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Optim Lett (2010) 4:311320DOI 10.1007/s11590-009-0146-5

ORIGINAL PAPER

Approximation algorithms for finding and partitioningunit-disk graphs into co-k-plexes

Balabhaskar Balasundaram Shyam Sundar Chandramouli Svyatoslav Trukhanov

Received: 15 June 2009 / Accepted: 17 September 2009 / Published online: 30 September 2009 Springer-Verlag 2009

Abstract This article studies a degree-bounded generalization of independent setscalled co-k-plexes. Constant factor approximation algorithms are developed for themaximum co-k-plex problem on unit-disk graphs. The related problem of minimumco-k-plex coloring that generalizes classical vertex coloring is also studied in thecontext of unit-disk graphs. We extend several classical approximation results forindependent sets in UDGs to co-k-plexes, and settle a recent conjecture on theapproximability of co-k-plex coloring in UDGs.

Keywords Unit-disk graph Independent set Graph coloring Co-k-plex k-dependent set Defective coloring t-improper coloring

1 Introduction

A simple approach to modeling the connectivity and interference characteristics ofa wireless network is using Unit-Disk Graphs (UDGs). In such graphs, the vertexset V represents the set of wireless nodes with associated disk centers cv for each

B. Balasundaram (B)School of Industrial Engineering and Management, Oklahoma State University,Stillwater, OK 74078, USAe-mail: baski.balasundaram@okstate.edu

S. S. ChandramouliIndian Institute of Technology-Madras, Chennai 600036, Indiae-mail: shyamsundar1988@gmail.com

S. TrukhanovMicrosoft Corporation, One Microsoft Way, Redmond, WA 98052-6399, USAe-mail: svytru@microsoft.com

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v V located in the Euclidean plane. Under the proximity model for UDGs, anedge (u, v) E exists between vertices u and v if the Euclidean distance betweenthe centers of corresponding disks is within a specified proximity threshold , that is cu cv . Mathematically this is equivalent under scaling to intersection graphsof unit circles in the Euclidean plane. Unlike many other geometric intersection graphmodels, UDGs are not necessarily perfect or planar as a cycle on five vertices and acomplete graph on five vertices can both be represented as UDGs.

Given the geometric representation (centers and proximity threshold), the UDGG = (V, E) is determined. However given a graph G, deriving a geometric repre-sentation or concluding none exists is NP-hard [5]. For this reason, in this articlewe assume that the UDG is given with its geometric representation. Given a graph,a clique is a subset of pairwise adjacent vertices, an independent set is a subset ofpairwise nonadjacent vertices, and a proper coloring partitions the vertex set intoindependent sets. These three ideas are extensively used in wireless network applica-tions as an edge in the UDG model indicates the possibility of interference. Hence,an independent set corresponds to wireless nodes that can transmit simultaneouslyand is used to find broadcasting sets [26]. A proper coloring is used in frequencyassignment problems to identify the number of frequency bands required for inter-ference free communication as each partition can be assigned the same frequencyband [16]. A clique, on the other hand, corresponds to a set of wireless nodes wheretransmission from any one node could be received by all others. For this reason, thisapproach is typically used to cluster wireless nodes by partitioning the UDG intocliques [24].

Cliques and independent sets are equivalent under graph complementation, and theytypically share complexity and inapproximability results for arbitrary graphs [15,18].However for UDGs, the maximum clique problem is polynomial time solvable whilethe maximum independent set problem is NP-hard [7], which is in curious analogyto planar graphs. The related minimum graph coloring problem [7], and the mini-mum clique partitioning problem [6] on UDGs are also known to be NP-hard. Hence,the research in this area has emphasized approximation algorithms and distributedheuristics. Recent surveys on these issues are available in [3,13,14].

In this article, we are motivated by applications in communication networks thatpermit relaxations of these classical models to be used. For instance, if a receiver candistinguish interference from limited number of sources while extracting the desiredsignal, then it could be part of a set in which it has a limited number of neighbors insteadof none. One such application in satellite communications is discussed in [19,20,23],where it would be sufficient to find a set of nodes that induce a subgraph of boundedmaximum degree, where this bound is specified by the user or determined by theapplication.

The remainder of this article is organized as follows. In Sect. 2, we provide for-mal definitions of all the relevant concepts and introduce the notations we use in thisarticle. A review of relevant literature is provided and our key contributions are iden-tified in Sect. 3. Approximation algorithms for the generalizations of independent setsand coloring in UDGs are presented in Sects. 4 and 5. We close with a summary inSect. 6.

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On co-k-plexes in unit-disk graphs 313

2 Definitions and notations

We always consider simple, finite, undirected graphs denoted by G = (V, E). Recallthat we denote the corresponding disk center of a vertex v V by cv R2 and theproximity threshold for the UDG model by . For a vertex v V , N (v) is its neighbor-hood and N [v] = N (v){v} is its closed neighborhood. We denote the complementarygraph by G = (V, E) and the subgraph induced by S V by G[S]. We denote by(G) and (G) the minimum and maximum vertex degrees in G, respectively.

A clique C is a subset of vertices such that for all i, j C , we have (i, j) E .The maximum clique problem is to find a largest cardinality clique in G and the cliquenumber (G) is the size of a maximum clique. An independent set I is a subset ofvertices such that for all i, j I, (i, j) / E . As before, the maximum independentset problem is to find an independent set of maximum cardinality. The independencenumber of a graph G is denoted by (G) and it is the size of a maximum independentset. A maximal clique (independent set) is one that is not a proper subset of anotherclique (independent set). Note that C V is a clique in G if and only if C is anindependent set in G and hence, (G) = (G).

A proper coloring of a graph is one in which every vertex is colored such that notwo vertices of the same color are adjacent. A graph is said to be t -colorable if itadmits a proper coloring with t colors. Under proper coloring, the vertices of the samecolor referred to as a color class, induce an independent set. The chromatic numberof the graph (G), is the minimum number of colors required to properly color G.Note that for any graph G, (G) (G), as different colors are required to color thevertices of a clique. A related problem is the minimum clique partitioning problem,which is to partition the given graph G into a minimum number of cliques, (G).Note that this is exactly the graph coloring problem on G and (G) = (G).

In this article, for notational convenience, we denote all graph invariants of inducedsubgraphs in the following manner. The independence number of G[S] for instance isdenoted by (S) instead of (G[S]). Next we describe the generalizations of interest,provide a background and a summary of relevant results from literature.

Definition 1 ([27]) Given a graph G = (V, E), a subset S V is a k-plex if

|N (v) S| |S| k v S,

that is (S) |S| k.Definition 2 Given a graph G = (V, E), a subset J V is a co-k-plex if

|N (v) J | k 1 v J,

that is (S) k 1.Note that 1-plexes and co-1-plexes are simply cliques and independent sets, respec-

tively. For k > 1, the clique definition is relaxed by a k-plex as it allows for at mostk 1 non-neighbors in S, while the co-k-plex is a relaxation of independent set def-inition as it allows for at most k 1 neighbors in J . Clearly, S is a k-plex in G

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if and only if S is a co-k-plex in G. The maximum co-k-plex problem is to find alargest cardinality co-k-plex in G, the cardinality of which is the co-k-plex numberof G denoted by k(G). The maximum k-plex problem is similarly defined with thek-plex number denoted by k(G). Maximal k-plexes and co-k-plexes are also definedsimilar to cliques and independent sets by inclusion. A natural partitioning extensionof this relaxation is the following.

Definition 3 A proper co-k-plex coloring of a graph G = (V, E) is an assignment ofcolors to V such that each vertex has at most k 1 neighbors in the same color class.

Clearly, co-1-plex coloring is classical graph coloring and this model offers arelaxation for k > 1. Note that co-k-plex coloring aims to partition the given graphinto co-k-plexes and hence it is equivalent to k-plex partitioning on the complementgraph. Recall the motivating application introduced in Sect. 1 for which the maximumco-k-plex problem and co-k-plex coloring using minimum number of colors are rele-vant. However, originally the k-plex model was introduced in [27] to model cohesivesubgroups in social network analysis. The overly restrictive and impractical definitionof cliques motivated the development of this and other clique relaxations in this field.The co-k-plex was introduced as the complementary structure of k-plex, and relatedco-k-plex coloring and k-plex partitioning were defined and studied in [2]. For adetailed discussion on the advantages of k-plex as a cluster model see [2,4]. Interest-ingly, the concepts of co-k-plex and co-k-plex coloring were independently developedin literature prior to the work of [2], seemingly with no connection to k-plexes, andnaturally under different nomenclature. A brief summary of the relevant results fromliterature follows.

3 Previous work and our contributions

The maximum co-k-plex problem was introduced under the name of maximumk-dependent set problem in [12] and the decision version was shown to beNP-complete on arbitrary graphs. A co-k-plex is precisely a (k 1)-dependent set.The study was furthered in [11] where among other results, the problem was shownto be NP-complete even for planar and bipartite graphs for fixed k 2. Note that themaximum co-1-plex problem (independent set) problem is polynomial time solvableon bipartite graphs through graph matching techniques [8]. Recently in [20,23], theproblem is shown to be NP-complete for every fixed k on UDGs and is shown toadmit a polynomial time approximation scheme (PTAS) on UDGs. However, no sim-ple constant factor approximation algorithms are available for this problem on UDGs.In this article, we develop the first constant factor approximation algorithm for thisproblem that is much simpler, quicker and distributable, when compared to the PTASwhich offers any desired approximation guarantee. It should be noted that the PTAS,similar to the well-known approach for independent sets in UDGs [22], is more com-plicated requiring exact solution of the problem by complete enumeration on smallersquares into which the UDG is subdivided on the Euclidean plane. The pursuit of aconstant factor approximation algorithm also led us to extend the classical work of

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On co-k-plexes in unit-disk graphs 315

Marathe et al. [25] on approximating independent sets in UDGs and Hochbaum [21]on approximating independent sets in claw-free graphs.

The minimum co-k-plex coloring problem has been studied under the name defec-tive coloring [1,9,10,17] and improper coloring [19,20,23] in the literature. Thedecision version of minimum co-k-plex coloring problem is also NP-complete forevery fixed k on UDGs [20,23]. Furthermore, a polynomial time six-approximationalgorithm for the problem on UDGs is developed in [20,23]. Havet et al. [20] conjec-ture the existence of a polynomial time three-approximate algorithm for this problemon UDGs. This conjecture is settled in this article by extending a classical bound onthe chromatic number of a graph due to Szekeres and Wilf [28].

4 Co-k-plexes in unit-disk graphs

Proposition 1 For any simple graph G = (V, E) the co-k-plex number k(G) andindependence number (G) are related as

k(G) k(G).

Proof Suppose (G) = m and k(G) km + 1. We will find an independent set Iin G of size at least m + 1, thereby deriving a contradiction. Let G(0) be a co-k-plexinduced in G of size km + 1. Apply the following greedy algorithm for a maximalindependent set in the graph G(0), with set I initially empty.

Pick any vertex v from V (G(i)), let I I {v} and G(i+1) G(i) N [v]. Incre-ment i and repeat until G(i+1) is null. Since G(0) is a co-k-plex, so are all G(i), and thus(G(i)) k 1. So |V (G(i+1))| = |V (G(i))|1|N (v)V (G(i)| |V (G(i))|k.After m iterations |I | = m and |V (G(m))| km + 1 km = 1. So the algorithm canadd at least one more vertex to independent set I , contradicting the initial assumption.

unionsqCorollary 1 For any simple graph G = (V, E) the k-plex number k(G) and cliquenumber (G) are related as

k(G) k(G).

Remark 1 Note that the bound is tight as k, m > 0 there exists a graph G with(G) = m and k(G) = km. One of the simplest example of such graphs is thedisjoint union of m cliques of size k, which also has a UDG representation.Remark 2 Note that the three-approximate algorithm for independent sets in UDGs[25] together with Proposition 1 implies a 3k-approximate algorithm for co-k-plexes inUDGs. The following results provide a more direct proof of existence of a 3k-approx-imate algorithm for co-k-plexes on UDGs by extending many interesting intermediateresults from Marathe et al. [25] and Hochbaum [21].

Corollary 2 Let G = (V, E) be a UDG. For any v V , the co-k-plex numberk(N (v)) 5k.

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Fig. 1 Sharpness of Corollary 2

Proof By Proposition 1 we have, k(N (v)) k(N (v)) 5k where the last inequal-ity follows from the result of Marathe et al. [25] that (N (v)) 5. unionsqRemark 3 Consider the proximity model of the UDG in Fig. 1 with vertex set V ={v0, v1, . . . , v5k} and required proximity . Then |N (v0)| = |V \{v0}| = 5k with fivegroups of k vertices, with their centers inside five sectors of angle each and at dis-tance . Clearly, we can choose > 3 and > 0 satisfying 5( + ) = 2 , sothat the vertices in each -sector forms a co-k-plex and there is no edge between two-sectors. Thus we have k(N (v0)) = 5k indicating the sharpness of Corollary 2.

Corollary 3 Let G = (V, E) be a UDG. Let v V be a vertex corresponding tominimum X-coordinate, then the co-k-plex number k(N (v)) 3k.

Corollary 3 is also a sharp extension of the result in [25] and can be proved alongsimilar lines as Corollary 2.

Definition 4 A graph G = (V, E) is said to be (p+1; k)-claw free if for every v V ,

k(N (v)) p.

A polynomial time p-approximation algorithm for finding independent sets in(p + 1; 1)-claw free graphs has been developed by Hochbaum [21], which has beenapplied to UDGs by Marathe et al. [25]. Note that...