Approximation Algorithms for Submodular Set Cover with Applications

IEICE TRANS. INF. & SYST., VOL.E83–D, NO.3 MARCH 2000

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INVITED SURVEY PAPER

Special Issue on Algorithm Engineering: Surveys

Approximation Algorithms for Submodular Set Cover with Applications Toshihiro FUJITO† , Member

SUMMARY The main problem considered is submodular set cover, the problem of minimizing a linear function under a nondecreasing submodular constraint, which generalizes both wellknown set cover and minimum matroid base problems. The problem is NP-hard, and two natural greedy heuristics are introduced along with analysis of their performance. As applications of these heuristics we consider various special cases of submodular set cover, including partial cover variants of set cover and vertex cover, and node-deletion problems for hereditary and matroidal properties. An approximation bound derived for each of them is either matching or generalizing the best existing bounds. key words: submodular function, set cover, approximation algorithms, greedy heuristics

1.

Introduction

We start with the set cover (SC ) problem. Given a finite set M and a family N of subsets of M , a subfamily S of N is called a set cover if every element of M appears in some subset in S; in other words, the union of all subsets in S coincides with M . Each set in N is associated with a (nonnegative) cost, and the cost of a family is the sum of costs of subsets in it. The set cover problem then asks to find a minimum cost set cover. As a special case when all the costs associated with sets are identical, it is called the unit cost set cover, and it is one of the basic NP-complete optimization problems presented by Karp [17]. The problem is also equivalent to the hitting set problem and the dominating set problem on graphs. This standard set cover can be extended into the following demand-supply form: Each element i demands bi units from a subfamily while each subset j can supply aij units to element i. It is now required to find a minimum subfamily such that subsets in it can collectively satisfy all the demands associated with elements of M , and this problem can be formulated by the following integer linear program     (IC) min cj xj : Ax ≥ b, x ∈ {0, 1}N   j∈N

where all the entries in A = (aij ) and b = (bi ) are nonnegative integers, and we call it integer cover (IC ). Here, column j of A corresponds to subset j in N , row Manuscript received July 5, 1999. Manuscript revised November 8, 1999. † The authors is with the Department of Electronics, Nagoya University, Nagoya-shi, 464–8603 Japan.

i to element i in M , and x is the characteristic vector of subfamily {j : xj = 1}. Notice that when set cover is modeled by (IC) A is a 0-1 matrix and b = 1. In the minimum weight spanning tree problem a spanning tree of minimum total weight is sought in a edge-weighted connected graph. It is well known that the spanning tree problem can be solved quite efficiently, in particular, by the greedy algorithm (aka Kruskal’s algorithm). In fact the minimum weight spanning tree is a special case of the minimum matroid base problem, and it is by now a classic result that the greedy algorithm attains optimality in the matroid base problem for any matroid [29]. Unlike the matroid base problem, however, either set cover or integer cover is NP-hard, and no polynomial time algorithm is expected for them. Nevertheless, it has been extensively studied, perhaps due to their importance in many applications, how to efficiently compute “near optimal” solutions for them. One of the most natural and universally applicable approaches to these hard problems is a greedy heuristic. The cost of a greedy solution is known to come always within a multiplica tive factor of H(maxj∈N |j|), or H(maxj∈N i∈M aij ) k for IC, of the optimum. Here, H(k) = i=1 1i is the kth harmonic number and bounded by 1 + ln k. This was shown first by Johnson [16] and Lovász [19] for the unit cost SC (i.e., c = 1), by Chv´ atal [4] for general SC, and then by Dobson for arbitrary nonnegative integer entries in A and b (i.e., integer cover) [5]. Despite its naive look recent results indicate that the performance of the greedy heuristic is a nearly best possible one. Assuming P=NP no polynomial time algorithm approximates set cover within a factor of log n for some constant > 0 [31], and the lower bound on approximation bounds becomes (1 − ) ln n, for any > 0, if NP ⊂ DTIME(nO(log log n) ) [7]. Meanwhile another approach was investigated for SC by Hochbaum and some LP duality based heuristics were designed [14]. The performance ratio derived therein is max i∈M |{j : i ∈ j}|, or in the form of (IC), maxi∈M j∈N aij . While this bound is inferior to the greedy bound in general, it becomes the best (or nearly best) ratio known for some special cases of SC, most notably the vertex cover (VC ) problem where, given a vertex-weighted graph G = (V, E), a minimum cost vertex set is sought so that every edge

FUJITO: APPROXIMATION ALGORITHMS FOR SUBMODULAR SET COVER WITH APPLICATIONS

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in G is incident to some vertex in the set. By taking M = E, jv = {e ∈ E : e is incident to v}, and N = {jv : v ∈ V }, VC can be seen as a special case of SC. It is special in the sense that each i ∈ M belongs to exactly two subsets in N . In the formulation of (IC) this means each row of A contains exactly two 1’s, and the Hochbaum’s bound gives an approximation factor maxi∈M j∈N aij = 2. For VC the Hochbaum’s heuristic is indeed superior to the greedy heuristic since the latter may possibly end up with a vertex cover of size Ω(log n) times larger than the minimum one [16]. Moreover, it is known to be NP-hard to approximate VC within a factor arbitrarily close to 76 [13]. Although there still exists a gap from this lower bound, the constant upper bound has remained at 2, despite a long period of extensive research, and currently the best bound log n known is only 2 − log 2 log n [2], [23]. The main subject of this survey is the submodular set cover (SSC ) problem, and it is a common generalization of SC, IC, and the matroid base problems. In light of those facts mentioned above it is quite natural to consider the greedy approach for SSC as well, and Wolsey analyzed its performance and generalized the aforementioned results for SC and IC in [34], which will be presented in Sect. 3.1. Paralleling the development in analysis of the greedy algorithm for SC, IC, and SSC, another natural heuristic for SSC has been designed based on the LP duality, and its performance generalizes the approximation bound of Hochbaum for SC (Sect. 3.2). The essential part of SSC comes mostly from its generality arising from high abstraction in its problem formulation, and the algorithmic study for it provides a uniform treatment for various problems of covering type. As stated above there exist two natural heuristics for SSC and their approximation bounds in general form are available to us. It is the aim of the survey to advertise this seemingly less known but very important problem, and to illuminate why it is worth knowing by formulating various problems in the form of SSC and then deriving their approximability results as applications of these heuristics. 2.

Preliminaries

The concept of a submodular function frequently plays a vital role in combinatorial theorems and algorithms, and its importance in discrete optimization has been well studied [6], [8], [20], [29]. Definition 1: Let f be a real valued function defined on all the subsets of a finite set N . • f is nondecreasing if f (S) ≤ f (T ) for S ⊆ T ⊆ N . • f is submodular if f (S)+f (T ) ≥ f (S∩T )+f (S∪T ) for S, T ⊆ N . The pair (N, f ) is called a submodular system on N , and

f (S) is the rank of S for any S ⊆ N . The following examples of a submodular function are most important ones for us: Example 1: One typical submodular function associated with graph G = (V, E) is a cut function. For X ⊆ V let δ(X) denote the set of edges in G with one end-vertex in X and the other in V − X. Then, |δ(X)| is submodular on subsets of N = V (G). When G is a directed graph and if δ − (X) (δ + (X)) denotes the set of edges entering X (leaving X, resp.), then |δ − (X)| (|δ + (X)|, resp.) is again submodular. Similarly, let δ(X) denote the set of edges incident to a vertex in X. Then |δ(X)| is submodular and nondecreasing on subsets of V (X). These constructions can be generalized to G with nonnegative real edge weights we , by taking the total weights on edges, instead of the number of them, in a cut. That is, the function w(δ(X)) = e∈δ(X) we is still submodular (and nondecreasing). Moreover, we could extend these functions to hypergraphs. Example 2: Given a finite set M and a family N of subsets of M , define f on subsets of N by f (S) = |∪j∈S j|, i.e., the number of elements covered by subsets in S. It can be easily seen that f is identical to |δ| of the previous example, when extended to the hypergraph representing M and N . Thus, f is submodular (and nondecreasing). Definition 2: A nondecreasing, submodular, and integer-valued function f with f (∅) = 0 is called a polymatroid function, and (N, f ) a polymatroid. Any set S ⊆ N is called • independent if f (S) = j∈S f (j) (assuming f (j) > 0, ∀j ∈ N ), and • spanning if f (S) = f (N ). If f additionally satisfies f ({j}) ≤ 1, ∀j ∈ N (which implies f (S) ≤ |S|, ∀S ⊆ N ), (N, f ) is a matroid. In a matroid the independence condition corresponds to that f (S) = |S|, and a maximal independent set in a matroid is called its base (for more on the matroid theory see, e.g., [30], [33]). Definition 3: For any polymatroid (N, f ) define another set function f d such that def f ({j}) − (f (N ) − f (N − S)). f d (S) = j∈S

Then f d is a polymatroid function and (N, f d ) is called the dual polymatroid of (N, f ). Example 3: Typical matroids associated with a graph G = (V, E) are the cycle matroid (E, r) and cocycle matroid (E, rd ), which are dual to each other. In the cycle matroid r(X) gives the number of edges in a maximal acyclic subgraph of G[X] = (V, X). In case of the cocycle matroid rd (X) is the maximum number of


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edges which can be deleted from G[X] without increasing the number of components in it. Example 4: Given a matroid (E, r) and a collection {Ak : k ∈ T } of subsets of E, define f on subsets of T such that f (S) = r(∪k∈S Ak ). Then, (T, f ) is a polymatroid. We describe the formal framework of the submodular set cover problem. A problem instance consists of a finite set N , a nonnegative cost cj associated with each element j ∈ N , and a nondecreasing submodular function f : 2N → ZZ+ (for simplicity f is assumed to be nonnegative integer valued). For a subset S ⊆ N the cost of S, c(S), is j∈S cj . The problem is then that of finding a spanning set of minimum cost, that is,     (SSC) min cj : f (S) = f (N ) .  S  j∈S

The matroid base problem is that of finding a minimum base in a given matroid (N, r). Obviously this is SSC defined with a very special polymatroid function r. Example 5: Recall the construction of f in Example 2, then clearly, the set cover problem is SSC on (N, f ). Recall the formulation (IC) and similarly define f (S) for the integer cover problem by   m   min aij , bi . f (S) =   i=1

Example 6: Let f be a polymatroid function defined on subsets of N . The problem of computing a largest independent subset X (i.e., f (X) = j∈X f ({j})) is of fundamental importance with many applications, including maximum hypergraph matching and maximum common independent set of k matroids. Observe now that f d (N − S) = f d (N ) iff f (S) + j∈N −S f ({j}) = j∈N f ({j}), or equivalently, f (S) = j∈S f ({j}). Thus, the problem of computing the largest independent set in (N, f ) is equivalent to the submodular cover problem on (N, f d ). Let f be a polymatroid function such that f ({j}) = k for all j ∈ N . It can be easily verified that the size of the maximum independent set in (N, f ) is equal to the maximum of the submodular function f (X)−(k −1)|X|, thus, the submodular function maximization. For other relations to submodular maximization see [25], [26], [35]. 3.

Approximation Algorithms

For any S ⊆ N the contraction of f onto N − S is the function fS defined on 2N −S s.t. fS (X) = f (X ∪ S) − f (S). It is easy to verify that if f is nondecreasing and submodular on N then so is fS on N − S; thus, we can define another submodular system (N − S, fS ) for any S ⊆ N . Let fS (j) denote fS ({j}). It was shown by Wolsey [34] that SSC can be formulated by the following integer linear programming: cj xj Min

j∈S

m

Then, f (N ) = i=1 bi and f (S) = f (N ) iff AxS ≥ b, assuming w.l.o.g. that (IC) has a solution, where xS is the characteristic vector of S. Also, it can be verified that f is nondecreasing and submodular, and hence, the integer cover is SSC on (N, f ). Many practically important optimization problems can be cast into some forms of submodular optimization problems, and most basic among them are the problems of submodular function minimization and maximization. The algorithmic study for the submodular function minimization has been rather successful, and it is now known not only that the problem can be solved in polynomial time using the ellipsoid method [11], [12], but also that purely combinatorial polynomial time algorithms exist when the submodular function is also symmetric [28] or posi-modular [24]. On the other hand the submodular function maximization is much harder. It contains NP-complete problems as special cases such as the max-cut problem, and in general it can be shown not to be solvable in “oracle-polynomial” time. The SSC problem was also introduced by Nemhauser and Wolsey [27] in relation to the submodular function maximization.

j∈N

s.t.

(IP)

fS (j)xj ≥ fS (N − S) S ⊆ N

j∈N −S

xj ∈ {0, 1}

j∈N

To see it let xT ∈ {0, 1}N be the characteristic vector of T ⊆ N . If T is an SSC solution, i.e., f (T ) = f (N ), xT is feasible to (IP) since fS (j)xTj = fS (j) j∈N −S

j∈T −S

≥ fS (T − S) = f (T ) − f (S) = f (N ) − f (S) = fS (N − S) for any S ⊆ N , where the inequality is due to submodularity of fS (and not to any particular choice of T ). Conversely, if xT is feasible to (IP), since it satisfies the constraint corresponding to T , fT (j)xTj ≥ fT (N − T ) = f (N ) − f (T ), 0= j∈N −T

and hence f (T ) = f (N ). Obtain the linear program relaxation of (IP) by replacing the integral constrain on xj by inequality xj ≥ 0


483 Initialize F = ∅. While F is not a solution cof SSC

(i.e., f (F ) < f (N)) do Add argminj∈N −F f j(j) to F . F Output F . Fig. 1

Greedy algorithm Greedy for SSC.

for all j ∈ N , and then its dual LP is: Max fS (N − S)yS

follows that by setting yF 0 = θ1 /H(maxj f (j)), yF t = (θt+1 − θt )/H(maxj f (j)), t = 1, . . . , T − 1, and yS = 0 for all other S’s, y becomes feasible to (D). On the other hand, the cost of a greedy solution can be rewritten, in terms of θt ’s and this y, as follows: c(F T ) =

T t=1 1

= θ (f (N ) − f (F 0 ))

S⊆N

s.t. (D)

+ fS (j)yS ≤ cj

S:j ∈S /

yS ≥ 0

j∈N S⊆N

The importance of these formulations is twofold, one in analysis of both the greedy and dual greedy heuristics and the other in design of the dual greedy heuristic.

θt (f (F t ) − f (F t−1 ))

T t=2

The algorithm Greedy, displayed in Fig. 1, starts with empty F and repeatedly adds elements to F until it becomes an SSC solution. The element j picked in each iteration is the “locally optimal” one in the sense that j maximizes the rank increase per cost among those not yet selected; in other words, the one minimizing cj cj f (F +j)−f (F ) = fF (j) . Theorem 1 (Wolsey [34]): When f is integer-valued with f (∅) = 0, the cost of a solution computed by Greedy is at most H(maxj∈N f ({j})) times the minimum cost. Proof (Sketch). The proof goes along the same line as in the one for set cover [4]. Let F t denote the set constructed by the t th iteration of Greedy, and suppose it terminates after T iterations (so, F 0 = ∅ and F T is c the output). Set θt = minj∈N −F t−1 f t−1j (j) . Using the F

T −1 facts that (θt )Tt=1 and (fF t (j))t=0 are both nonnegative, and respectively monotonically nondecreasing and nonincreasing sequences, it can be shown that

θ1 fF 0 (j) + (θ2 − θ1 )fF 1 (j) + · · · + (θT − θT −1 )fF T −1 (j) ≤ ( max θt fF t−1 (j))H(fF 0 (j)) 1≤t≤T

for any j ∈ N , and thus, θ1 fF 0 (j) + · · · + (θT − θT −1 )fF T −1 (j) ≤ ( max θt fF t−1 (j))H(f (j)) 1≤t≤T

≤ cj H(f (j)) ≤ cj H(max f (k)) k

since fF 0 (j) = f (j) and θ fF t−1 (j) ≤ cj , ∀t, due to the greedy choice made at each iteration by Greedy. It t

= H(max f (j)) yF 0 fF 0 (N − F 0 ) j

+

T

yF t−1 fF t−1 (N − F

t=2

= H(max f (j)) j

3.1 Greedy Heuristic

(θt − θt−1 )(f (N ) − f (F t−1 ))

t−1

)

fS (N − S)yS

S⊆N

So, the greedy cost is at most H(maxj f (j)) times the objective value of (D) for y, and the proof is completed by the weak duality theorem of LP. ✷ Notice that when applied to set cover where f (S) = | ∪j∈S j| the bound obtained is H(maxj |j|), and this is the result of [4], [16], [19]. 3.2 Dual Greedy Heuristic The second heuristic for SSC, called Dual runs in two phases. In the first phase the dual variables in (D) are increased in a greedy fashion (thus called “dual greedy”). At the same time it is recorded in what order the dual constraints become binding during this process. Note that each dual constraint corresponds to some j ∈ N , one-to-one, and the ordering of j’s is this way determined. Using this ordering, an actual SSC solution is constructed in the second phase, where certain minimality conditions are enforced. This is the part missing in the original heuristic of Hochbaum for set cover, but it turns out to be essential to do this operation in bounding approximation factors in some applications as will be seen later. More specifically, suppose y is feasible to (D) and consider which yS ’s can be increased. We say a constraint is “binding” if it is satisfied by equality, and clearly, the value of yS participating in a binding constraint cannot be increased. Let F be the set of all j’s whose corresponding constraints are binding, so that any constraint for j not in F is not yet binding. Then, yF can be increased since it appears only in non-binding constraints. Besides, F is the smallest set with this property since any F smaller than F must exclude some j corresponding to a binding constraint. Observe next that the coefficients fS (N − S) =


484 Initialize F = ∅, y = 0, l = 0. While F is not a solution of SSC (i.e., f (F ) < f (N)) do l ← l + 1. / F the dual constraint /* Increase yF until for some j ∈ corresponding to j becomes */

binding cj −

Let jl ← argminj∈N −F

=

cj −

Set yF ←

=

1≤k≤l−1

cj − l

cj − l

S:j ∈S,S / =F

fS (j)yS

fF (j)

fF (j)yF k k

fF (j)

S:jl ∈S,S / =F fF (jl )

1≤k≤l−1

fS (jl )yS

fF (jl )yF k

k

fF (jl )

Add jl into F (and let Fl ← F ). For k = l downto 1 do if F − {jk } is a solution of SSC then remove jk from F . Output F . Fig. 2

Dual greedy algorithm Dual for SSC.

f (N ) − f (S) in the dual objective function are monotonically decreasing in S. From these facts the following heuristic for (D), starting with y = 0 and F = ∅, is indeed greedy. Increase yF as much as possible so that the dual constraint for j becomes binding for some j in N −F . Add all such elements to F , and repeat. For the later analysis it is pointed out that when the constraint for j becomes binding as a result of increased yS for some S, we have fS (j)yS = cj (1) S:j ∈S /

In the second phase an SSC solution is constructed based on F and the order in which j’s are added to F during the first phase. In doing so it is made sure that F satisfies certain minimality conditions to be used in the later analysis. Let F0 = ∅ ⊆ F1 ⊆ . . . ⊆ FT denote (intermediate) F ’s constructed, in this order, by each iteration of the first phase (So, yS > 0 iff S is one of these F ’s). For any S ⊆ N we say X ⊆ N − S is a minimal SSC in (N −S, fS ) if X is an SSC in (N −S, fS ) but no proper subset of X is (of course a minimal SSC is easy to locate). A final SSC F is then constructed in F in such a way that F − Ft is a minimal SSC in (N −Ft , fFt ) for all t = 0, 1, . . . , T −1. This construction can be implemented simply by processing elements in F , one by one, in reversal of the order in which they were added to F during the first phase, and removing any of them whenever it is found to be redundant for an SSC solution. For more details see Fig. 2. We now turn to analysis of the approximation performance. The cost of a dual greedy solution, c(F ), is estimated in terms of the dual cost corresponding to y constructed simultaneously. Recall Eq. (1), and then c(F ) is cj = fS (j)yS j∈F

/ j∈F S:j ∈S

=

S⊆N

 

 fS (j) yS

(2)

j∈F −S

When (2) is compared with the dual objective function in (D), term by term, it can be seen that (by the weak duality theorem of LP) the approximation ratio is bounded by the maximum ratio between j∈F −S fS (j) and fS (N − S) for any S ⊂ F with nonzero yS ; in other words, j∈F −Ft fFt (j) max 1≤t≤T −1 fFt (N − Ft ) Recall furthermore the effect of the second phase; F −Ft is a minimal SSC in (N −Ft , fFt ) for all t = 0, 1, . . . , T − 1. Therefore, the performance ratio of Dual can be estimated by the following combinatorial bound. Theorem 2: The cost of an SSC computed by Dual is at most

j∈X fS (j) max (3) fS (N − S) times the minimum cost, where max is taken over any S ⊆ N and any minimal SSC X in (N − S, fS ). (1) Side Remarks What happens to these algorithms and analysis when f is actually a matroid rank function ? Of course Greedy reduces to the standard greedy heuristic for the minimum weight matroid base, and interesting enough, so does the dual greedy Dual. To see it observe only that it repeatedly selects, in the first phase, j with minimum cj from N − F so that F + j is independent (note that for any j such that Fk + j is independent in (N, f ), fFt (j) = 1 for all 0 ≤ t ≤ k − 1, and so, S:j ∈S,S / =F fS (j)yS = 1≤t≤k−1 fFt (j)yFt is constant for such j). Not surprisingly now the bound given in Theorem 2 also reduces to one (i.e., optimal) since in any matroid a minimal spanning set is always a maximum independent set. Thus, the algorithm Dual can be thought of as another form of generalization from the greedy algorithm for the minimum weight matroid base. Likewise it is no hard to derive from (3) the maximum row sum bound of Hochbaum for SC as a special case because, for any set cover S, |j| j∈S ≤ maxi∈M |{j : i ∈ j}|. |M | 4.

Applications

4.1 Partial Cover Problems A standard covering problem usually asks for a complete coverage, and all the given elements are required to be covered by a solution. A natural generalization


485

arises when this requirement is dropped and an additional input is given instead, to specify what fraction of the ground set is to be covered. We call this type of problem partial cover and such a relaxation is applicable to almost any problem of covering type. Consider such a variant of SC, in which instead of the entire ground set M , only p-fraction of it is asked to be covered by subsets for some 0 ≤ p ≤ 1. The greedy heuristic for SC can be easily adapted to this version, modifying only its stopping criteria. The partial SC was first studied by Kearns [18] in relation to learning, and the performance of the greedy heuristic was estimated to be at most 2H(|M |) + 3. Later Slav´ık showed that it is bounded by H(min{maxj∈N |j|, p|M |}) [32]. We shall see that the same bound follows from the Wolsey’s theorem (Theorem 1) for a more general SSC and this is because the partial SSC problem is also SSC. Let f and g be submodular set-functions s.t. f − g is nondecreasing (or nonincreasing). Then, min{f, g} is also submodular (see, e.g., [20, Proposition 2.2]). Since f is here nondecreasing, min{f, k} is also nondecreasing for any fixed k, f (∅) ≤ k ≤ f (N ). Thus, (N, min{f, k}) is a submodular system and S is an SSC here iff f (S) ≥ k, that is, a partial cover of (N, f ). Since maxj∈N min{f ({j}), k} = min{maxj∈N f ({j}), k}, Corollary 3: For f : 2N → ZZ+ with f (∅) = 0 and 0 ≤ k ≤ f (N ), the algorithm Greedy finds a partial SSC of (N, f ) with an approximation factor of H(min{maxj∈N f ({j}), k}). For partial SC where k = p|M |, this bound is H(min{maxj∈N |j|, p|M |}). Consider next the partial vertex cover problem, in which a minimum cost set of vertices covering k edges (instead of all) is sought, for some specified integer k. This “truncated” version of VC was shown to remain approximable with a factor of 2, the best constant bound known for VC, first in [9] and then independently in [3], whose algorithm requires solving a linear program. Later the time complexity of these algo|2 rithms was improved to O(|E||V | log |V | log |V |E| ) [15], and then to O(|V |2 ) [1]. The problem can be slightly generalized when each edge e in a given graph is associated with a nonnegative capacity we and it is required to obtain a minimum vertex set C such that the total capacity of edges covered by C exceeds the specified threshold q. We call this the capacitated partial vertex cover problem. By letting f (S) = min{w(δ(S)), q}, where w(δ(S)) = e∈δ(S) we , the problem reduces to SSC on (V, f ). It can be shown then that in any graph G = (V, E) with edge capacities we , any minimal partial vertex cover C satisfies the following bound: w(δ(v)) ≤ 2q. v∈C

Using this in application of Theorem 2 (Eq. (3)),

Theorem 4: The capacitated partial cover problem can be approximated by Dual within a factor of 2. The problem was considered also by Bar-Yehuda and, using the local-ratio technique, the same approximation bound was obtained [1]. 4.2 Node-Deletion Problems The node-deletion problem for a graph property π (denoted ND(π)) is that of finding a minimum cost vertex set, given a vertex-weighted graph G, whose deletion (along with all the incident edges) from G leaves a subgraph satisfying the property π. A graph property π is hereditary if every subgraph of a graph satisfying π also satisfies π. A number of well-studied graph properties are hereditary such as independent set, planar, bipartite, acyclic, degree-constrained, circular-arc, circle graph, chordal, comparability, permutation, perfect. Naturally, many well known graph problems fall into this class of problems when desired graph properties are specified appropriately. Lewis and Yannakakis proved, however, that whenever π is nontrivial and hereditary on induced subgraphs ND(π) is NP-hard [21]. Any graph which does not satisfy hereditary π is called a forbidden graph for π, and it is a minimal one if, additionally, every proper subgraph of it satisfies π. Any hereditary property π is equivalently characterized by the set of all the minimal forbidden graphs for π. As observed in [22] whenever hereditary π has only a finite number of minimal forbidden graphs ND(π) can be efficiently approximated within some constant factor of the optimum, and examples of such properties include transitive, partial order, symmetric, antisymmetric, tournament, line graph, interval and others. Actually in this case, by setting up the set cover instance M = {a minimal forbidden graph for π in G} and N = {a set of minimal forbidden graphs containing v : v ∈ V }, the Hochbaum’s bound for SC provides a constant factor; that is, Theorem 5: The Hochbaum’s heuristic computes a solution for ND(π) in G whose cost is at most d times larger than the minimum cost, where d is the size of a largest minimal forbidden graph for π existing in G. We now treat a more specific type of hereditary graph properties. We say that hereditary π is matroidal if, on any graph G = (V, E), the edge sets of subgraphs of G satisfying π form the set of independent sets of a matroid defined on E. Such a matroid is said to be derived from the property π on E. Example 7: The feedback vertex set problem asks to find a minimum cost vertex set C in a vertex-weighted graph G such that every cycle in G passes through a vertex in C. This problem corresponds to ND(π) for π = “acyclic,” and this property is matroidal because the cycle matroid is derived from π on any graph.


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For the matroid (E, r) derived from some matroidal property on G = (V, E), denote its dual matroid by (E, rd ). Define a set function f such that def

f (S) = rd (δ(S)) for S ⊆ V . Then, f is clearly nondecreasing but also submodular (recall Example 4), and (V, f ) is a submodular system. Since C ⊆ V is a solution of ND(π) iff the edge set of G − C = G[V − C], which is E − δ(C), is independent in (E, r), it follows that C is a solution exactly when δ(C) is spanning in (E, rd ) because in any matroid a subset is independent iff its complement is spanning in the dual matroid; that is, f (C) = rd (δ(C)) = rd (E) = f (V ). Thus, ND(π) for any matroidal π can be modeled by the SSC by means of f defined this way. Theorem 6: For matroidal π Greedy approximates ND(π) in G = (V, E) within a factor of H(maxv∈V rd (δ(v))) ≤ H(maxv∈V |δ(v)|). For some matroidal properties π the rank associated with any minimal solution can be shown bounded, within a constant factor, by the rank associated with the minimum solution. For instance, consider the cocycle matroid (E, rd ) defined on G = (V, E), and let C be any minimal feedback vertex set in G. Then, the following relation exists: rd (δ(v)) ≤ 2rd (E) (4) v∈C

Thus, j∈C f (j) ≤ 2f (V ) for any minimal C in any G. Plugging this into Eq. (3) and applying Theorem 2 to SSC on (V, f ), Theorem 7: The algorithm Dual approximates the feedback vertex set problem within a factor of 2. The relation (4) is now known to hold for many other matroidal properties. Definition 4: A graph G is uniformly k-sparse iff in any nonempty subgraph H of G the number of edges exceeds the number of vertices by at most k. The property “uniformly k-sparse” can be shown matroidal for all integer k, and even more, Eq. (4) holds for them for any k, and hence, Theorem 8 [10]: When π is “uniformly k-sparse” for any fixed k, Dual computes a solution for ND(π) of cost at most twice the minimum cost. 4.3 Extension to Hypergraph Problems As indicated in Example 1 the function f = |δ| extended to hypergraphs remains submodular, and all the previous SSC problems defined based on the cut function δ are still SSC’s when redefined on hypergraphs. While the approximability of Greedy is rather immune to such modification, the performance of Dual tends to be highly sensitive to it. Fortunately, though, the approximation factor of 2 on graphs tends to go up only to

the reasonable ones on hypergraphs. Let ∆(G) denote the cardinality of the largest edge in a hypergraph G. Then, it can be shown that Dual computes a solution for either capacitated partial vertex cover or feedback vertex set of cost at most ∆(G) times the optimal cost in any hypergraph G. When it is taken into account that no polynomial time approximation algorithm with a constant factor better than ∆(G) is known even for the ordinary vertex cover, these bounds are acceptable as reasonably good ones. References [1] R. Bar-Yehuda, “Using homogeneous weights for approximating the partial cover problem,” Proc. 10th Annual ACM-SIAM Symp. on Discrete Algorithms, pp.71–75, 1999. [2] R. Bar-Yehuda and S. Even, “A local-ratio theorem for approximating the weighted vertex cover problem,” Annals of Discrete Mathematics, vol.25, pp.27–46, North-Holland, 1985. [3] N.H. Bshouty and L. Burroughs, “Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem,” Proc. 15th Symp. on Theoretical Aspects of Computer Science, pp.298–308, 1998. [4] V. Chv´ atal, “A greedy heuristic for the set-covering problem,” Mathematics of Operations Research, vol.4, no.3, pp.233–235, 1979. [5] G. Dobson, “Worst-case analysis of greedy heuristics for integer programming with nonnegative data,” Mathematics of Operations Research, vol.7, no.4, pp.515–531, 1982. [6] J. Edmonds, “Submodular functions, matroids, and certain polyhedra,” in Combinatorial Structures and Their Applications, eds. R. Guy, H. Hanani, N. Sauer, and J. Sch¨ onheim, pp.69–87, 1970. [7] U. Feige, “A threshold of ln n for approximating set cover,” Proc. 28th Annual ACM Symp. on Theory of Computing, pp.314–318, 1996. [8] S. Fujishige, “Submodular functions and optimization,” Annals of Discrete Mathematics, vol.47, North Holland, 1991. [9] T. Fujito, “A unified local ratio approximation of nodedeletion problems,” Proc. 4th European Symp. on Algorithms, pp.167–178, 1996. [10] T. Fujito. “Approximating node-deletion problems for matroidal properties,” J. Algorithms, vol.31, pp.211–227, 1999. [11] M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization,” Combinatorica, vol.1, pp.169–197, 1988. [12] M. Gr¨ otschel, L. Lov´ asz, and A. Schrijver, “Geometric methods in combinatorial optimization,” in Progress in Combinatorial Optimization, ed. W.R. Pulleyblank, pp.167–183, Academic Press, 1984. [13] J. H˚ astad, “Some optimal in-approximability results,” Proc. 29th Annual ACM Symp. on Theory of Computing, pp.1–10, 1997. [14] D.S. Hochbaum, “Approximation algorithms for the set covering and vertex cover problems,” SIAM J. Computing, vol.11, pp.555–556, 1982. [15] D.S. Hochbaum, “The t-vertex cover problem: Extending the half integrality framework with budget constraints,” Proc. APPROX’98 1st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, pp.111–122, 1998.


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Toshihiro Fujito received his B.E. and M.E. degrees in Mechanical Engineering from Kyoto University, in 1981 and 1983, respectively. He received his M.S. and Ph.D. degrees in Computer Science from Pennsylvania State University, in 1986 and 1994, respectively. He joined the Department of Electronics at Nagoya University as an assistant professor in 1999, prior to which he had been a research associate at Hiroshima University. His research interests are in the area of algorithms for combinatorial optimization problems.