hence, we obtain an O(polylog w∗ ) approximation ratio for the unit-length O(log β w∗ )-spanner problem, contradicting Corollary 6.3. This completes the proof of the claim for the first ratio type. The claim for the second ratio type follows by analogous considerations.

21

Now let us turn to the unit-length k-spanner augmentation problem. Assuming an algorithm A(G, k) 1 guaranteeing an approximation ratio of the first type, we can substitute k = log β w∗ , and because 1 1 w∗ ≤ n2 it holds that k ≤ 2 β · log β n for sufficiently large n, contradicting Thm. 6.2. Similarly, The nonexistence of an approximation algorithm with a ratio of the second type is proven by substituting k = w∗1−δ for any 0 < δ < 1. Finally, we establish that the unweighted DJ problem, the unweighted AC problem, the unweighted C-S problem and the unweighted directed k-spanner problem do not enjoy the ratio degradation property. The proof of these theorems is analogous to that of Theorem 6.7 and based on Theorems 6.4, 6.5 and 6.6. Theorem 6.10 For any α, β > 0 and 0 < ǫ1 , ǫ2 < 1, and for any problem Π from among the DJ, AC, C-S or directed k-spanner problems, there is no algorithm A(G, k) that approximates Π on every n-vertex 1

graph G and for every sufficiently large k with ratio O(kα · n kβ ) or O(2log

7

ǫ1

k

1

· n logǫ2 k ).

The Uniform Spanner Problem

The uniform k-spanner problem is a version of the general k-spanner problem in which the weight and length of each edge are equal. The weights/lengths of different edges may be different. There are two possible objectives for the problem. One is to find the minimum weight k-spanner. Another requires to minimize the number of edges in the k-spanner. Obviously, the latter is equivalent to the unit-weight 1 k-spanner problem. For both problems there are algorithms with approximation ratio of O(n k ), c.f. [3, 13]. Observe that since the edge lengths need not be integral, we may consider the problem with non-integral stretch. In this section, we prove by reduction from MIN-REP that for 1 < k ≤ 3 the problem of finding the 1−ǫ 1−ǫ ∗ minimum weight k-spanner cannot be approximated with ratio 2log n or 2log w , for any 0 < ǫ < 1. This is done in three stages. First we show the claim for 1 < k ≤ 2. Then we extend it to 1 < k < 3, and finally prove it for k = 3.

7.1

Stretch Requirement 1 < k ≤ 2

The reduction is as follows. Fix 0 < ǫ ≤ 1 and let x = max{ǫ−1 , |E| + 2rN 2 + 1} .

(18)

¯ = (V¯ , E) ¯ Given an instance of MIN-REP problem (V1 , V2 , E, {Ai }, {Bi }), we construct an instance G of the general (1 + ǫ)-spanner problem as follows. 1. All the original edges (ali , bm j ) become unit weight (and length) edges. 2. Create cliques of unit weight edges inside each Ai and Bj . Denote this set of edges D. 3. For every ali ∈ Ai , connect si to ali with edges of weight x. m 4. For every bm j ∈ Bj , connect tj to bj with edges of weight x.

5. For every (Ai , Bj ) ∈ H, create an edge (si , tj ) of weight (2x + 1)/(1 + ǫ). ¯ of weight ω(H), we construct a MIN-REP cover C of Given the spanner H for the instance G size approximately ω(H)/x in two stages. First, for every spanner H we construct a spanner H ′ of approximately the same size that does not use (si , tj ) edges. We call such a spanner a proper spanner. Next, from H ′ we build a MIN-REP cover of size approximately ω(H ′ )/x. 22

Lemma 7.1 For every (1 + ǫ)-spanner H, there is a polynomial time constructible proper (1 + ǫ)-spanner H ′ such that ω(H ′ ) ≤ (1 + ǫ) · ω(H). Proof: If there is no (si , tj ) edge in H, then H is the required spanner. Otherwise, consider some edge (si , tj ) ∈ H. By the construction, there is a superedge (Ai , Bj ) ∈ H, hence there exists a path m ¯ Psi ,tj = (si , ali ), (ali , bm j ), (bj , tj ) in E. The weight of the path is 2x + 1. Recall that 2x + 1 . 1+ǫ

ω(si , tj ) =

Hence the path Psi ,tj (1 + ǫ)-spans the edge (si , tj ). Consequently, the desired proper spanner H ′ is obtained from H by removing every edge (si , tj ) in it and replacing it with the appropriate path Psi ,tj . Clearly, the weight of H ′ grows by a factor of at most 1 + ǫ. It remains to argue that H ′ is still a (1 + ǫ)-spanner, i.e., that removing (si , tj ) could not unspan any edge. Indeed, (si , tj ) might participate in a path that spans some edge (sl , tr ), but with stretch greater or m l equal than 3. Similarly, it could be used to span some (si , ali ) edge through a path (si , tj ), (tj , bm j ), (bj , ai ), but the length of such a path is 3 2x + 1 ≥ 2x + , x+1+ 1+ǫ 2 l whereas l(si , ai ) = x, hence the stretch factor is greater than 2. Given a proper spanner H ′ we construct a REP-cover C of size close to ω(H ′ )/x by letting l m ′ C = {ali , bm j | (si , ai ), (bj , tj ) ∈ H } .

(19)

¯ and |C| ≤ (1 + ǫ)ω(H)/x . Lemma 7.2 C defined by (19) is a REP-cover for G, ¯ H ′ is a proper spanner, hence Proof: Consider some superedge (Ai , Bj ) ∈ H. Then (si , tj ) ∈ E. ′ ′ (si , tj ) ∈ H , and H (1 + ǫ)-spans the edge. By the construction, the only possibility to span the edge with stretch smaller than or equal to 1 + ǫ is by using a path P of the form m P = (si , ali ), (ali , bm j ), (bj , tj ) .

(20)

Indeed, another possibility is by a path P ′ of the form m P ′ = (si , ali ), (ali , api ), (api , bm j ), (bj , tj ) ,

but the length of such a path is 2x + 2, whereas l(si , tj ) =

2x + 1 . 1+ǫ

Hence l(si , tj ) · (1 + ǫ) < 2x + 2, i.e. the path P ′ does not (1 + ǫ) span (si , tj ). Thus a path of the l m form P must be contained in H ′ . Hence, by (19) ali , bm j ∈ C. Note that (ai , bj ) ∈ E, by (20). Thus the super-edge (Ai , Bj ) is REP-covered by C. It remains to prove the bound on the size of C. It follows from (19) that there is a one-to-one correspondence between the nodes of the cover C and the edges of weight x of the spanner H ′ . Indeed, S S for any edge of weight x in H ′ , we insert one vertex from V = Ai ∪ Bi into the cover C and no two edges of weight x intersect at a vertex from V . Hence |C| equals the number of edges of weight x in H ′ , so |C| ≤ ω(H ′ )/x . The lemma follows, by Lemma 7.1. ¯ by letting Conversely, given a REP-cover C we construct a (1 + ǫ)-spanner H for G ¯ l m H = E ∪ D ∪ {(si , ali ), (bm j , tj ) ∈ E | ai , bj ∈ C} . 23

(21)

¯ and ω(H) < 2x · |C| . Lemma 7.3 H is a (1 + ǫ)-spanner for G, Proof: The edges of E ∪ D are self-spanned. Consider some edge (si , tj ). There exists a superedge m l m l (Ai , Bj ) ∈ H. Thus there exist nodes ali , bm j ∈ C, such that ai ∈ Ai and bj ∈ Bj . Hence (si , ai ), (bj , tj ) are in H and (ali , bm j ) ∈ E ⊆ H. So the whole spanning path is in H and it is easy to verify that the path (1 + ǫ)-spans the edge (si , tj ). Consider some edge (si , ali ) such that ali is not in the cover C. Without loss of generality, there exists a supernode Bj such that the superedge (Ai , Bj ) is in H. Hence there exists a node api in the set Ai such that (si , api ) ∈ H. Note also that (api , ali ) ∈ D ⊆ H. Hence the path (si , api ), (api , ali ) is contained in H. Since its length is 1 + x, it (1 + ǫ)-spans the x-long edge (si , ali ), since x > ǫ−1 by (18). Finally, by construction, D = 2rN 2 . Hence, by (18) |H| = |E| + |D| + |C|x < x + |C|x ≤ 2x|C|, since |C| ≥ 1. Theorem 7.4 For any 0 < ǫ < 1 and 0 < ǫ′ ≤ 1 it is quasi-NP-hard to approximate the uniform 1−ǫ 1−ǫ ∗ (1 + ǫ′ )-spanner problem with ratio 2log n or 2log w . 1−ǫ

Proof: Let us start with the ratio 2log n . Suppose for contradiction that there is such an approximation algorithm for the (1 + ǫ′ )-spanner problem. Given an instance of MIN-REP, we build the corresponding instance of the spanner problem and approximate it. Note that the number of nodes in the spanner instance is 2n and so we can ignore the constant factor of two in our analysis. We obtain a spanner H such that 1−ǫ ω(H) ≤ ω(H ∗ ) · 2log n . Using H we build a proper spanner H ′ , such that

ω(H ′ ) ≤ 2 · ω(H ∗ ) · 2log

1−ǫ

n

.

From H ′ we build a REP-cover C such that |C| ≤ 2 ·

1−ǫ ω(H ∗ ) · (1 + ǫ′ ) · 2log n . x

(22)

Let C˜ be an optimal REP-cover. We claim that ˜ · 2log1−ǫ n . |C| ≤ |C| Indeed, by Lemma 7.3 and inequality (22) ˜ > |C|

ω(H ∗ ) |C| ω(H) ≥ ≥ . 1−ǫ ′ 2x 2x 4(1 + ǫ ) · 2log n

Hence |C| < 4(1 + ǫ′ ) · 2log

1−ǫ

Obviously, for every ǫ′′ such that 0 < ǫ′′ < ǫ < 1,

n

1−ǫ · C˜ < 23+log n .

′′

3 + log1−ǫ n < log1−ǫ n. 1−ǫ′′

n -approximation of the optimal REP-cover for some 0 < ǫ′′ < 1, contradicting the Thus C is a 2log hardness of MIN-REP problem.

24

Now suppose there is a 2log

1−ǫ

w ∗ -approximation

algorithm for some 0 < ǫ < 1. Then

w∗ ≤ nx ≤ n3 and therefore 2log

1−ǫ

w∗

1−ǫ ·log1−ǫ

≤ 23

n

< 2log

1−ǫ′′

n

,

for every 0 < ǫ′′ < ǫ < 1 and for any sufficiently large n. The claim now follows from Theorem 7.4. We remark that the same hardness result can be established for the (1+ ǫ(n))-spanner problem, where ǫ(·) is any positive function that tends to zero as n goes to infinity.

7.2

Stretch Requirement 2 < k < 3

Next, we extend the above result to k = 2 + ǫ for 0 < ǫ < 1. Note that this could not be done directly by ¯ dividing the weight of (si , tj ) by more than 2, since modifying Part 5 of the definition of the instance G (2x + 1)/k < x for k > 2. Then (si , tj ) could be used to span edges of the type (tj , bm j ), and this would cause the construction to fail. To overcome the above difficulty, we slightly modify the construction. Instead of each edge (si , ali ) of weight x we will now have two edges (si , sli ), (sli , ali ), both of weight x. Analogously, the edge (tj , bm j ) is m , bm ). Set x = max {ǫ−1 , |E| + 4rN 2 + 1}. Additionally, we create cliques between replaced by (tj , tm ), (t j j j the nodes sli having the same index i for every i = 1, 2, .., r and between the nodes tm j having the same index j for every j = 1, 2, .., r. Finally, we change the weights of (si , tj ) edges to ω(si , tj ) = (4x+1)/(2+ǫ). Under the same definition of a proper spanner we can now prove the following lemma analogously to the proof of Lemma 7.1. Lemma 7.5 For every (2 + ǫ)-spanner H, there is a polynomial time constructible proper (2 + ǫ)-spanner H ′ such that ω(H ′ ) ≤ (2 + ǫ) · ω(H) . The construction of a REP-cover C from a proper spanner H ′ is done in analogous way to (19). Specifically, l l m m ′ C = {ali , bm (23) j | (si , ai ), (bj , tj ) ∈ H } . A claim analogous to Lemma 7.2 is proved in the same way exactly. The only obvious change is in the bound on the size of C, which becomes |C| ≤ (2 + ǫ)ω(H)/x . Given a REP-cover C we construct a (2 + ǫ)-spanner H analogously to (21). Specifically, m m ¯ l m H = E ∪ D ∪ {{(si , sli ), (sli , ali )}, {(tj , tm j ), (tj , bj )} ∈ E | ai , bj ∈ C} .

(24)

The proof that H is a spanner is analogous to that of Lemma 7.3. Then we bound the weight of H by ω(H) < 3x · |C|. Using all the above we generalize Theorem 7.4 as follows. Theorem 7.6 For any 0 < ǫ < 1 and 0 < ǫ′ < 2 it is quasi-NP-hard to approximate the uniform 1−ǫ 1−ǫ ∗ (1 + ǫ′ )-spanner problem with ratio 2log n or 2log w .

7.3

Stretch Requirement k = 3

Finally, we extend the result to the case of k = 3, by another modification of the construction. The modification is similar to the one done for the proof of Theorem 3.4. Specifically, we add to the construction a node pij and edges (si , pij ), (pij , tj ) for every pair i, j such that the superedge (Ai , Bj ) is in the supergraph H. The weights of the edges (si , pij ) are set to 1 and the weights of edges (pij , tj ) are set to be (4x + 1)/3, where x is defined by x = max {ǫ−1 , |E| + |E(H)| + 4rN 2 + 1}. Using this construction we generalize Theorems 7.4 and 7.6 to 25

Theorem 7.7 For any 0 < ǫ < 1 and 1 < k ≤ 3 it is quasi-NP-hard to approximate the uniform 1−ǫ 1−ǫ ∗ k-spanner problem with ratio 2log n or 2log w . We observe that the construction enables us to strengthen Theorem 7.7, as follows. Theorem 7.8 For any 0 < ǫ < 1 and k = 3 + n−3/2 it is quasi-NP-hard to approximate the uniform 1−ǫ k-spanner problem with ratio 2log n . Proof: In order that no edge of type (pij , tj ) would be (3 + δ)-spanned by a path that consists edges of the same type and/or edges of type (sr , prm ), it must hold that 3·

4x + 1 > 4x + 1 , 3+δ

i.e., δ< Observe that by choosing d =

3 2

we minimize the

12 . 4x − 3

max{rN 2 , nd } = max{nd−1 · n(2−d)·2 , nd } . We conclude the section with two hardness results that can be easily obtained by combining the previously established results. The first concerns the directed uniform k-spanner problem and can be proved by combining the Theorems 4.9 and 7.8. Theorem 7.9 For any 0 < δ, ǫ < 1 and every 1 < k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ directed uniform k-spanner problem with a 2log n ratio. Next we show a hardness result concerning the general k-spanner problem, where both length and weight are arbitrary and may be different. The proof of the following theorem is straightforward by combining Theorems 4.4 and 7.7. Theorem 7.10 For any 0 < δ, ǫ < 1 and every 1 < k = O(n1−δ ) it is quasi-NP-hard to approximate the 1−ǫ general k-spanner problem with a 2log n ratio.

8 8.1

The Unit-Weight k-Spanner Problem The Basic Unit-Weight Problem

When considering a uniform spanner, it makes sense to minimize not only its weight, but also its number of edges. This problem is equivalent to the unit-weight k-spanner problem. We proceed with giving a hardness result for the unit-weight k-spanner problem. This problem was studied in [3] and it was shown there that the problem admits an n1/k approximation ratio. However, not even a constant ratio approximation threshold was known for the range of the stretch requirement 1 < k < 2. We show that 1−ǫ it is quasi-NP-hard to approximate the unit-weight k-spanner problem with a ratio of 2log n , for any 0 < ǫ < 1, for 1 < k < 3. For k ≥ 2 the only known hardness result was log n inapproximability following from [27], hence our result significantly improves the previously known result of the range 2 ≤ k < 3 too. ¯ is built This hardness result is proven by a modification of the reduction of Section 4. The graph G exactly as done there, and all the edges are assigned a unit length, except the EH edges which are assigned a length of 3/k and the EQ edges which are assigned a k − 1 length. The latter is required for k-spanning 26

the edges of EsA (and EtB ) that are not used by the spanner. Note that since 1 < k < 3, l(spi , tpj ) > 1. Hence these edges cannot participate in any path that k-spans some other edge. Indeed, a path which is contained in EH can only 3-span an edge from EH , but k < 3. A path containing an EH edge and two additional unit-length edges can span another unit-length edge with a stretch of 2 + 3/k > 3 > k. ¯ edge at all. Hence, Finally, a path containing an EH edge and only one additional edge cannot span any G analogously to Lemmas 3.3 and 4.8 we show ¯ has a k-spanner H ′ with no edges of EH , which satisfies ω(H ′ ) ≤ 3ω(H ∗ ), Lemma 8.1 The graph G ∗ where H is an optimal spanner. A spanner is proper if it satisfies the condition of the lemma. We observe that the only way for a l m p proper spanner to k-span an EH edge e = (spi , tpj ) is by a path (spi , ali ), (ali , bm j ), (bj , tj ), for some ai ∈ Ai , m bj ∈ Bj . The crucial point is that EQ edges cannot be used for k-spanning EH edges, because any spanning path for e that contains at least one EQ edge has length at least 4 and thus the stretch of the 4 > k. Hence we establish spanning path is 3/k Theorem 8.2 For any ǫ > 0 and stretch requirement 1 < k < 3, it is quasi-NP-hard to approximate the 1−ǫ unit-weight k-spanner problem with ratio 2log n .

8.2

The DJ and AS Unit-Weight Problems

In this section we extend the analysis to the widest possible range of the stretch requirement, whereas restricting our attention to the client-server version of the problem. Specifically, we show that the DJ version (and thus the client-server version) of this problem is quasi-NP-hard to approximate with ratio 1−ǫ 2log n for any 0 < ǫ < 1 and 0 ≤ k < ∞. Also we show that the AC version is quasi-NP-hard to 1−ǫ approximate with ratio 2log n for any 0 < ǫ < 1 and 0 ≤ k < 3. Theorem 8.3 For any 0 ≤ k < ∞ and 0 < ǫ < 1 it is quasi-NP-hard to approximate the DJ unit-weight 1−ǫ k-spanner problem with ratio 2log n , even when restricted to two possible edge lengths. Proof: Again we reduce MIN-REP to this problem. The construction is similar to the one used in Theorem 7.4, but with the following changes. l(p)

1. We have no cliques among the nodes ali , bm j , si

m(p)

and tj

.

2. The lengths of all the edges, except (si , tj ), are set to 1. 3. The lengths of the (si , tj ) edges are set to (2x + 1)/k. 4. The client and server sets are defined to be C = {(si , tj ) | (Ai , Bj ) ∈ H} , ¯\C . S = E Given a spanner H, we build a REP-cover C by C = {ali , bm j ∈V | l(1)

{(si , si

l(1)

), (si

l(2)

, si

l(x−1)

), . . . , (si

r(x−1)

, ali )}, {(tj , tj

r(x−1)

), (tj

r(x−2)

, tj

r(1)

), . . . , (tj

, bm j )} ⊆ H} .

Note that there is a slight inconsistency with the previous construction, where we have inserted to the cover every node that has an incident edge which belongs to the spanner. Here we require that the whole 27

l path from the node ali to si (symmetrically, from bm j to tj ) is in the spanner in order to insert ai to the cover. In fact, we could use this approach previously too, but the one used was sufficient, since for each node ali in C there necessarily was an edge of weight x in the spanner. Here this is not the case, which necessitated the change in the construction.

Lemma 8.4 C is a REP-cover and |C| ≤ ω(H)/x . Proof: The proof is obvious if we note that the only alternative path for spanning the edge (si , tj ) is l(1)

(si , ai

l(x−1)

), .., (si

s(x−1)

p s m p s , ali ), (ali , bm j ), (bj , ai ), (ai , bj ), (bj , tj

s(x−1)

), .., (tj

, tj ) ,

and its length is

2x + 1 ·k . k Also by definition of C, every node in the cover corresponds to x-long (and hence x-weight) path from the node to si or tj . These paths are edge-disjoint. The inequality is strict due to the E edges (edges of type (ali , bm j )). 2x + 3 >

Observe that an important difference between this case and the uniform directed spanner case, is that a spanner in this instance cannot include any (si , tj ) edges, since they are not in S. Hence we do not need the notion of a proper spanner (as every spanner is proper), and hence we avoid the 1 + ǫ factor increase of spanner’s weight, incurred in the construction of Section 7.1 upon the transition from a non-proper spanner to a proper one. This difference is what enables us to prove the theorem for every positive k < ∞ 1−ǫ and not only k = O(2log n ) (as in Theorem 6.2). Given a REP-cover, we construct a C-S spanner by l(1)

H = E ∪ {{(si , ai

l(x−1)

), .., (si

ali , bm j ∈ C} .

r(x−1)

, ali )}, {(tj , tj

r(x−1)

), (tj

r(x−2)

, tj

r(1)

), .., (tj

, bm j )} | (25)

¯ and ω(H) ≤ 2x · |C| . Lemma 8.5 H is a DJ k-spanner for G Proof: The fact that H is a k-spanner is obvious. The weight of H is bounded by ω(H) = |E| + x · |C| ≤ x + x · |C| = x · (|C| + 1) ≤ 2x · |C| . Now we are ready to prove Theorem 8.3. 1−ǫ Suppose, for the sake of contradiction, that there is a 2log n -approximation algorithm for the DJ unit-weight k-spanner problem, for some 0 < k < ∞ and 0 < ǫ < 1. Let H be the resulting spanner. By Lemma 8.4 there is a polynomial time constructible REP-cover C such that |C| ≤

ω(H) . x

˜ such Let C˜ be any REP-cover. Then by Lemma 8.5 there is a polynomial time constructible spanner H that ˜ ≤ 2x · |C| ˜ . ω(H) Hence ˜ ≥ |C|

˜ ω(H) ω(H˜∗ ) |C| , ≥ ≥ 1−ǫ 2x 2x 2 · 2log n 28

contradicting the hardness of approximating MIN-REP. Observe that the only restriction imposed on k is k < ∞. Indeed, for infinite stretch (i.e., a spanner guaranteeing merely connectivity), the edges of type (si , tj ) could be spanned by paths of type l(1)

(si , ai

l(x−1)

), .., (si

s(x−1)

p s m p s , ali ), (ali , bm j ), (bj , ai ), (ai , bj ), (bj , tj

s(x−1)

), .., (tj

, tj ) ,

and the construction would fail. Note also that the same construction enables us to establish hardness for the unit-weight all-server (AS) k-spanner problem, but only for 0 < k < 3. Theorem 8.6 For any 0 < k < 3 and 0 < ǫ < 1, it is quasi-NP-hard to approximate the unit-weight AS 1−ǫ k-spanner problem with ratio 2log n , even when restricted to two possible edge lengths.

9

Relations between spanner and red-blue problems

In [26] Arora and Lund introduced a classification of all the problems according to their approximation threshold. The class III of problems is defined there to be the class of problems whose inapproximability 1−ǫ threshold is known to be Ω(2log n ) for any 0 < ǫ < 1 and not known to be higher. The class IV is defined to be the class of the problems for which Ω(nδ )-inapproximability is known for some 0 < δ < 1. The “canonical” problem for class III, i.e., the problem to which the majority of the problems of this class reduce, is the Label-Cover problem. The MINREP, MAXREP and Symmetric Label-Cover problems considered above are all variants of the Label-Cover problem. A reduction from the Red-Blue problem, defined below, to the minimization variant of the Label-Cover problem was presented in [16]. In this section we present a reduction from the directed κ-spanner problem to the Red-Blue problem. It therefore 1−ǫ follows that a stronger hardness result for the directed κ-spanner than the Ω(2log n )-inapproximability, shown by us in Section 5, i.e., a result of Ω(nδ )-inapproximabilty for some 0 < δ < 1, would imply that the majority of the problems known today to be in Class III are in fact in Class IV. The Red-Blue problem is the following natural generalization of the set-cover problem. We are given a universe of elements U , partitioned into a set R of red elements and a set B of blue elements, R ∪ B = U , and a collection of sets over the universe, S = {S1 , S2 , ..., Sk }. For any subcollection T ⊆ S, let U (T ) =

[

Si ∈T

Si ,

B(T ) = U (T ) ∩ B,

R(T ) = U (T ) ∩ R .

The goal is to choose a subcollection T of S that covers all the elements of B (i.e., s.t. B ⊆ B(T )) while minimizing |R(T )|, the number of red elements in T . 1−ǫ This problem was shown to be Ω(2log n )-inapproximable by [16], [11] and [24] independently (the strongest result is due to [16] which establishes this inapproximability under the weakest N P 6= P assumption). Theorem 9.1 For any α > 0 and integer κ ≥ 2, if the Red-Blue problem has an O(nα ) approximation algorithm then the directed κ-spanner problem has an O(nακ ) approximation ratio. Proof: The reduction is as follows. Given an instance of G = (V, E) of the directed κ-spanner problem, construct an instance (U, S), U = B ∪R, of the Red-Blue problem as follows. For each arc e ∈ E build two elements in the universe, the ”client” copy be in B and the ”server” copy re in R, i.e., R = {re | e ∈ E} and B = {be | e ∈ E}. Every l-long path P for l ≤ κ in G is associated with a set SP containing all the blue elements which are spanned by P and all the red elements {re | e ∈ P }. 29

After obtaining a solution T for the Red-Blue problem, create a spanner H by taking into it all the edges e corresponding to the red elements re that are covered by the RB-cover T . H is a κ-spanner, because for every edge e there is a blue element be ∈ B and there is a set Si in the RB-solution T that contains it. Thus there is an l-long path, l ≤ κ, that spans the edge e. The size of the spanner is equal to the number of red elements covered by the RB-solution, i.e., |H| = |R(T )|. In the opposite direction, given a κ-spanner H, build an RB-cover T in the following way. For any 2 ≤ l ≤ κ, for any path P of length l contained in the spanner H, take the set Si ∈ S corresponding to the path into the RB-cover T . It is easy to see that an RB-cover T constructed in this way satisfies |R(T )| = |H| . Note that the size of the Red-Blue instance created by the reduction is O(|S|) = O(nκ ). It follows that any approximation algorithm A with ratio nα for the Red-Blue problem can be transformed into an approximation algorithm A′ with ratio nκα for the directed κ-spanner problem, by applying the above reduction.

10

Open questions

The following remain as open questions for further research. 1. Is the basic k-spanner problem strongly inapproximable? 2. Is any of the variants of the spanner problem in the class IV (i.e., Ω(nδ )-inapproximable for some δ > 0)? 3. Does the additive κ-spanner problem enjoy the ratio degradation property? Or in other words, what is the complexity of the additive κ-spanner problem for κ = Ω(log n)? 4. What is the complexity of the additive 1-spanner problem (i.e., the (α, β)-spanner problem for α = β = 1)? Is it strongly inapproximable? Is its directed variant strongly inapproximable?

Acknowledgements The authors would like to thank Uri Feige, Ran Raz, Oded Regev, Guy Kortsarz and Oleg Hasanov for helpful discussions, an anonymous referee for helpful remarks, and Madhav Marathe, Irit Dinur and Evgeny Dodis for clarifications concerning their papers.

30

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