Approximating Maximum Diameter-Bounded Subgraph in Unit Disk Graphs

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1 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs A. Kaim Abu-Affash 1 Softwae Engineeing Depatment, Shamoon College of Engineeing Bee-Sheva 84100, Isael abuaa1@sce.ac.il Paz Cami Depatment of Compute Science, Ben-Guion Univesity Bee-Sheva 8410, Isael camip@cs.bgu.ac.il Anil Maheshwai School of Compute Science, Caleton Univesity Ottawa, Canada anil@scs.caleton.ca Pat Moin 4 School of Compute Science, Caleton Univesity Ottawa, Canada moin@scs.caleton.ca Michiel Smid School of Compute Science, Caleton Univesity Ottawa, Canada michiel@scs.caleton.ca Shakha Smoodinsky 6 Depatment of Mathematics, Ben-Guion Univesity Bee-Sheva 8410, Isael shakha@math.bgu.ac.il Abstact We conside a well studied genealization of the maximum clique poblem which is defined as follows. Given a gaph G on n vetices and an intege d 1, in the maximum diamete-bounded subgaph poblem (MaxDBS fo shot), the goal is to find a (vetex) maximum subgaph of G of diamete at most d. Fo d = 1, this poblem is equivalent to the maximum clique poblem and thus it is NP-had to appoximate it within a facto n 1 ɛ, fo any ɛ > 0. Moeove, it is known that, fo any d, it is NP-had to appoximate MaxDBS within a facto n 1/ ɛ, fo any ɛ > 0. In this pape we focus on MaxDBS fo the class of unit disk gaphs. We povide a polynomialtime constant-facto appoximation algoithm fo the poblem. The appoximation atio of ou algoithm does not depend on the diamete d. Even though the algoithm itself is simple, its analysis is athe involved. We combine tools fom the theoy of hypegaphs with bounded VC-dimension, k-quasi plana gaphs, factional Helly theoems and seveal geometic popeties of unit disk gaphs. 1 Wok was suppoted by Gant fom the United States Isael Binational Science Foundation. Wok was suppoted by Gant fom the United States Isael Binational Science Foundation. Wok was suppoted by NSERC 4 Wok was suppoted by NSERC Wok was suppoted by NSERC 6 Wok was patially suppoted by Gant 6/16 fom the Isael Science Foundation A. Kaim Abu-Affash, Paz Cami, Anil Maheshwai, Pat Moin, Michiel Smid, and Shakha Smoodinsky; licensed unde Ceative Commons License CC-BY 4th Intenational Symposium on Computational Geomety (SoCG 018). Editos: Bettina Speckmann and Csaba D. Tóth; Aticle No. ; pp. :1 :1 Leibniz Intenational Poceedings in Infomatics Schloss Dagstuhl Leibniz-Zentum fü Infomatik, Dagstuhl Publishing, Gemany

2 : Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs 01 ACM Subject Classification Theoy of computation Computational geomety, Theoy of computation Numeic appoximation algoithms Keywods and phases Appoximation algoithms, maximum diamete-bounded subgaph, unit disk gaphs, factional Helly theoem, VC-dimension Digital Object Identifie 10.40/LIPIcs.SoCG.018. Acknowledgements The authos would like to thank the Fields Institute fo hosting the wokshop in Ottawa and fo thei financial suppot. 1 Intoduction Computing a maximum size clique in a gaph is one of the fundamental poblems in theoetical compute science [7]. It is not only NP-had but even had to appoximate within a facto of n 1 ɛ, fo any ɛ > 0, unless P = NP [16]. A clique, equivalently, is a subgaph of diamete 1. A natual, well studied, genealization of the maximum clique poblem is the maximum diamete-bounded subgaph poblem (MaxDBS fo shot), in which the goal is to compute a maximum (vetex) subgaph of diamete d, fo a given d. A subgaph with diamete d is sometimes efeed to in the liteatue as a d-club. MaxDBS is also known to have hadness of appoximation of n 1/ ɛ, unless P = NP []. In this pape, we study MaxDBS in the class of unit disk gaphs. A unit disk gaph is defined as the intesection gaph of disks of equal (e.g., unit) diamete in the plane. Unit disk gaphs povide a gaph-theoetic model fo ad hoc wieless netwoks, whee two wieless nodes can communicate if they ae within the unit Euclidean distance away fom each othe. Many classical NP-Complete poblems including chomatic numbe, independent set and dominating set ae still NP-complete even fo unit disk gaphs [1, 14]. Howeve, the class of unit disk gaphs is one of the non-tivial classes of gaphs fo which the maximum clique poblem is in P. Indeed, in a celebated esult, Clak, Colboun and Johnson [1] povide a beautiful polynomial time algoithm to compute the maximum clique in unit disk gaphs. Unfotunately, we do not know how to extend this algoithm to the MaxDBS poblem. Ou main contibution in this pape is a polynomial-time algoithm that appoximates MaxDBS within a constant facto in unit disk gaphs. Moeove, ou constant facto appoximation atio does not depend on the diamete d. To the best of ou knowledge, it is not known whethe this poblem is NP-had. Ou algoithm woks as follows: Given a unit disk gaph G = (V, E) and an intege d, compute fo each vetex v V the BFS-tee of adius d d+1 (o if d is odd) centeed at v, and etun the tee of maximum size. Although, the algoithm is vey natual and simple, its analysis fo unit disk gaphs is athe involved. We note that this algoithm might pefom vey bad fo abitay gaphs. Fo example, a -subdivision of a clique with n vetices is the gaph obtained by subdividing each edge of the clique K n into a path of length. It is easily seen that such a gaph with n + ( n ) vetices has diamete 4 but evey BFS tee of adius contains at most n vetices. 1.1 Related wok MaxDBS has been studied extensively in geneal gaphs in the last two decades. Boujolly et al. [8] showed that MaxDBS is NP-had. Balasundaam et al. [6] poved that fo any d, MaxDBS is NP-had in gaphs of diamete d + 1. Asahio et al. [] showed that, fo any ɛ > 0 and d, it is NP-had to appoximate MaxDBS within a facto of n 1/ ɛ,

3 A. K. Abu-Affash, P. Cami, A. Maheshwai, P. Moin, M. Smid, and S. Smoodinsky : and they gave an n 1/ -appoximation algoithm fo the poblem. Chang et al. [11] povide an algoithm that finds a maximum subgaph of diamete d in O(1.6 n poly(n)) time. Thee ae moe esults on solving MaxDBS by using vaious intege and linea pogamming fomulations [4, 6, 9, 8, 10, 1]. Asahio et al. [] studied the MaxDBS in othe subclasses of gaphs, including chodal gaphs, inteval gaphs, and s-patite gaphs. Fo chodal gaphs, they showed that the poblem can be solved in polynomial-time fo odd d s, and cannot be appoximated within facto n 1/ ɛ, fo any ɛ > 0 fo even d s. Fo inteval gaphs, they showed that the poblem can be solved in polynomial-time. Fo s-patite gaphs, they showed that the poblem cannot be appoximated (unless P = NP ) within a facto of n 1/ ɛ, fo any ɛ > 0, when s = and d, and when s and d. Fo unit disk gaphs, the hadness of MaxDBS is still open, fo d we ae not awae of any pevious wok. As mentioned aleady, fo d = 1, the poblem is equivalent to the maximum clique poblem and it can be solved in polynomial-time [1]. 1. Motivation MaxDBS is a elaxation of the maximum clique poblem and is motivated by cluste-detection that aise in wide vaiety applications. Fo instance, finding clustes in netwoks helps in undestanding and analyzing the stuctue of the netwok. Anothe well studied notion is that of a d-clique [6, 0, 1]. A d-clique of a gaph G is a subset S of vetices of G, such that, fo evey two vetices in S, the shotest distance between them in G is at most d. Clealy, evey d-club is a d-clique, but not vice vesa, as shown in the example given by Alba [] in Figue 1. v v 4 v 1 v 6 v v Figue 1 S = {v 1, v, v, v 4, v } is a -clique but not a -club since the gaph induced by S has a diamete. Peliminaies Let V be a finite set of points in the plane. Fo two points u, v V, let uv denote the Euclidean distance between u and v. The unit disk gaph on V is the undiected gaph G = (V, E), such that (u, v) E if and only if uv 1. The following lemma (though vey simple) tuns out to be cucial to pove ou main esult. Lemma 1. Fo evey two cossing edges (a, b) and (c, d) in G, at least one of the edges (a, c) and (b, d) is in G, and at least one of the edges (a, d) and (b, c) is in G; see Figue fo an illustation. Poof. To pove the lemma, it suffices to show that min{ ac, bd } 1 and min{ ad, bc } 1; see Figue. Let x be the intesection point of (a, b) and (c, d). By the tiangle inequality, ac ax + xc and bd bx + xd. Thus, ac + bd ab + cd. Theefoe, min{ ac, bd } 1. By a simila agument, we pove that min{ ad, bc } 1. S o C G 0 1 8

4 :4 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs a d x c b Figue An illustation fo the poof of Lemma 1..1 Tool box A ange space (o a set system) (X, R) is a pai consisting of a set X of objects (called the space) and a family R of subsets of X (called anges). We say that a subset A of X is shatteed by R, if fo each subset A of A, thee exsist a ange R R, such that A R = A. The Vapnik-Chevonenkis dimension (o VC-dimension fo shot) of a ange space (X, R) is the size of the lagest (finite) shatteed subset of X; see [1] fo examples of ange spaces of bounded VC-dimension. The dual ange space of (X, R) is a ange space (Y, R ), whee Y = {y R : R R} and, fo each x X, the set {y R : x R} is a ange in R. It is well known [17] that, if the VC-dimension of (X, R) is k, then the VC-dimension of the dual ange space (Y, R ) is at most k. A ange space (X, R) has factional Helly numbe k, if fo evey α > 0, thee exists β > 0, such that if at least α ( ) R k subsets of size k of R have a non-empty intesection, then thee exists an element of X that is contained in at least β R sets of R. In [18], Matoušek poved the following theoem showing that evey ange space of bounded VC-dimension has a factional Helly popety. Theoem ([18]). Let (X, R) be a ange space such that the VC-dimension of the dual ange space of (X, R) is at most k 1. Then, (X, R) has a factional Helly numbe k. A ange space (X, R) satisfies the (p, q)-popety if among evey p anges of R some q have a non-empty intesection. Matoušek [18] established the following (p, q)-theoem fo ange spaces of bounded VC-dimension. Theoem ([18]). Let (X, R) be a ange space such that the VC-dimension of the dual ange space of (X, R) is at most k 1 and let p k. Then, thee exists a constant t (depending only on p and k), such that if (X, R) satisfies the (p, k)-popety, then thee exists a subset X of X of size at most t intesecting all the anges of R, i.e., X R, fo evey R R. A (simple) topological gaph is a gaph dawn in the plane, such that its vetices ae epesented by a set of distinct points and its edges ae Jodan acs connecting the coesponding points, so that (i) each edge does not contain any othe vetex as an inteio point, (ii) evey pai of edges intesect at most once, and (iii) no thee edges have a common intesection point. Agawal et al. [] showed that any topological gaph with n vetices and without k paiwise cossing edges has O(n log k 6 n) edges. This bound was futhe impoved to O(n log k 8 n) by Ackeman [1]. Hence, if G is a complete topological gaph on n vetices and without k paiwise cossing edges, then ( n ) = n(n 1)/ c n log k 8 n, whee c is the constant in the big O, depending only on k. This implies that k log(n 1) c log log n + 4, whee c is a constant depending on c. Theefoe, we have the following coollay.

5 A. K. Abu-Affash, P. Cami, A. Maheshwai, P. Moin, M. Smid, and S. Smoodinsky : Coollay 4. Any complete topological gaph on n vetices contains at least log(n 1) c log log n + 4 paiwise cossing edges, whee c is a constant. Appoximation algoithm Let G = (V, E) be the unit disk gaph of a set of points V in the plane. Fo two vetices u and v in V, let d(u, v) denote the shotest (hop) distance between u and v in G. 7 Assuming that G is connected, the diamete of G is defined as the maximum (hop) distance between any two vetices in G, i.e., max u,v V d(u, v). A subgaph of G is called d-club if its diamete is equal to d. Let G opt denote a maximum d-club of G. In this section, we fist pesent a polynomial-time appoximation algoithm that computes a d-club of size at least c times the size of G opt, whee c is a constant and d is even. Late, we show how to genealize this algoithm fo odd d s. Set = d. Fo a vetex u V, let T (u) denote the tee of cente u and adius that contains all vetices of distance at most fom u in G. Namely, a vetex v is in T (u) if and only if d(u, v). Given a vetex u V, T (u) can be computed using the beadth fist seach (BFS) algoithm in O( V + E ) time. Ou algoithm computes all the tees of adius centeed at vetices of G and etuns a tee T of the maximum size, i.e., the tee that contains the maximum numbe of vetices. It is clea that T is a d-club of G and can be computed in polynomial time. Let V opt denote the set of vetices of G opt and n = V opt denote the size of G opt. In the following, we pove that T contains at least cn vetices, whee c is a constant. Let T (G opt ) denote the set of all tees of adius centeed at vetices of G opt, and let T be a tee in T (G opt ) that contains the maximum numbe of vetices of G opt among tees of T (G opt ). Fist, obseve that the size of T is at least as the size of T, since G opt is a subgaph of G. Theefoe, it is sufficient to pove that T contains at least cn vetices. Let (X, R) be the ange space whee X = V opt and fo each tee T (u) in T (G opt ), the set {v V opt : v T (u)} is a ange in R. Thus, in the dual ange space (Y, R ) of (X, R), we have Y = {y T(u) : u V opt }, and, fo each point v V opt, the set {y T(u) : v T (u)} is a ange in R. Obsevation. (X, R) and (Y, R ) ae isomophic. Poof. Since T (u) contains v if and only if T (v) contains u, we have {y T(u) : v T (u)} = {y T(u) : u T (v)}. Now, if we set y T(u) = u, then we obtain that X = Y and {y T(u) : v T (u)} = {u V opt : v T (u)} = {u V opt : u T (v)}, fo each v V opt. Theefoe, fo each v V opt, the set {u V opt : u T (v)} is a ange in R, which implies that R = R. Theoem 6. T contains at least cn vetices, whee c is a constant. Poof. The poof plan is as follows. We show (late in Section 4) that the VC-dimension of the ange space (V opt, T (G opt )) is 4. Thus, by Obsevation, the VC-dimension of the dual 7 Note that fo any ɛ > 0, it could hold that the Euclidean distance between u and v is 1 + ɛ but they ae in diffeent connected components of G and hence d(u, v) is not necessaily bounded. S o C G 0 1 8

6 :6 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs u j u j u i u j u i u j u i u i (a) (b) Figue (a) δ(u i, u j) and δ(u i, u j ) intesect in moe than one point, and (b) eplacing subpaths of δ(u i, u j ) by subpaths of δ(u i, u j) between the intesection points. ange space of (V opt, T (G opt )) is also 4. Then, we use Coollay 4 to show that thee exists a constant m, such that (V opt, T (G opt )) satisfies the (m, )-popety, which means that at least ( ) ( n m / n ( m ) = n ( ) / m ) subsets of tees of T (G opt ) shae a common point. Thus, by Theoem, thee exists a point that is contained in at least βn tees of T (G opt ), and theefoe, thee is a tee in T (G opt ) that contains at least βn points of V opt, which poves that c β > 0. Let m be an intege such that log(m 1) c log log m + 4 6, whee c is a constant. Let A be a set of m tees of T (G opt ) and let C = {u 1, u,..., u m } be the centes of the tees of A. Fo two points u i, u j C, let δ(u i, u j ) be a shotest path between u i and u j in G opt, and let d(u i, u j ) be the length of δ(u i, u j ). Fo evey fou distinct points u i, u j, u i, u j C, we assume that the intesection of the paths δ(u i, u j ) and δ(u i, u j ) is eithe empty o a path (othewise, we eplace evey subpath of δ(u i, u j ) between evey two consecutive intesection points of the paths by the subpath of δ(u i, u j ) between the same points; see Figue ). Since d(u i, u j ), thee is at least one point u i,j on δ(u i, u j ) that is contained in T (u i ) and in T (u j ). We now constuct a dawing of a complete gaph H on the points of C in which the edges ae dawn as the Jodan acs δ(u i, u j ). Notice that, H is not necessaily a topological gaph. Howeve, we can tansfom it into a topological gaph H, such that, fo evey fou distinct points u i, u j, u i, u j C, δ(u i, u j ) and δ(u i, u j ) ae cossing in H if and only if they ae cossing in H. This tansfomation is obtained using standad opeations as in [1] and we omit the technical details hee. Since H is a complete topological gaph on m vetices, by Coollay 4, H has at least 6 paiwise cossing edges. Let P = {δ(u 1, u 1 ), δ(u, u ),..., δ(u 6, u 6 )} be the set of the coesponding 6 paiwise cossing paths in H. Lemma 7. Let δ(u i, u i ) and δ(u j, u j ) be two cossing paths of P and let x be thei intesection point. Assume, w.l.o.g., that x is between u i,i and u i, and between u j,j and u j ; see Figue 4. Then, eithe T (u i ) contains u j,j o T (u j ) contains u i,i. Poof. We distinguish between two cases. Case 1: x is a point of V opt ; see Figue 4(a). Assume, w.l.o.g., that d(u i, x) d(u j, x). Thus, d(u i, u j,j ) d(u i, x) + d(x, u j,j ) d(u j, x) + d(x, u j,j ) = d(u j, u j,j ). Theefoe, u j,j is of distance at most fom u i and, hence, is contained in T (u i ).

7 A. K. Abu-Affash, P. Cami, A. Maheshwai, P. Moin, M. Smid, and S. Smoodinsky :7 u j u j,j u i u j u j,j d a u i x b x c u i,i u j u i,i u j u i u i (a) (b) Figue 4 δ(u i, u i ) and δ(u j, u j ) intesect at x. (a) x is a point of V opt, and (b) x is an intesection point of the edges (a, b) and (c, d). Case : x is not a point of V opt. Thus, x is an intesection point of two edges (a, b) and (c, d) of G. Assume, w.l.o.g., that a is between x and u 1 and c is between x and u ; see Figue 4(b). If d(u i, a) = d(u j, c), then, by Lemma 1, at least one of the edges (a, d) and (b, c) is in G opt. If (a, d) is in G opt, then d(a, u j,j ) = d(c, u j,j ), and hence, d(u i, u j,j ) d(u i, a) + d(a, u j,j ) = d(u j, c) + d(c, u j,j ) = d(u j, u j,j ). Theefoe, u j,j is of distance at most fom u i and, hence, is contained in T (u i ). If (b, c) is in G opt, then d(a, u i,i ) = d(c, u i,i ), and hence, d(u j, u i,i ) d(u j, c) + d(c, u j,j ) = d(u i, a) + d(a, u i,i ) = d(u i, u i,i ). Theefoe, u i,i is of distance at most fom u j and, hence, is contained in T (u j ). Othewise, assume, w.l.o.g., that d(u i, a) < d(u j, c). By Lemma 1, at least one of the edges (a, c) and (b, d) is in G opt. If (a, c) is in G opt, then d(u i, c) d(u j, c). Hence, d(u i, u j,j ) d(u i, c) + d(c, u j,j ) d(u j, c) + d(c, u j,j ) = d(u j, u j,j ). If (b, d) is in G opt, then d(u i, d) d(u j, d). Hence, d(u i, u j,j ) d(u i, d) + d(d, u j,j ) d(u j, d) + d(d, u j,j ) = d(u j, u j,j ). In both cases, u j,j is of distance at most fom u i and, hence, is contained in T (u i ). Lemma 8. (V opt, T (G opt )) satisfies the (m, )-popety. Poof. By Lemma 7, fo evey two paths δ(u i, u i ) and δ(u j, u j ) in P, eithe at least one of the tees T (u i ) and T (u i ) contains u j,j o at least one of the tees T (u j ) and T (u j ) contains u i,i. We constuct a diected gaph on the vetices {u 1, u,..., u 6 }, such that thee is a diected edge fom u i to u j if and only if at least one of the tees T (u i ) and T (u i ) contains u j,j. Since we have 6 paiwise cossing paths, thee ae at least 1 edges in this gaph, which means that thee is a vetex u l in this gaph, 1 l 6, of in-degee at least. Hence, thee is a point u l,l that is coveed by at least othe tees, in addition to the tees T (u l ) and T (u l ). Thus, u l,l is contained in at least tees of A. Theefoe, (V opt, T (G opt )) satisfies the (m, )-popety. S o C G 0 1 8

8 :8 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs Since (V opt, T (G opt )) satisfies the (m, )-popety, out of evey m tees of T (G opt ) thee ae tees that shae a common point. Thus, thee ae at least ( ) ( n m / n ( m ) = n ( ) / m ) sets containing tees that shae a common point. Moeove, in Theoem 1 (Section 4), we show that the VC-diminsion of the ange space (V opt, T (G opt )) is 4. Thus, by Obsevation, the VC-dimension of the dual ange space of (V, T (G)) is also 4 and theefoe, by Theoem, the factional Helly numbe of (V opt, T (G opt )) is. Now, by setting α = 1/ ( ) m (in the factional Helly theoem), we have a point u V opt that is contained in at least βn tees of T (G opt ), which means that T (u) contains at least βn points of V opt, whee β > 0. Theefoe, c β > 0, which completes the poof of the theoem. Coollay 9. Fo any even d, evey d-club in any unit disk gaph can be coveed by a constant-numbe of tees of adius d. Poof. To pove the coollay, we show that thee exists a constant ρ, such that G opt can be coveed by at most ρ tees of T (G opt ). By Theoem 1, the VC-dimension of the dual ange space of (V opt, T (G opt )) is 4, and, by Lemma 8, (V opt, T (G opt )) satisfies the (m, )-popety. Theefoe, by Theoem, thee exists a set of at most t tees of T (G opt ) that cove all vetices of V opt, which poves that ρ t. Uppe bound on c We show, in Figue, a unit disk gaph G on n vetices fo which the tee computed by ou algoithm does not contain moe than n. G contains n = 16 points and its diamete is d =. Each tee of adius in G coves at most 6 points. This poves that c 8. To show that c 1 n 16, we locate six cliques of size 6 on the points a, b, c, a, b, and c. Now, each tee of adiuns can cove at most cliques. Theefoe, fo sufficiently lage n, we have c 1. c a b a c Figue G contains 16 points and its diamete is d = Each tee of adius coves at most 6 points. b

9 A. K. Abu-Affash, P. Cami, A. Maheshwai, P. Moin, M. Smid, and S. Smoodinsky :9 Genealization fo odd d In this section, we extend ou algoithm to appoximate MaxDBS fo odd d s. Given a unit disk gaph G of a set V of points in the plane and an odd intege d, let G d be a maximum d-club and let G d+1 be a maximum (d + 1)-club of G. Let n d and n d+1 be the sizes of G d and G d+1, espectively, and obseve that n d+1 n d. We set = d+1 and we use ou algoithm to compute a tee T (u) of size at least cn d+1 cn d. Notice that T (u) is a (d + 1)-club but may not be a d-club. In the following lemma, we show that thee is a subtee of T (u) of diamete d 1 that contains at least 1/1 of the vetices of T (u). Lemma 10. The vetices of tee T (u) can be coveed by at most 1 tees of adius 1. Poof. Let V (u) be the set of vetices of T (u) and let D (u) = {v V (u) : d(u, v) = }, i.e., the set of all vetices of V (u) of distance two fom u. Let I be a maximal independent set of D (u). By the packing agument in unit disk gaphs, we have I 1. Let v be a vetex in V (u), and let δ(u, v) be a shotest path between u and v in T (u). Since d(u, v), thee is at least one vetex u D (u), such that evey vetex in δ(u, v) is of distance at most fom u. Hence, thee is at least one vetex x I, such that evey vetex in δ(u, v) is of distance at most 1 fom x. Thus, evey vetex in V (u) is contained in T 1 (x), fo some x I, and theefoe, V (u) is coveed by x I T 1(x). By Lemma 10, we can find a tee T 1(x) that contains at least 1/1 of the vetices of T (u). Since = d+1, the diamete of T 1(x) is at most ( 1) = d 1. Theefoe, T 1(x) is a d-club of G and its size is at least c 1 n d. The following theoem summaizes the esult of this section. Theoem 11. Given a unit disk gaph G in the plane and an intege d, one can find c in polynomial time a d-club of G of size at least 1 the size of a maximum d-club of G, whee 0 < c 1. 4 The VC-Dimension of (V opt, T (G opt )) In this section, we pove the following theoem. Theoem 1. The ange space (V opt, T (G opt )) has VC-dimension 4. Poof. We fist pove that the VC-dimension of T (G opt ) is at most 4. Fo the sake of contadiction, suppose that thee exist a set of points P and a subset S = {u 1, u, u, u 4, u } of V opt, such that S is shatteed by T (G opt ). Thus, fo each 1 i < j, thee is a tee T (c i,j ) in T (G opt ), such that T (c i,j ) S = {u i, u j }. Let P i,j be the path between u i and u j in T (c i,j ). Note that P i,j S = {u i, u j }. Moeove, since S contains five points, by planaity constaints, at least two paths P i,j and P k,l, fo distinct two pais (i, j) and (k, l), intesect. Assume, w.l.o.g., that P 1, and P,4 intesect and let x be thei intesection point. Assume also that x is between u 1 and c 1,, and between u and c,4 ; see Figue 6. Lemma 1. Eithe u 1 is contained in T (c,4 ) o u is contained in T (c 1, ). Poof. The poof is simila to the poof of Lemma 7. We distinguish between two cases. Case 1: x is a point of V opt ; see Figue 6(a). Assume, w.l.o.g., that d(x, u 1 ) d(x, u ). Thus, d(c,4, u 1 ) d(c,4, x) + d(x, u 1 ) d(c,4, x) + d(x, u ) = d(c,4, u ). S o C G 0 1 8

10 :10 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs c,4 c 1, c,4 c 1, x d x a b c u 4 u 1 u u u 4 u 1 u u (a) (b) Figue 6 P 1, and P,4 intesect at x. (a) x is a point of V opt, and (b) x is an intesection point of the edges (a, b) and (c, d). Theefoe, u 1 is of distance at most fom c,4 and, hence, is contained in T (c,4 ). Case : x is not a point of V opt. Thus, x is an intesection point of two edges (a, b) and (c, d) of G. Assume, w.l.o.g., that a is between x and u 1 and c is between x and u ; see Figue 6(b). If d(a, u 1 ) = d(c, u ), then, by Lemma 1, at least one of the edges (a, d) and (b, c) is in G opt. If (a, d) is in G opt, then d(c,4, a) = d(c,4, c), and hence, d(c,4, u 1 ) d(c,4, a) + d(a, u 1 ) = d(c,4, c) + d(c, u ) = d(c,4, u ). Theefoe, u 1 is of distance at most fom c,4 and, hence, is contained in T (c,4 ). If (b, c) is in G opt, then d(c 1,, a) = d(c 1,, c), and hence, d(c 1,, u ) d(c 1,, c) + d(c, u ) = d(c 1,, a) + d(a, u 1 ) = d(c 1,, u 1 ). Theefoe, u is of distance at most fom c 1, and, hence, is contained in T (c 1, ). Othewise, assume, w.l.o.g., that d(a, u 1 ) < d(c, u ). By Lemma 1, at least one of the edges (a, c) and (b, d) is in G opt. If (b, d) is in G opt, then d(c,4, b) d(c,4, c) and d(b, u 1 ) d(c, u ). Hence, d(c,4, u 1 ) d(c,4, b) + d(b, u 1 ) d(c,4, c) + d(c, u ) = d(c,4, u ). If (a, c) is in G opt, then d(c, u 1 ) d(c, u ). Thus, d(c,4, u 1 ) d(c,4, c) + d(c, u 1 ) d(c,4, c) + d(c, u ) = d(c,4, u ). In both cases, u 1 is of distance at most fom c,4 and, hence, is contained in T (c,4 ). Since T (c,4 ) S = {u, u 4 } and T (c 1, ) S = {u 1, u }, we have a contadiction. Theefoe, the VC-dimension of (V opt, T (G opt )) is at most 4. To pove that the VC-dimension of (V opt, T (G opt )) is at least 4, we show in Figue 7 a unit disk gaph on a set of points V of diamete and a subset S = {a, b, c, d} of V, such that S can be shatteed by T (G opt ). The distance between evey two points of S is. Fo each subset S S, S T (v S ) = S, and S T (a) = S. Concluding emaks In this pape, we conside the poblem of computing a maximum subgaph of diamete d. We pesent the fist constant-facto appoximation algoithm fo the poblem in unit disk gaphs, fo any d.

11 A. K. Abu-Affash, P. Cami, A. Maheshwai, P. Moin, M. Smid, and S. Smoodinsky :11 v φ 1 v {b,c,d} v {d} 1 d 1 v {a,d} v {b,d} 6 v {a,b,d} v {a,c,d} v {c,d} a 1 v {a} v {a,b} b v {a,c} 6 1 c 1 v {b} v {a,b,c} v {c} v {b,c} Figue 7 Shatteing the points a, b, c, and d. S T (v S ) = S, fo each S S, and S T (a) = S. Ou algoithm is simple and efficient, howeve, its analysis is not tivial and based on tools fom the theoy of hypegaphs with bounded VC-dimension, k-quasi plana gaphs, factional Helly theoems and seveal geometic popeties of unit disk gaphs. Unfotunately, the constant obtained is athe lage. On the othe hand, the most impotant featue of ou algoithm is that its appoximation facto is a constant independent of the diamete d. It is vey easy to obtain an O(d ) appoximation facto. Indeed, by a packing agument, any gaph of diamete d can be coveed by O(d ) cliques and as mentioned aleady the max-clique poblem is in P. Moeove, ou algoithm woks also fo an abstact input of the unit disk gaph without the geometic epesentation. It emains an open poblem to detemine whethe MaxDBS fo unit disk gaphs is in P fo d. Anothe inteesting fact to note is that the shotest path metic on unit disk gaphs does not have the so-called constant doubling dimension. It is easily seen that ou algoithm has a constant facto appoximation fo gaph families with constant doubling dimension. Recall that a d-clique of a gaph G is a set S of vetices of G, such that, fo evey two vetices in S, the shotest distance between them in G is at most d. Finding the maximum d-clique poblem is closely elated to MaxDBS. Unfotunately, ou algoithm can not be diectly extended to the maximum d-clique poblem. Except fo the 1 -appoximation algoithm of Pattillo et al. [19], fo d =, thee is no elated wok discussing the maximum d-clique poblem in unit disk gaphs. Hence, appoximating the maximum d-clique poblem in unit disk gaphs is also an inteesting open poblem. Refeences 1 E. Ackeman. On the maximum numbe of edges in topological gaphs with no fou paiwise cossing edges. Discete Comput. Geom., 41:6 7, 009. P. K. Agawal, B. Aonov, J. Pach, and M. Shai. Quasi-plana gaphs have linea numbe of edges. Combinatoica, 17:1 9, R. D. Alba. A gaph-theoetic definition of a sociometic clique. J. Math. Sociol., :11 16, 197. S o C G 0 1 8

12 :1 Appoximating Maximum Diamete-Bounded Subgaph in Unit Disk Gaphs 4 M. T. Almeida and F. D. Cavalho. Intege models and uppe bounds fo the -club poblem. Netwoks, 60:1 166, 01. Y. Asahio, E. Miyano, and K. Samizo. Appoximating maximum diamete-bounded subgaphs. In LATIN, LNCS 604, pages 61 66, B. Balasundaam, S. Butenko, and Tukhanov S. Novel appoaches fo analyzing biological netwoks. J. Combin. Optim., 10: 9, I. M. Bomze, M. Budinich, P. M. Padalos, and M. Pelillo. The maximum clique poblem. Handbook of Combinatoial Optimization, pages 1 74, Kluwe Academic Publishes, J.-M. Boujolly, G. Lapote, and G. Pesant. An exact algoithm fo the maximum k-club poblem in an undiected gaph. Euopean J. Ope. Res., 18:1 8, A. Buchanan and H. Salemi. Pasimonious fomulations fo low-diamete clustes F. D. Cavalho and M. T. Almeida. Uppe bounds and heuistics fo the -club poblem. Euopean J. Ope. Res., 10: , M.-S. Chang, L.-J. Hung, C.-R. Lin, and P.-C. Su. Finding lage k-clubs in undiected gaphs. Computing, 9:79 78, V. Chepoi, B. Estellon, and Y. Vaxès. Coveing plana gaphs with a fixed numbe of balls. Discete Comput. Geom., 7:7 44, B. N. Clak, C. J. Colboun, and D. S. Johnson. Unit disk gaphs. Discete Math., 86:16 177, A. Gäf, M. Stumpf, and G. Weißenfels. On coloing unit disk gaphs. Algoithmica, 0():77 9, S. Ha-Peled. Geometic Appoximation Algoithms. Ameican Mathematical Society, Boston, USA, J. Håstad. Clique is had to appoximate within n 1 ɛ. In FOCS, pages 67 66, J. Matoušek. Lectues on Discete Geomety. Spinge, New Yok, USA, J. Matoušek. Bounded VC-dimension implies a factional Helly theoem. Discete Comput. Geom., 1:1, J. Pattillo, Y. Wang, and S. Butenko. Appoximating -cliques in unit disk gaphs. Discete Appl. Math., 166: , J. Pattillo, N. Youssef, and S. Butenko. On clique elaxation models in netwok analysis. Euopean J. Ope. Res., 6:9 18, A. Veemyev and V. Boginski. Identifying lage obust netwok clustes via new compact fomulations of maximum k-club poblems. Euopean J. Ope. Res., 18:16 6, 01.