Graph Labelling and Assignment of Non-Reusable ... - CiteSeerX

Graph Labelling and Assignment of Non-Reusable Frequencies? Dimitris A. Fotakis1;2 1

Paul G. Spirakis1;2

Computer Engineering and Informatics Department University of Patras, 265 00 Rion, Patras, Greece 2 Computer Technology Institute | CTI Kolokotroni 3, 262 21 Patras, Greece Email: [email protected], [email protected]

Abstract. In this work, we study Graph Radio Labelling, that is to assign unique integer labels

to all the vertices of a graph, such that adjacent vertices get labels at distance at least 2. The objective is to minimize the label span. Radio labelling models the assignment of frequencies, in case that the transmitters are not allowed to operate at the same channel. At rst, we show that radio labelling is related to TSP(1,2), and it is NP -complete and MAX{ SNP-hard. Then, we present a polynomial time algorithm for computing an optimal radio labelling, given a coloring of the graph with constant number of colors. Thus, we prove that radio labelling in planar graphs is in P . Also, we describe an NC algorithm that, given an arbitrary graph G(V;E ), computes a radio labelling of value at most jV j plus the cardinality of the Maximum Clique (Chromatic Number) of G. Hence, it achieves an approximation ratio of 3 2 , and an absolute performance guarantee of (G) without assuming a near optimal coloring. Additionally, we show the existence of a constant
? < 1, such that approximating radio labelling in graphs with maximum degree at most
jV j,

? , is essentially as hard as in general instances. We also obtain similar results for TSP(1,2). In particular, our parallel algorithm is the rst 3 -approximation NC algorithm for TSP(1,2). 2

?

This work was partially supported by ESPRIT LTR Project no. 20244|ALCOM{IT.

1 Introduction In the problem of frequency assignment, we are given a set of transmitters (radio stations) and a set of interference constraints on pairs of transmitters, and we seek an assignment of frequencies (radio channels) to the transmitters so as to avoid interference, i.e. an assignment that ful lls all the interference constraints. The objective is to minimize the maximum channel used (frequency span). If we assume that frequency channels are uniformly distributed in the spectrum, the frequency span determines the bandwidth allocated for the assignment. The Frequency Assignment Problem (FAP) is fundamental in the areas of telecommunications and design of radio networks (cf. [Hal80]). The most common model for an instance of frequency assignment is the interference graph. Each vertex of an interference graph represents a transmitter, and each edge represents an interference constraint between the adjacent transmitters. In the following, we consider that interference constraints are represented by an unweighted interference graph implicitly, in the sense that an edge denotes spacial adjacency (and potential interference) between the corresponding transmitters. Thus, the interference graph represents the intersections between the areas of coverage of the dierent transmitters (see also [HLS96]). In the majority of practical applications, the interference graphs are either planar or have a small (not necessarily constant) chromatic number, because of the spacial distribution of the transmitters. Moreover, the interference graphs are sometimes assumed to have a very special structure (e.g. lattices, grids [HLS96, Lee97, GL98]), but this assumption seems quite restrictive. Under the consideration that a pair of adjacent transmitters should be assigned dierent frequencies, the problem of frequency assignment is equivalent to the problem of coloring the interference graph with the minimum number of colors. In graph coloring, we assign colors to the vertices of a graph, such that no two adjacent vertices get the same color. Eventhough it is NP -complete to nd the chromatic number of a planar graph [GJ79], and to approximate general instances of graph coloring [LY94], it is well known that any planar graph can be colored with at most four colors, that is within 1 from the optimal (e.g. [HS78]). Multicoloring (i.e. each vertex gets more than one colors), distributed, and on-line coloring of planar graphs (e.g. [PPS97]) have been extensively used for modelling various practical problems related to frequency assignment. It was observed many years ago [Hal80] that graph coloring provides a very limited model for frequency assignment. Suppose that a radio receiver is tuned to a signal on a channel c0 , broadcasted by its local transmitter (i.e. the one at closest distance). Reception will be degraded if there is excessive interference from other transmitters in the vinicity. First, there is \co-channel" interference, due to reuse of the channel c0 at nearby sites; but there are also contributions from sites using channels near c0, since in practice neither transmitters nor receivers operate exclusively within the frequencies of their assigned channels. Recently, the limitations of graph coloring as a model for FAP have also been observed in the framework of mobile communications (cf. [NS97]). Additionally, in some practical applications, the transmitters cover a local or metropolitan area. In this case, graph coloring is not an appropriate model at all, because the transmitters are not 2

allowed to operate at the same frequency (non-reusable frequencies). To ensure acceptable signal quality, constraints should be imposed on the allowed channel separations between pairs of potentially interfering transmitters. A realistic model for FAP in unweighted interference graphs is the -coloring problem [Hal80, Ray85]. In -coloring, two vertices are allowed to get colors at distance x,   x  0, i their distance in the interference graph is at least  ? x + 1. The objective is to minimize the color span. An exact algorithm for -coloring in lattices is described in [HLS96]. However, we are not aware of a polynomial time approximation algorithm of guaranteed performance, even for unweighted planar graphs, and for small values of  (e.g.  = 2). In this paper, we study Graph Radio Labelling, which is the equivalent of 2-coloring in the context of the assignment of non-reusable frequencies. In particular, a radio labelling of an interference graph is an assignment of distinct labels/channels to all the transmitters, such that adjacent transmitters get labels at distance at least 2 (i.e. non-neighboring channels). The objective is to minimize the maximum label used (label span). The de nition of radio labelling is communicated to us by [Har97]. Radio labelling is an appropriate model for practical applications, where the transmitters are not allowed to operate at the same channel, and a single value can be used for the spectral separation between the channels assigned to potentially interfering transmitters. Competitive algorithms and lower bounds for on-line radio labelling are described in [FPPS97]. Radio labelling and related combinatorial problems, that model the assignment of non-reusable frequencies, are widely used for obtaining lower bounds on the optimal values of general FAP instances. Since FAP is an intractable optimization problem and approximation algorithms of guaranteed quality are not known, lower bounds are necessary in order to assess the quality of the assignments found by heuristic algorithms. Two approaches are mainly used for computing lower bounds [SH97]. The rst approach is based on a transformation from frequency assignment to the Travelling Salesman Problem (TSP) [Ray85]. In general, lower bounds obtained from TSP are poor, because the corresponding instances contain many edges of zero length, and many interference constraints are violated. The second approach is based on the observation that certain subgraphs (e.g. kernels [ASH97]) of the interference graph determine the value of the frequency span. Moreover, the majority of vertices of such subgraphs are assigned distinct channels. Thus, good lower bounds can be obtained by computing near optimal radio labellings for such subgraphs.

1.1 Summary of Results We show that radio labelling and 2-coloring remain NP -complete for graphs with diameter 2. We also prove that radio labelling is equivalent to the problem of nding a spanning path of minimum length in a complete graph, where the edge lengths are either 1 or 2 (HP(1,2)). Obviously, HP(1,2) is the path analogue of TSP(1,2). Therefore, radio labelling is MAX{SNP-hard and approximable in polynomial time within 67 [PY93]. In a recent survey paper [GL98], the relation of radio labelling with path covering is also observed. 3

Then, we present a polynomial time algorithm that computes an optimal radio labelling of a graph, given a coloring with constant number of colors. Thus, we prove that radio labelling is in P for planar graphs, and for graphs colorable with constant number of colors in polynomial time. Although planar interference graphs are quite typical in the context of FAP, this is the very rst result concerning the complexity of radio labelling in planar graphs. We present an ecient NC approximation algorithm that, given an arbitrary graph G(V; E ), computes a radio labelling of value at most jV j plus the cardinality of the Maximum Clique of G. Hence, this algorithm approximates radio labelling within a factor of 23 . Given a graph G(V; E ) with chromatic number (G), our NC algorithm operates without assuming a near optimal coloring or any knowledge on (G), and it computes a radio labelling of value within an additive term of (G) from the optimal. Therefore, it outperforms all known sequential approximation algorithms, when restricted to graphs with chromatic number (clique number) less than jV6 j . This is important because typical interference graphs have small chromatic number. On the other hand, if we excude some classes of graphs (e.g. planar), we do not know how to compute an approximate coloring with reasonable number of colors, even for graphs with constant chromatic number (cf. [KMS94]). Consequently, our NC approximation algorithm complements the aforementioned exact algorithm, which assumes the existence of a near optimal coloring. Additionally, we prove that there exists a constant
? < 1, such that for all
2 [
?; 1), radio labelling in graphs G(V; E ) with maximum degree
(G) 
jV j is essentially as hard to approximate as in general graphs. On the other hand, if
(G) < n2 , then an optimal radio labelling of value jV j can be easily computed. Since radio labelling is closely related to HP(1,2) and TSP(1,2), the results above can be translated to similar results for the latter problems. In particular, given a partition of a graph with constant number of cliques, we show how to decide in polynomial time if this graph is Hamiltonian. We are not aware of a similar algorithm that exploits partition into cliques so as to decide Hamiltonicity. Also, our NC algorithms approximates TSP(1,2) and HP(1,2) within a factor of 23 . This is the rst 32 -approximation NC algorithm for TSP(1,2). Up to now, the best known parallel algorithm for metric TSP and all the special cases is the algorithm of Christo des (an RNC implementation for metric TSP is described in [DSST97]). However, since it requires the computation of a perfect matching of minimum weight (maximum cardinality matching for TSP(1,2)), it is in RNC , but it is not known to be in NC . Additionally, we show that dense instances of TSP(1,2) and HP(1,2) are essentially as hard to approximate as general instances, as opposed to the success in aproximating dense instances of many combinatorial optimization problems (cf. [Kar97] for a survey). Very recently, a dierent proof of the same result appears independently in [FVK98].

2 Preliminaries Given a graph G(V; E ), (G) denotes the chromatic number of G, MISG is the cardinality of the Maximum Independent Set, MCG is the cardinality of the Maximum Clique, and d(u; v ) denotes the 4

length of the shortest path between u; v 2 V . For all graphs G, MCG  (G). The complementary graph G is a graph on the same vertex set V that contains an edge fu; v g 2 V  V , i fu; v g 62 E . Clearly, MISG = MCG . A graph G(V; E ) is called
-bounded for some constant 1 >
> 0, if the maximum degree
(G) is no more than
jV j. Also, a graph G(V; E ) is called  -dense for some constant 1 >  > 0, if the minimum degree  (G) is at least  jV j. In the Traveling Salesman Problem (TSP), we are given n nodes, and for each pair of distinct nodes fi; j g a distance di;j . The objective is to nd a salesman tour (i.e. a simple cycle that visits each node exactly once) of minimum length. TSP with distances one and two (TSP(1,2)) is a special case of TSP restricted to complete graphs, where all edge lengths are either 1 or 2; clearly, TSP(1,2) is a special case of metric TSP, since the edge lengths always satisfy the triangle inequality. Alternatively, TSP(1,2) is a generalization of the Hamiltonian Cycle problem, where each edge of the input graph has length 1, and each non-edge has length 2. In this case, we seek the tour (simple spanning cycle) with the fewest possible non-edges. An interesting variant of TSP(1,2) is the Hamiltonian Path with distances 1 and 2 (HP(1,2)), where we seek a simple spanning path of minimum possible length. Obviously, both TSP(1,2) and HP(1,2) are NP -complete, as generalizations of the Hamiltonian Cycle and Path problems. Moreover, they are MAX{SNP-hard, even if the graph formed by the edges of length 1 has maximum degree at most 4, and there exists a polynomial time 67 -approximation algorithm [PY93]. In the sequel, the graphs formed by the edges of length 1 are sometimes used for de ning instances of TSP(1,2) and HP(1,2).

De nition 1 (-labelling). Given a graph G(V; E ), the problem of -labelling,   l2, is tom com . The pute a function  : V 7! f1; : : :;  g, such that, for all v; u 2 V , j (v ) ?  (u)j  d v;u value of a valid -labelling is the maximum label used (label span). (

)

This paper is devoted to the study of 2-labelling, which is also called radio labelling or distance2 labelling. Obviously, radio labelling is the analogue of 2-coloring in the context of non-reusable frequencies. In particular, for graphs G with diameter 2, radio labelling is equivalent to 2-coloring. Furthermore, both problems are NP -complete for such graphs, eventhough the coloring of G2 is trivial.

Lemma 2. Radio labelling and 2-coloring restricted to graphs with diameter 2 are NP -complete. Proof. Given any graph G(V; E ) with diameter 2, a 2-coloring of G is a function 2 : V 7! f1; : : :;  g such that, for all v; u 2 V , j2(v ) ? 2 (u)j  2, i fv; ug 2 E ; and j2 (v ) ? 2 (u)j  1, otherwise. Clearly, this is an instance of radio labelling. Since radio labelling is in NP , and it is equivalent to Hamiltonian Path in the complementary graph (see also the proof of Lemma 3), we conclude the proof by showing that Hamiltonian Path remains NP -complete for complements of graphs with diameter 2. Let G0(V 0; E 0) be any graph, and s; t 2 V 0 be two non-adjacent vertices. The problem of deciding if G0 contains a Hamiltonian Path, that starts from s and ends to t, is NP -complete (Hamiltonian Path between Two Vertices [GJ79]). Let G(c)(V 0 [ fvs ; vtg; E 0 [ f(s; vs); (t; vt)g) be the graph

5

obtained from G0 by adding two independent vertices vs ; vt , and connecting vs to s and vt to t. Hamiltonian Path in G(c) is NP -complete, because G(c) contains a Hamiltonian Path i G0 contains a spanning path between s and t. Moreover, the following observations for the complementary graph G(c) (i.e. G) justify that diam(G(c)) = 2. 1. The vertex pairs (s; t); (vs; vt); (s; vt) and (t; vs) are connected by edges. 2. Any pair of vertices u; w 2 V 0 ?fs; tg are at distance at most two, because they are connected to both vs ; vt. 3. Any vertex u 2 V 0 [ fvs g ? fs; tg is at distance at most two from s, because both u and s are connected to vt . 4. Any vertex u 2 V 0 [ fvt g ? fs; tg is at distance at most two from t, because both u and t are connected to vs . ut Apart from the intractability, the previous lemma justi es the application of radio labelling to certain interference subgraphs for obtaining good lower bounds for frequency assingment. Radio labelling can be thought as a vertex arrangement problem. In particular, given a vertex ordering v1; v2; : : :; vn, a valid radio labelling can be computed as follows: 2(v1) = 1, and for i = 1; : : :; n ? 1, 2(vi+1) = 2(vi) + 1, if fvi; vi+1g 62 E , and 2(vi+1) = 2(vi) + 2, otherwise. Conversely, any radio labelling implies a vertex arrangement, in the sense that v precedes u, i 2(v ) < 2(u). Throughout this paper, we only consider radio labellings that are minimal with respect to the corresponding vertex arrangement, i.e. given a radio labelling L, there does not exist another radio labelling L0 that corresponds to the same vertex arrangement and 2(L0 ) < 2(L). In fact, the following is a consequence of the discussion above.

Lemma 3. Radio labelling is equivalent to HP(1,2) in the complementary graph. Proof. Given an instance of radio labelling, i.e. a graph G(V; E ), the corresponding instance of HP(1,2) is a complete graph G^ on the vertex set V , and the distance function d^ is de ned for all v; u 2 V , v 6= u, by 8 < d^(v; u) = : 1 if (v; u) 62 E 2 if (v; u) 2 E Additionally, if we are given a valid radio labelling of value   n, we can easily compute a spanning path of G^ of length exactly  ? 1. In particular, we construct a spanning path of G^ , in which the vertices are arranged in increasing order of their labels. We claim that the length of the spanning path up to any vertex of label i is exactly i ? 1. We prove the claim by induction on i. The claim is clearly true for the rst vertex (i = 1). We assume that it is true for any vertex v of label i, and let u be the next vertex in the spanning path. If the label of u is i + 1, the edge fv; ug is not present in G, and d^(v; u) = 1. Thus, the length of the path up to the vertex u is exactly i. If the label of u is i + 2, there does not exist a vertex of label i + 1. Therefore, the edge fv; ug 2 E , and d^(v; u) = 2.

6

Consequently, the path up to u has length exactly i + 1. Obviously, the resultant spanning path has length exactly  ? 1. ^ d^) of HP(1,2), we can construct an instance G(V; E ) Conversely, if we are given an instance (G; of radio labelling by only connecting the vertex pairs that are at distance 2. Furthermore, given a spanning path of G^ of length  ? 1  n ? 1, we can easily compute a valid radio labelling of G of value exactly  , as follows: We select (arbitrarily) an end vertex v of the path, and we assign the label 1 to v . Then, the label of each vertex u 2 V is one plus the distance of u from v in the spanning path. Alternatively, if the label 2(w) is assigned to a vertex w, and u is the next vertex in the spanning path, the label of u is 2(u) = 2(w) + d^(w; u). Obviously, the last vertex of the path gets the label  , and this procedure assigns unique labels from the set f1; : : :;  g to the vertices of G. Since an edge fv; ug is present in E , i d^(v; u) = 2, the vertices that are adjacent in E are at a distance at least two in the spanning path. Therefore, if fv; ug 2 E , then j(v ) ? (u)j  2, and the resultant radio labelling is valid. ut The previous lemma implies that radio labeling is NP -complete, and MAX{SNP-hard, even if the minimum degree of the input graph G(V; E ) is at least jV j ? 5. Moreover, it is approximable in polynomial time within 67 [PY93]. On the other hand, given a graph G(V; E ) with maximum degree
(G) < jV2 j , a radio labelling of value jV j can always be computed in polynomial time, because of Dirac's Theorem. Given a coloring of a graph G(V; E ) with  colors, it is easy to nd a radio labelling of value at most jV j +  ? 1. Therefore, 2(G)  jV j + (G) ? 1. However, it usually is NP -hard even to approximate the value of (G) within a reasonable factor.

3 An Exact Algorithm for Constant Number of Color Classes Then, we show that, given a graph G(V; E ), jV j = n, and a coloring of G with constant number of colors, we can decide in polynomial time if there exists a radio labelling for G of value n. Therefore, we can compute an optimal radio labelling of G in polynomial time. The class consisting of the graphs colorable with constant number of colors in polynomial time is non trivial. An interesting subclass is planar graphs. The class of planar graphs is very important for the frequency assignment problem, because the interference graphs are planar in many practical applications. Moreover, the complexity of nding an optimal radio labelling in such graphs is open, despite the fact that a near optimal radio labelling can be easily obtained from a coloring with constant number of colors. Radio labelling is equivalent to Hamiltonian Path in the complementary graph, and a coloring with constant number of colors corresponds to a partition of the complementary graph into constant number of cliques. In the following, the technical part of the proof is presented by means of Hamiltonian Paths (actually Cycles) in the complementary graph. The main reason is that well understood graph theoretic tools (e.g. spanning trees, eulerian trails) can naturally be applied to 7

the context of Hamiltonian Cycles and partitions into cliques. As a result, the arguments are better motivated and the presentation is more consistent. Another (technical) reason is that we are able to show how to decide in polynomial time if the complementary graph contains a Hamiltonian Cycle. Notice that, given a graph and a Hamiltonian Path, it is NP -complete to decide if the graph is Hamiltonian (cf. [GJ79]).

3.1 Hamiltonian Cycles and Partitions into Cliques Throughout this section, let G(V; E ) be a connected graph,  > 1, and C = fC1; C2; : : :; C g be a partition of V into  cliques.

Theorem 4. Given a graph G(V; E ), jV j = n, and a partition of V into  cliques, there exists a deterministic algorithm that runs in time O n ? and decides if G is Hamiltonian. If G is (2

1)

Hamiltonian, the algorithm outputs a Hamiltonian Cycle.

Proof. Given a set M  E of inter-clique edges, the clique graph T (C; M ) contains exactly  vertices, that correspond to the cliques of C , and represents how the edges of M connect the dierent cliques. For any spanning subgraph H (V; EH) of G, there exists a clique graph TH (C; MH ) that corresponds to H . Obviously, if H is connected, then TH is connected, and if H is a Hamiltonian Cycle, then TH is euclidean. However, the converse statements are not always true. On the other hand, any clique graph T (C; M ) corresponds to a family H of spanning subgraphs of G, that share M as the set of inter-clique edges. The graphs of H may dier from each other with respect to the edges connecting vertices that belong to the same clique. A clique graph T is an HC-clique graph, if the family H contains at least one spanning cycle (Figure 1.b). Given a clique graph T (C; M ), we color an edge red, if it shares a vertex of G with another edge of M . Otherwise, we color it blue. The corresponding edges of G are colored with the same colors, while the remaining edges (E ? M ) are colored black. Additionally, we color red each vertex w 2 V , which is the common end vertex of two or more (red) edges. We color blue each vertex w 2 V , which is adjacent to exactly one edge of M (red or blue). The remaining vertices of G are colored black (Figure 1.a). Notice that the coloring of the vertices and the edges of G is completely determined by the set M of inter-clique edges. Let H be any spanning cycle of G and T be the corresponding HC-clique graph. Obviously, red vertices cannot be exploited for visiting any black vertices belonging to the same clique. Let H visit a clique Ci through a vertex v, and leaves Ci through w. Then v; w 2 Ci consist a pair of blue vertices. Clearly, H can use some black edges for visiting some black vertices of Ci . A blue pass through a clique Ci is a simple path P of length at least one, that entirely consists of non-red vertices of Ci . We say that a clique Ci is covered by M , if all the vertices of Ci have degree at most 2 in M , and the existence of a non-red vertex implies the existence of at least one blue vertex pair. The following proposition characterizes the class of HC-clique graphs.

8

Proposition 5. A connected, eulerian, clique graph T (C; M ) is an HC-graph i it ful lls the following property (HC-Property):

(HC1) For any i = 1; : : :; , Ci is covered by M ; and (HC2) There exists an eulerian trail R for T such that: For any red vertex v 2 V , R traverses the red edges that are incident to v in consecutive order (i.e. R passes through any red vertex exactly once using the corresponding red edge pair). Proof. By de nition, if T is an HC-graph, then there exists a cycle H that spans V . Additionally, the set of inter-clique edges of H is exactly M . Since any vertex of V has degree 2 in H , all the cliques Ci are covered by M . Also, for any red vertex v , H traverses the red edges that are incident to v consecutively. Clearly, if we remove the black vertices and edges from H , then we obtain a traversal of the edges of M that ful lls (HC2). Moreover, since H is a spanning cycle, the resultant traversal is an eulerian trail for T . Conversely, we will show that given a connected, eulerian clique graph T (C; M ), that ful lls the HC-Property, we can easily construct a Hamiltonian Cycle H , such that the set of inter-clique edges of H is exactly M . Let R be an eulerian trail for T that ful lls (HC2). The cycle H will follow the trail R. Since T is connected and all the cliques are covered by M , the trail passes through all the cliques Ci 2 C ; i = 1; : : :; . The Property (HC2) implies that every time R visits a red vertex v, H passes through v using the corresponding red edge pair. Since R is an eulerian trail, all the red vertices are included in H with degree 2. The rst time that R passes through a blue vertex pair v; w 2 Ci (case (a)), a path consisting of all the black vertices of Ci is imposed between v and w in H . The Property (HC1) implies that all the black vertices are included in H with degree 2. If R passes through a blue vertex pair v; w 2 Ci, and Ci does not contain any unvisited black vertices, then the black edge fv; wg is added to H . Since R is an eulerian trail that ful lls (HC2), all the blue vertices are included in H with degree 2. Moreover, H is connected and the inter-clique edges of H are exactly the edges of M . ut

The proof of Proposition 5 implies a polynomial time, deterministic procedure for deciding if a clique graph T (C; M ) is an HC-clique graph. Moreover, in case that T is an HC-graph, this procedure outputs a Hamiltonian Cycle. The time complexity of this deterministic decision procedure is polynomial in n and . For convenience, we consider that an HC-clique graph can be decided in time O(n ). It remains to show how to exploit the partition of G into  cliques so as to decide Hamiltonicity. Intuitively, if G is Hamiltonian, there should exist a spanning cycle H that takes advantage of the cliques of G. Namely, H is not expected to pass through the cliques many times. Clearly, if we prove that G is Hamiltonian i it contains a spanning cycle that passes through the cliques a small number of times, then this cycle could be found using exhaustive search. The previous discussion implies that this is equivalent to the existence of an HC-clique graph with a small number of edges. The following technical lemmas formalizes the aforementioned intuition. 9

Lemma 6. Let B  2 be some integer only depending on , such that the following is true for any graph G(V; E ), any partition of V into  cliques: If G is Hamiltonian and jV j > B , G contains at

least one Hamiltonian Cycle not entirely consisting of red vertices (inter-clique edges). Then, for any graph G(V; E ) and any partition of V into  cliques, G is Hamiltonian i it contains a Hamiltonian Cycle with at most B inter-clique edges. Proof. Let H be the Hamiltonian Cycle of G containing the minimum number of inter-clique edges and M be the corresponding set of inter-clique edges. Assume that jM j > B . Because of (i), H cannot entirely consist of red vertices. Therefore, H should contain at least one blue vertex pair. We substitute any blue pass of H through a clique Ci with a single (red) super-vertex v^, that is also considered to belong to the clique Ci . Hence, v^ is connected to all the remaining vertices of Ci . These substitutions result in a cycle H 0 that entirely consists of red vertices, and contains exactly the same set M of inter-clique edges with H . Obviously, the substitutions of all the blue passes of H with single red super-vertices belonging to the corresponding cliques result in a graph G0(V 0; E 0) dierent from G. However, since H 0 is a Hamiltonian Cycle, G0 is also Hamiltonian, and V 0 is partioned into  cliques. Moreover, it is not hard to verify that, if H 0 is any Hamiltonian Cycle of G0, then the reverse substitutions of all the red super-vertices v^ with the corresponding blue passes result in a Hamiltonian Cycle of G, that contains exactly the same set of inter-clique edges with H 0 (Figure 1). Since H 0 is a Hamiltonian Cycle that entirely consists of red vertices and contains more than B inter-clique edges, the statement (i) implies that there exists another Hamiltonian Cycle of G0, that contains strictly less inter-clique edges than H 0. Therefore, there exists a Hamiltonian Cycle of G, that contains strictly less inter-clique edges than H . Obviously, this contradicts the selection of H . ut

Consequently, for proving that G is Hamiltonian i there exists a Hamiltonian Cycle using at most B inter-clique edges, it only suces to prove the same bound for red Hamiltonian Cycles (i.e. Hamiltonian Cycles that entirely consist of inter-clique edges). Intuitively, the class of graphs for which all the Hamiltonian Cycles entirely consist of inter-clique edges should be very restricted.

Lemma 7. Given a graph G(V; E ) and a partition of V into  cliques, G is Hamiltonian i there exists an HC-clique graph T (C; M ), such that jM j  ( ? 1). Proof. By de nition, the existence of an HC-clique graph for G implies that G is Hamiltonian (independently of the number of edges of M ). Thus, (2) implies (1) trivially. Conversely, let H be the Hamiltonian Cycle of G that contains the minimum number of interclique edges, and T (C; M ) be the corresponding HC-clique graph. If jM j  ( ? 1), then we are done. Otherwise, Lemma 6 implies that it suces to consider only Hamiltonian Cycles H , that entirely consist of inter-clique edges. Consequently, assume that the coloring of V under M entirely consists of red vertices, and consider an arbitrary orientation of the Hamiltonian Cycle H (e.g. a traversal of the edges of H in

10

01

10 10 01 10 10 (a) A graph partitioned into 3 cliques

1 0 0 1

11 00 00 11 11 00 00 11 01

01

(b) An HC-clique graph

A black vertex A red vertex A blue vertex

(c) A red Hamiltonian Cycle

01 1 0 10 1 0

0101 10

A blue edge A red edge A black edge A black edge of the cycle An edge that may be removed

11 00 (d) A Hamiltonian Cycle using less red edges

11 00

(e) An HC-graph with less edges

Fig. 1. A decrease in the number of inter-clique edges contained by a Hamiltonian Cycle. the clockwise direction). If there exist a pair of cliques Ci and Cj and four vertices v1; v2 2 Ci and u1; u2 2 Cj , such that both vx are followed by ux (x = 1; 2) in a traversal of H , then the (black) edges fv1; v2g and fu1 ; u2g can be exploited in order to obtain a Hamiltonian Cycle containing less inter-clique edges than H . Clearly, this would be a contradiction to the selection of H . The previous situation can be avoided, only if for all i = 1; : : :; , and j = 1; : : :; , j 6= i, at most one vertex vj 2 Ci is followed by a vertex u of the clique Cj in a traversal of H . Obviously, this restriction results in Hamiltonian Cycles H that contain at most ( ? 1) inter-clique edges. Hence, any Hamiltonian Cycle containing more than ( ? 1) inter-clique edges cannot be a Hamiltonian Cycle with the minimum number of inter-clique edges. This contradicts the selection of H and the corresponding HC-clique graph T . ut 



Lemma 7 implies that we can decide if G is Hamiltonian in time O n(2?1) , because the number of the dierent edge sets containing at most ( ? 1) inter-clique edges is at most n2(?1) , and we can decide if a clique graph is an HC-graph in time O(n ). ut Then, we prove that if G contains a blue Hamiltonian Cycle (i.e. a Hamiltonian Cycle that does not contain any red vertices and edges), then there exists a blue Hamiltonian Cycle using at most 2( ? 1) inter-clique edges. 11

1

2

... ...

3

k-1

k

Fig. 2. An HC-graph that contains exactly 2( ? 1) inter-clique edges.

Lemma 8. Given a graph G(V; E ) and a partition of V into  cliques, if G contains a blue Hamiltonian Cycle, then there exists a (blue) HC-clique graph T (C; M ), such that jM j  2( ? 1). Proof. Let H be the blue Hamiltonian Cycle of G that contains the minimum number of interclique edges, and T (C; M ) be the corresponding HC-clique graph. If jM j  2( ? 1), then we are done. Notice that, since red vertices cannot be created by removing edges, any eulerian, connected, spanning subgraph of T is an HC-graph that corresponds to a blue Hamiltonian Cycle of G. Let ST (C; MS ) be any spanning tree of T . Since MS consists of ?1 edges, the graph T (S ) (C; M ? MS ), obtained by removing the edges of the spanning tree from T , contains at least  edges. Therefore, T (S ) contains a simple cycle L. The removal of the edges of L does not aect connectivity (the edges of L do not touch the spanning tree), and subtracts 2 from the degrees of the involved vertices. Clearly, the graph T 0(C; M ? L) is an HC-graph with less edges than T . This contradicts the selection of T and H . ut

There exist HC-graphs that contain exactly 2( ? 1) edges and correspond to a Hamiltonian Cycle using the minimum number of inter-clique edges (Figure 2). Therefore, the bound of 2( ? 1) is tight. However, even if we allow red vertices, we are not able to construct examples of HC-clique graphs that contain more than 2( ? 1) edges, and correspond to Hamiltonian Cycles using the minimum number of inter-clique edges. Therefore, we conjecture that the bound of 2( ? 1) holds for any graph and any partition into  cliques. An inductive (on the cardinality of V ) application of Lemma 6 suggests that this conjecture is equivalent to the following:

Conjecture 9. For any Hamiltonian graph G(V; E ) of 2 ? 1 vertices and any partition of V into  cliques, there exists at least one Hamiltonian Cycle not entirely consisting of inter-clique edges. The previous conjecture is very similar to a theorem proved by C.A.B. Smith in 1946. It states that the number of Hamiltonian Cycles that contain any given edge of a cubic graph (i.e. a simple regular graph of degree 3) is even. A simple algebraic proof of this theorem can be found in [Ber85]. If we only consider Hamiltonian graphs, Smith's Theorem can be applied to graphs G(V; E ) consisting of a Hamiltonian Cycle and a perfect matching. On the other hand, Conjecture 9 has a weaker conclusion than Smith's Theorem, but it is applicable to the graphs consisting of a red Hamiltonian Cycle and  cliques of arbitrary cardinalities. Remark. Notice that our algorithm works for all graphs, integers  > 1, and partitions into  cliques. Also, any graph can be partitioned into a (non-constant) number of cliques in polynomial 12

time (e.g. maximum matching). Hence, any (deterministic) algorithm that (i) exploits a partition into  cliques for deciding Hamiltonicity, and (ii) works for all graphs, integers  > 1, and partitions into  cliques (as our algorithm does), should run in time exponential in  (e.g. nO() , 2O() ), unless NP has (deterministic) subexponential simulations.

3.2 A Reduction from Radio Labelling to Hamiltonian Cycle Lemma 10. Given a graph G(V; E), jV j = n, and a coloring of G with  colors, an optimal radio labelling of G can be computed in O n  time. (2 +1)

Proof. Let G(V; E) be the complement of the input graph G. Obviously, 2(G)  n +  ? 1. Therefore, at most  ? 1 channels remain unused by an optimal radio labelling. Thus, any optimal solution to the HP(1,2) instance, that corresponds to G, (see also the proof of Lemma 3) contains at most  ? 1 non-edges (of G), and any optimal tour (simple spanning cycle) contains at most  nonedges (of G). Then, we show how to compute an optimal TSP(1,2) solution to the complementary graph G. Obviously, we can delete an edge of maximum length from the optimal tour so as to obtain an optimal spanning path. ?
Let A be the algorithm of Theorem 4. We call A at most M = O n2 times, with input the graphs Gi(V; E [ Ni ), i = 1; : : :; M . The sets Ni are all possible subsets of non-edges of G with at most  elements, including the empty one. Let Gi be a Hamiltonian graph that corresponds to a set Ni of minimum cardinality. Obviously, the Hamiltonian Cycle produced by A(Gi ) is an optimal tour for G. ut

Since any planar graph can be colored with constant number of colors in polynomial time, the following theorem is an immediate consequence of Lemma 10.

Theorem 11. An optimal radio labelling of a planar graph can be computed in polynomial time. Remarks. Theorem 11 suggests that an approximation algorithm for 2-coloring in planar graphs may be obtained as follows: At rst, (partially) decompose the graph to subgraphs, such that almost all vertices of each subgraph get distinct colors by an optimal (or near optimal) assignment; Then, compute an optimal radio labelling for these (planar) subgraphs; Finally, combine the partial assignments to obtain a near optimal assignment. Obviously, Conjecture 9 implies that, given a graph G(V; E ) and a coloring of V with  colors, optimal solutions for radio labelling can be computed in time nO() . ut

4 An N C Approximation Algorithm for Arbitrary Graphs Then, we present an NC algorithm that, for any graph G(V; E ), produces a radio labelling of value at most jV j + MCG ? 1, and a clique of cardinality at least 2(G) ? jV j + 1. Since 2(G)  2MCG , our algorithm achieves in NC an approximation guarantee of 23 for arbitrary graphs. Moreover, 13

it outperforms all known sequential approximation algorithms, when restricted to graphs with chromatic number (clique number) less than jV6 j . If we consider graphs with small chromatic number (e.g. constant, poly-logarithmic), then our NC algorithm always produces a radio labelling within a small additive term of the optimal, without assuming a near optimal coloring or some knowledge on the actual chromatic number of the underlying graph. In this sense, the NC approximation algorithm complements the exact algorithm presented in the previous section, which assumes a near optimal coloring. As before, for clarity of presentation, the NC approximation algorithm is described and analyzed in the context of Hamiltonian Paths and Cycles. In particular, we obtain the rst 23 -approximation NC algorithm for HP(1,2) and TSP(1,2). Up to now, the best known parallel algorithm for TSP(1,2) and all the special cases of metric TSP is the algorithm of Christo des [Chr76], that is based on the computation of a minimum weighted matching (maximum cardinality matching for TSP(1,2)), which is in RNC , but it is not known to be in NC .

4.1 The Algorithm for HP(1,2) and TSP(1,2) Let G(V; E ) be any graph of TSP(1,2) instance.

n vertices, and TSPG be the optimal value of the corresponding

Theorem 12. The following inequalities hold for any graph G(V; E ) of n vertices: 2MISG  TSPG  n + MISG

(1)

Moreover, a tour and an independent set that ful ll (1) can be computed in NC . Proof. The main part of the proof consists of an algorithm that computes a tour and an independent set of G, that ful ll (1). Correctness: The algorithm proceeds in phases, such that the following holds for the ith phase: For all vertices v , v 2 V i implies that degP (v )  1. Moreover, for each path p 2 P , exactly one of the end vertices of p is in V i . Therefore, at the end of the algorithm, the set P consists of simple paths (of 1-edges) and isolated vertices. If the isolated vertices are considered as paths of length 0 (trivial paths), the edge set P covers V with vertex disjoint simple paths. Additionally, the vertex sets S i are independent sets, because M i is a maximal matching in Gi (V i ; E i). Performance: The performance of the algorithm is determined by the number of edges (of length 1) that are contained in the set P at the end of the algorithm. Initially, the set P is empty. At each phase i, the edges of M i are added to P in step (*). Let jM i j be the number of edges of a maximal matching M i of the graph Gi . The algorithm runs for K + 1 phases, i.e. i = 1; : : :; K + 1, where K P will be xed later. Clearly, jP j = Ki=1 jM i j.

14

Algorithm TSP-MIS Input: A graph G(V;E ). Output: A set P of edges that cover V with vertex disjoint simple paths.

P := ;; i := 0;

A set S ? of independent vertices.

repeat

(*)

i := i + 1; /* ith phase */ V i := fv 2 V : degP (v) = 0g; For any non-trivial path p 2 P , add exactly one of Ei := fv;wg 2 E : v 2 V i ^ w 2 V i ; Find a Maximal Matching M i in Gi(V i ;E i); P := P [ M i; Si := fv 2 V i : degM (v) = 0g; until M i is empty or i > K ; S? := Si of maximum cardinality; return(P , S ? ); Fig.3. The Algorithm TSP-MIS.

the end vertices of

p to V i;

i

Since the vertices of V i that are not covered by a maximal matching M i form an independent set, we obtain that jV ij ? MISG  jM ij  jV ij 2 2 Furthermore, for any pair of vertices of V i that are matched by M i , exactly one vertex is added to V i+1. Therefore, jV ij  jV i+1j = jV ij ? jM ij  jV i j + MISG 2 2 The following inequality holds for all i  1, and can be proved by induction on i. 1 i jV i+1j  jV j + (2 2?i 1)MISG If K = dlog ne, then inequality (2) implies that 1 jV K+1j  jV j + (n ? 1)MISG < MIS + 1

n

(2)

G

By summing the equalities jV i+1 j = jV i j ? jM i j, for 1  i  K + 1, we obtain that K X K +1 1 jV j = jV j ? jM ij = jV j ? jP j i=1

Therefore, we obtain that jP j  n ? MISG , because if K = dlog ne, then jV K +1 j  MISG . Since the set P consists of simple paths of total length at least n ? MISG , we can construct a spanning cycle from P by adding no more than MISG edges of length 2. The length of the resultant tour will be no more than n + MISG . Therefore, TSPG  n + MISG . 15

Complexity: The algorithm runs for at most dlog ne +1 phases. Since the connected components of P are simple paths, the computation of Gi can be implemented in NC . The complexity of

the algorithm is dominated by the computation of the maximal matching. There exist a CRCW PRAM algorithm that produces a maximal matching in O(log3 n) time using O(n2 ) processors [IS86]. Moreover, the sequential complexity of TSP-MIS is O(n2 log n). Next, we prove that 2MISG  TSPG . Let TSPG = n + , for any 0   n, and let  : [n] 7! V be a permutation that corresponds to an optimal tour of G. Obviously,  de nes a set fp1; p2; : : :; p g of simple paths, which consist of 1-edges and are linked to a spanning cycle by 2-edges. Clearly, if a path pi consists of jpij vertices, it cannot contribute to any independent set more than jpi2j+1 vertices, if jpi j is odd, and jp2i j vertices, if jpi j is even. Therefore, the cardinality of any independent set in G cannot be greater than the sum of the cardinalities of the independent sets of pi 's. Consequently, MISG 



X



jpij 

i=1 2

jp j + 1 X i i=1 2

G  n +2 = TSP 2

It remains to prove that the algorithm TSP-MIS computes in NC an independent set of cardinality at least TSPG ? n. Assume that TSPG ? n = > 0. We run TSP-MIS for K = dlog ne phases, and we output the set S ? , namely the set S i, i = 1; : : :; K + 1, of maximum cardinality. Clearly, S i's are sets of independent vertices. If jS ?j = 0 < , we can substitute the MISG with 0 in the analysis on the size of P . Therefore, if jS ij < , for all 1  i  K + 1, the resultant set P would contain more than n ? edges. This is a contradiction, because it implies that we can compute a super-optimal tour. Consequently, the algorithm TSP-MIS always produces an independent set of cardinality at least TSPG ? n. There exist instances such that TSPG = n + MISG ? 1. For example, for any x > 1, consider a graph G consisting of a clique of n ? x vertices and an independent set of x vertices, all of them connected to the same vertex of the clique. Clearly, MISG = x + 1, and TSPG = n + x. Also, the inequality 2MISG  TSPG becomes tight for many instances. It is worth mentioning the Cn graph (a simple cycle of n vertices), and the complete bipartite graph, where the sizes of the classes are n and (1 ?  )n, for any  > 0. ut

4.2 Some Consequences of Theorem 12 Obviously, the arguments above also apply to HP(1,2) and to radio labelling in the complementary graph G. Therefore, the following are immediate consequences of Theorem 12:

Corollary 13. There exists an NC algorithm that runs in a CRCW PRAM in time O(log n) using O(n ) processors and approximates, 4

2

1. Radio labelling, HP(1,2), and TSP(1,2) within 23 .

16

2. HP(1,2) and TSP(1,2) restricted to graphs G, such that MISG  n, within (1 + ). 3. Radio labelling restricted to graphs G, such that MCG  n ((G)  n), within (1 + ). 2 4. Maximum Clique restricted to graphs G, such that 2(G)  (1 + )n, within 1+ .

5 Graphs with Bounded Maximum Degree Eventhough radio labelling is better approximable in graphs with bounded chromatic and clique numbers, we show that this is not the case for graphs with bounded maximum degree. In particular, we prove that there exists a constant
? < 1, such that for all
2 [
?; 1),
-bounded instances of radio labelling (i.e.
(G) 
n) are essentially as hard to approximate as general instances. On the other hand, an optimal radio labelling of value jV j can be found in polynomial time for any graph G(V; E ) with maximum degree less than jV2 j .

Theorem 14. There exist a constant
? < 1, such that, for all 1 >

?,
-bounded instances of radio labelling do not admit a PTAS, unless P = NP . Proof. Let G(V; E ) be any graph of n vertices. Clearly, n  2 (G)  2n. For any constant 1 >
> 1 , let  = 1?

, and Kn be the complete graph on n vertices. We construct a graph G0 by adding 2 Kn to G. The vertices of G are not connected to the vertices of Kn. Formally, the new graph is G0(V [ Kn; E 0), where E 0 = E [ffu; vg : 8u; v 2 Kng

Obviously, since  = 1?

, the maximum degree of any vertex of G0 is bounded by
n. By the construction of G0 , if we are given any radio labelling LG of G0, we can nd a radio labelling LG of G of value 2(LG )  2(LG ). (Ignore the labels assigned to the vertices of Kn in LG . This always results in a valid radio labelling of G.) Conversely, given any radio labelling LG of G, we can construct a radio labelling LG of G0 , such that the vertex arrangements implied by the labellings LG and LG are the same with respect to the vertices of G. Moreover, the value of the labelling LG is 0

0

0

0

0

0

 (LG ) = n + max(n; n ?  (LG)) 2

2

0

Let  be a label not assigned by LG , and u; v 2 V be the vertices assigned the labels  ? 1 and  +1, i.e. LG (u) =  ? 1 and LG (v ) =  + 1. Given a label not assigned by LG , such a vertex pair exists, because we only consider labellings of minimal value. Then, an arbitrary vertex of Kn is assigned the label . Obviously, if the labels not assigned by LG are at most n, then all the labels up to (1+  )n will be assigned by LG . Consequently, 2(LG ) = (1+  )n. Otherwise (2(LG )  (1+  )n), the labels assigned to Kn cannot increase the value of the resulting radio labelling. Therefore, 2(LG ) = 2(LG). Therefore, the optimal values 2(G) and 2(G0) ful ll the following inequalities: 0

0

0

 (G0) ? n   (G)   (G0)  (1 + ) (G) 2

2

2

17

2

Assume that for some constant  > 0, radio labelling is approximable within (1 + ) for
bounded graphs. Given any graph G, we can construct a
-bounded graph G0 by adding a clique on n vertices,  = 1?

, and nd an (1 + ) approximate solution LG to this instance. Then, we can transform LG to a radio labelling LG for the original graph G by ignoring the labels assigned to the additional vertices. Clearly, the following inequalities hold: 0

0

 (LG)   (LG )  (1 + ) (G0)  (1 + )(1 + ) (G) Since radio labelling is MAX{SNP-hard, there exists a constant ? > 0, such that radio labelling is not approximable within (1 + ? ), unless P = NP [ALMSS92]. Let ? >  > 0 be any small constant, and
? = ? < 1. If
-bounded instances of radio labelling admitted a PTAS for some

?, then general instances would be approximable within (1 + ? ). 2

2

0

2

2

1+ 1+

It is straight forward to verify that the same arguments can be applied even if a small constant number of edges are placed between the vertices of G and the vertices of Kn so as G0 to be a connected graph. ut Moreover, the previous transformation implies that approximating radio labelling in
-bounded instances is essentially as hard as approximating radio labelling in general instances. This becomes clear by considering instances with large optimal values, namely graphs G such that 2(G)    1?
1 +
n, for some constant 1 >

? . Then, the previous transformation is an L-reduction from such instances of radio labelling to
-bounded instances. A similar result from HP(1,2) and TSP(1,2) follows from the fact that radio labelling is equivalent to HP(1,2) in the complementary graph.

Corollary 15. There exist a constant ? > 0, such that, for all 0 <   ?, -dense instances of HP(1,2) and TSP(1,2) do not admit a PTAS, unless P = NP .

6 Conclusions An interesting direction for research is the desing of a polynomial time approximation algorithm for 2-coloring in planar graphs. Since 2-coloring is widely used as a model for frequency assignment, the design of an approximation algorithm of guaranteed performance is of major importance. One approach may be to decompose the planar graph to subgraphs, such that almost all vertices of each subgraph get distinct colors by an optimal (or near optimal) assignment. Then, our exact algorithm can be used for computing an optimal radio labelling for each of the resulting subgraphs. Obviously, the decomposition of the planar graph and the combination of the partial assignments to a near optimal solution require an appropriate planar separator theorem. Another direction for further research is the conjecture that, given a graph G(V; E ) and a partition of V into  cliques, G is Hamiltonian i there exists a Hamiltonian Cycle containing at most 2( ? 1) inter-clique edges. This would imply a nO() algorithm for the radio labelling of a graph G, given a coloring of G with  colors. In this paper, we prove the conjectured bound for 18

graphs and partitions that contain at least one Hamiltonian Cycle H , such that any vertex of V has degree at most 1 in the set of inter-clique edges of H . Moreover, we show that the conjectured bound is equivalent to a generalized version of Smith's Theorem on the number of Hamiltonian Cycles contained by cubic graphs.

References [ASH97] S.M. Allen, D.H. Smith, and S. Hurley. Lower Bounding Techniques for Frequency Assignment. Submitted to Discrete Mathematics, 1997. [ALMSS92] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof veri cation and hardness of approximation problems. Proc. of the 33th Annual IEEE Symposium on Foundations of Computer Science, pp. 14{23, 1992. [Ber85] C. Berge. Graphs (second edition). North Holland, 1985. [Chr76] N. Christo des. Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem. J.F. Traub (editor) Symposium on new directions and recent results in algorithms and complexity, page 441, 1976. [DSST97] J. Daz, M.J. Serna, P. Spirakis, and J. Toran. Paradigms for Fast Parallel Approximability. Cambridge University Press, 1997. [FVK98] W. Fernandez de la Vega and M. Karpinski. On Approximation Hardness of Dense TSP and other Path Problems. Available from http://cs.uni-bonn.de/info5/publications/CS-1998-en.html, 1998. [FPPS97] D. Fotakis, G.E. Pantziou, G. Pentaris, and P. Spirakis. Assignment of Frequencies in Mobile and Radio Networks. Invited talk in DIMACS Workshop on Networks and Distributed Computing, 1997. [GJ79] M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP Completeness. Freeman, San Francisco, 1979. [GL98] J. Griggs and D. Liu. Minimum Span Channel Assignments. To appear in Recent Advances in Radio Channel Assignment. Satelite Symposium to 9th SIAM Conf. on Discrete Mathematics, July 1998. [Hal80] W.K. Hale. Frequency Assignment: Theory and Applications. Proceedings of the IEEE 68(12), pp. 1497{1514, 1980. [Har97] F. Harary. Personal Communication, 1997. [HS78] E. Horowitz and S. Sahni. Fundamentals of Computer Algorithms. Pitman, 1978. [HLS96] J. van den Heuvel, R.A. Leese, and M.A. Shepherd. Graph Labelling and Radio Channel Assignment. Available from http://www.maths.ox.ac.uk/users/gowerr/preprints.html, 1996. [IS86] A. Israeli and Y. Shiloach. An improved algorithm for maximal matching. Information Processing Letters 33, pp. 57{60, 1986. [KMS94] D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semide nite programming. Proc. of the 35th IEEE Symposium on Foundations of Computer Science, pp. 2{13, 1994.

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[Kar97] M. Karpinski. Polynomial Time Approximation Schemes for Some Dense Instances of NP -hard Optimization Problems. Proc. of the 1st Symposium on Randomization and Approximation Techniques in Computer Science, pp. 1{14, 1997. [Lee97] R.A. Leese. A uni ed approach to the assignment of radio channels on a regular hexagonal grid. To appear in IEEE Transactions on Vehicular Technology, 1997. [LY94] C. Lund and M. Yannakakis. On the Hardness of Approximating Minimization Problems. Journal of Association for Computing Machinery 41, pp. 960{981, 1994. [NS97] L. Narayanan and S. Shende. Static Frequency Assignment in Cellular Networks. Proc. of the 4th International Colloquium on Structural Information and Communication Complexity, 1997. [PPS97] G. Pantziou, G. Pentaris, and P. Spirakis. Competitive Control in Mobile Networks. Proc. of the 8th International Symposium on Algorithms and Computation, pp. 404{413, 1997. [PY93] C.H. Papadimitriou and M. Yannakakis. The Traveling Salesman Problem with Distances One and Two. Mathematics of Operations Research 18(1), pp. 1{11, 1993. [Ray85] A. Raychaudhuri. Intersection assignments, T-colourings and powers of graphs. PhD Thesis, Rutgers University, 1985. [SH97] D.H. Smith and S. Hurley. Bounds for the frequency assignment problem. Discrete Mathematics, 168, pp. 571{582, 1997.

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