Finding Large Clique Minors is Hard

Finding Large Clique Minors is Hard David Eppstein Computer Science Department Univ ersity of California, Irvine eppstein@uci.edu Abstract. W e prov e that it is NP-complete, giv en a graph G and a parameter h , to determine whether G contains a complete graph K h as a minor . 1 Introduction The Hadwiger number of a graph G is the number of vertices in the largest clique that is a minor of G ; that is, that can be formed by contracting some edges and deleting others. Equi valently , it is the largest number of verte x-disjoint connected subgraphs that one can find in G such that for each two subgraphs S i and S j there is an edge v i v j in G with v i ∈ S i and v j ∈ S j . In 1943, Hugo Hadwiger conjectured that in any graph the Hadwiger number is greater than or equal to the chromatic number [6], and this important conjecture remains open in general, although it is known to be true when the chromatic number is at most six [10]. The Hadwiger number is also closely associated with the sparseness of the giv en graph: if G has Hadwiger number h , ev ery subgraph of G has a vertex with degree O ( h √ log h ) . It follows from this fact that, if G has n vertices, it has O ( nh √ log h ) edges [8]. Gi ven its graph-theoretic importance, it is natural to ask for the computational complexity of the Had- wiger number . In this light, Alon et al. [1] observe that the Hadwiger number is fixed parameter tractable: for any constant h , there is a polynomial-time algorithm that either computes the Hadwiger number or deter- mines that it is greater than h , and the exponent in the polynomial time bound of this algorithm is independent of h , due to standard results in graph minor theory . Ho wev er , this is not a polynomial time algorithm for the Hadwiger number problem because its running time includes a f actor e xponential or worse in h . In addition, as Alon et al. show , the Hadwiger number may be approximated in polynomial time more accurately than the problem of finding the lar gest clique subgraph of a gi ven graph: they provide a polynomial time approx- imation algorithm for the Hadwiger number of an n -verte x graph with approximation ratio O ( √ n ) , whereas it is NP-hard to approximate the clique number to within a factor better than n 1 − ε for any ε > 0 [14]. T o classify the problem of computing the Hadwiger number in complexity theoretic terms, we need to consider a decision version of the problem: given a graph G , and a number h , is the Hadwiger number of G greater than or equal to h ? 1 W e call this decision problem the Hadwiger number pr oblem . Unsurprisingly , it turns out to be NP-complete. A statement of its NP-hardness was made without proof by Chandran and Si vadasan [2]. Ho wev er , we hav e been unable to find a clear proof of the NP-completeness of the Had- wiger number problem in the literature. Our goal in this short paper is to fill this gap by providing an NP- completeness proof of the standard type (a polynomial-time many-one reduction from a known NP-complete problem) for the Hadwiger number problem. 2 Reduction from domatic number Recall that a verte x v dominates a vertex w if v = w or v and w are adjacent; a dominating set of a graph G is a set of vertices such that, for ev ery verte x w in G , some member of the set dominates w . The domatic 1 Chandran and Sivadasan [2] formulate a different decision problem, in which the positiv e instances are those with small Had- wiger number , but this is in CoNP rather than being in NP . , 2 , 32 Fig. 1. A confluent drawing [4] of our NP-completeness reduction. T wo vertices are connected by an edge in G 0 if and only if there is a smooth (possibly self-intersecting) path between the circles representing them in the drawing. In this example, a 5-c ycle with domatic number 2 is transformed into a 37-verte x graph with Hadwiger number 32. One possible 32-vertex clique minor is formed by contracting the two shaded sets of vertices into single supervertices, removing the remaining middle-layer vertex, and combining the two supervertices with the 30 bottom-layer v ertices. number of a graph G is the maximum number of disjoint dominating sets that can be found in G [3]. In the domatic number pr oblem , we are giv en a graph G and a number d , and asked to determine whether the domatic number of G is at least d ; that is, whether G contains at least d disjoint dominating sets. This problem is known to be NP-complete e ven for d = 3: that is, it is difficult to determine whether the vertices of a gi ven graph may be partitioned into three dominating sets [5]. W e begin by describing a polynomial-time many-one reduction from an instance ( G , d ) of the domatic number problem into an instance ( G 0 , h ) of the Hadwiger number problem. W e may assume without loss of generality that no vertex of G is adjacent to all others, for if v is such a verte x we may take one of the dominating sets to be the one-verte x set { v } , simplify the problem in polynomial time by deleting v from G and subtracting one from d , and use the remaining simplified problem as the basis for the transformation. As we will show , with this assumption, the instance ( G , d ) may be translated in polynomial time to an equi valent instance ( G 0 , h ) of the Hadwiger number problem. W e also assume that the vertices of G are numbered arbitrarily as v i for 1 ≤ i ≤ n . T o perform this translation, construct G 0 in three layers: – The top layer is a d -v ertex clique with v ertices t i for 1 ≤ i ≤ d . – The middle layer is an n -verte x independent set with vertices m i for 1 ≤ i ≤ n . – The bottom layer is an n ( n + 1 ) -vertex clique with v ertices b i , j for 1 ≤ i ≤ n and 1 ≤ j ≤ n + 1. – Every pair of one top and one middle v ertex is connected by an edge. – Middle verte x m i and bottom verte x b j , k are connected by an edge if and only if either i = j or G has an edge v i v j . That is, there is an edge from m i to b j , k if and only if v i dominates v j . W e let h = n ( n + 1 ) + d . This reduction is illustrated in Figure 1. 2 Lemma 1. Let G 0 be as constructed above, and let S be a connected nonempty subset of the vertices of G 0 . Then at least one of the following thr ee possibilities is true: 1. S consists only of a single middle vertex. 2. S contains a top vertex. 3. S contains a bottom vertex. Pr oof. If S consists only of middle vertices, it can hav e only one of them, because the middle vertices form an independent set. If on the other hand S does not consist only of middle vertices, it must contain a top verte x or a bottom verte x. u t Lemma 2. Let vertex v i in G have de gr ee d i . Then middle vertex m i in G 0 has degr ee ( d i + 1 )( n + 1 ) + d . Pr oof. V ertex m i is connected to ( d i + 1 )( n + 1 ) bottom vertices: the n + 1 vertices b i , j (for 1 ≤ j ≤ n + 1) and the d i ( n + 1 ) vertices b i 0 , j (for 1 ≤ j ≤ n + 1) such that v i and v 0 i are neighbors. In addition it is connected to all d top vertices. u t Lemma 3. Let G 0 be as constructed above, suppose that G 0 has Hadwiger number at least h, and let S i ( 1 ≤ i ≤ h) be a family of disjoint mutually-adjacent subgraphs forming an h-verte x clique minor in G 0 . Then each subgraph S i has exactly one non-middle vertex and each non-middle vertex belongs to exactly one subgraph S i . Pr oof. A middle vertex in G 0 has degree at most ( n − 1 )( n + 1 ) + d < h − 1 by Lemma 2 and by the as- sumption that G has no vertex that is adjacent to all other vertices. If there were a subgraph S i consisting only of a single middle vertex, it would not have enough neighbors to be adjacent to all h − 1 of the other subgraphs, so we may infer that such subgraphs do not e xist and apply Lemma 1 to conclude that each subgraph S i contains at least one non-middle vertex. But there are h subgraphs, and h non-middle vertices, so each subgraph S i must contain exactly one non-middle v ertex and each such verte x must belong to one of these subgraphs. u t Lemma 4. Let G 0 be as constructed above, suppose that G 0 has Hadwiger number at least h, and let S i ( 1 ≤ i ≤ h) be a family of disjoint mutually-adjacent subgraphs forming an h-verte x clique minor in G 0 . Then, for each i, ther e is a bottom vertex b i , j that forms a single-vertex subgraph in the family . Pr oof. By Lemma 3, each set that contains more than one vertex contains a middle v ertex. But there are only n middle v ertices, so at most n disjoint sets can contain middle vertices. Since there are n + 1 bottom v ertices b i , j , and (by Lemma 3 again) each belongs to a different subgraph, one of the n + 1 subgraphs containing these bottom vertices must have no middle vertices. Since it contains only one non-middle vertex, it must form a single-verte x subgraph. u t Lemma 5. Let G 0 be as constructed above, suppose that G 0 has Hadwiger number at least h, let S i ( 1 ≤ i ≤ h) be a family of disjoint mutually-adjacent subgraphs forming an h-vertex clique minor in G 0 , and let t i be a top vertex belonging to set S i . Then the set D i = { v j | m j ∈ S i } is a dominating set in G. Pr oof. Let v k be any verte x in G , and let b k , k 0 be a bottom vertex in G 0 that forms a single-vertex subgraph in the family of disjoint subgraphs; b k , k 0 is guaranteed to exist by Lemma 4. Then S i must contain a vertex adjacent to b k , k 0 ; this verte x must be a middle vertex m j of G 0 , because top vertices are not adjacent to bottom vertices and S i contains top vertex t i as its only non-middle vertex. In order for middle vertex m j to be adjacent to bottom vertex b k , k 0 , the vertex v j in G that corresponds to m j must dominate v k . Thus, for e very verte x v k in G , there is a verte x v j in D i that dominates v k ; therefore, D i is a dominating set. u t 3 Lemma 6. Let ( G , d ) be given and ( G 0 , h ) be as constructed above. Then G has domatic number at least d if and only if G 0 has Hadwiger number at least h. Pr oof. First, suppose that G has domatic number at least d ; we must show that in this case the Hadwiger number is at least h . W e can form a family of mutually-adjacent connected subgraphs S i in G 0 , as follows: for each bottom vertex b j , k form a subgraph consisting of that single vertex, and for each dominating set D i form a subgraph S i consisting of a single top vertex together with the middle vertices in G 0 that correspond to vertices in D i . There are n ( n + 1 ) bottom vertices, and d sets containing a top vertex, so these subgraphs form a clique minor with n ( n + 1 ) + d = h vertices as desired. Con versely , suppose that G 0 has Hadwiger number at least h = n ( n + 1 ) + d ; that is, that it has this man y disjoint mutually-adjacent connected subgraphs S i ; we must show that, in this case, G has domatic number at least d . Each subgraph S i must include exactly one top or bottom vertex by Lemma 3, together with possibly some middle vertices. For each top verte x t i , the set D i is a dominating set in G by Lemma 5; these sets are disjoint because the y correspond to the disjoint partition of the middle v ertices in G 0 gi ven by the subgraphs S i . Thus, we hav e found d disjoint dominating sets D i in G , so G has domatic number at least d . u t Theorem 1. The Hadwiger number pr oblem is NP-complete. Pr oof. The construction of ( G 0 , h ) from ( G , d ) may easily be implemented in polynomial time, and by Lemma 6 it forms a valid polynomial-time many-one reduction from the domatic number problem to the Hadwiger number problem. This reduction (together with the known fact that domatic number is NP- complete and the easy observ ation that the Hadwiger number problem is in NP) completes the proof of NP-completeness. u t 3 Alternati ve reduction from disjoint paths Seymour [12] has suggested that it should be straightforward to prove NP-completeness of the Hadwiger number problem via an alternative reduction, from disjoint paths. W e briefly outline such a reduction here. In the vertex-disjoint paths problem [11], the input consists of a graph G and a collection of pairs of vertices ( s i , t i ) in G ; the output should be positive if there exists a collection of vertex-disjoint paths in G having each pair of terminals as endpoints, and negati ve otherwise. Although the vertex-disjoint paths problem is fixed- parameter tractable with the number of terminal pairs as parameter , it is NP-complete when this number may be arbitrarily large, e ven when G is cubic and planar [9]. An instance of this problem may be reduced to the Hadwiger number problem as follows. Let n be the number of vertices in G , and k be the number of terminal pairs in the instance. Let K be an ( n + 1 ) -clique from which k non-adjacent edges u i v i hav e been removed, and form a new graph G 0 with 2 n + 1 − 2 k vertices as a union of G and K in which u i is identified with s i and v i is identified with t i . Then, a positive solution to the disjoint paths problem in G leads to the existence of an ( n + 1 ) -vertex clique minor in G 0 , by using the paths to replace each missing edge. Con versely , if G 0 has an ( n + 1 ) -v ertex clique minor, corresponding to a collection of n + 1 disjoint connected and pairwise adjacent subgraphs of G 0 , then each of these subgraphs must contain exactly one vertex of K (for any set of vertices of G 0 \ K has at most n − 1 neighbors), and the adjacency between the two subgraphs containing u i and v i can be used to find a path in G connecting s i and t i that uses only vertices drawn from these two subgraphs. Therefore, the gi ven verte x-disjoint paths problem instance ( G , k ) is a positive instance if and only if ( G 0 , n + 1 ) is a positive instance of the Hadwiger number problem. 4 Conclusions W e have shown that the Hadwiger number problem is NP-complete. It is natural to ask whether the problem is also hard to approximate. V ery strong inapproximability results are known for the superficially similar 4 maximum clique problem [7]. Alon et al. 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