List Partitions of Chordal Graphs

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1 List Partitions of Chordal Graphs Tomás Feder 268 Waverley St., Palo Alto, CA 94301, USA Pavol Hell School of Computing Science Simon Fraser University Burnaby, B.C., Canada V5A 1S6 Sulamita Klein IM and COPPE-Sistemas Universidade Federal do Rio de Janeiro Caixa Postal 68511, , Rio de Janeiro, RJ, Brasil Loana Tito Nogueira COPPE-Sistemas, Universidade Federal do Rio de Janeiro Caixa Postal 68511, , Rio de Janeiro, RJ, Brasil and Fábio Protti IM and NCE Universidade Federal do Rio de Janeiro Caixa Postal 2324, , Rio de Janeiro, RJ, Brasil 1

2 Abstract In an earlier paper we gave efficient algorithms for partitioning chordal graphs into k independent sets and l cliques. This is a natural generalization of the problem of recognizing split graphs, and is NPcomplete for graphs in general, unless k 2 and l 2. (Split graphs have k = l = 1.) In this paper we expand our focus and consider general M-partitions, also known as trigraph homomorphisms, for the class of chordal graphs. For each symmetric matrix M over 0, 1,, the M-partition problem seeks a partition of the input graph into independent sets, cliques, or arbitrary sets, with some pairs of sets being required to have no edges, or to have all edges joining them, as encoded in the matrix M. Such partitions generalize graph colorings and homomorphisms, and arise frequently in the theory of graph perfection. We show that many M- partition problems that are NP-complete in general become solvable in polynomial time for chordal graphs, even in the presence of lists. On the other hand, we show that there are M-partition problems that remain NP-complete even for chordal graphs. We also discuss forbidden subgraph characterizations for the existence of M-partitions. 1 Introduction The M-partition problem was introduced in [8]. Let M be a symmetric m m matrix with entries M i,j {0,1, }. An instance of the M-partition problem is a graph G. A solution for the instance is a partition of vertices in G into m parts, corresponding to the rows (and columns) of the matrix M, such that for distinct vertices x and y of the graph G, placed in parts i and j (possibly with i = j) respectively, we have the following: if M(i,j) = 0, then xy is not an edge of G; if M(i,j) = 1, then xy is an edge of G. (If M(i,j) =, then xy may or may not be an edge in G.) An instance of the list M-partition problem is a graph G, together with a collection of lists L(x),x V (G), each list being a set of parts. A solution for the instance of list M-partition is a solution for the corresponding M- partition, such that each vertex x is placed in a part i L(x). List M-partitions generalize list colorings, retractions, and list homomorphisms [7], and are of interest in the theory of perfect graphs [3, 4]. Many well-known problems seeking, say, clique cutsets, homogeneous sets, 2

3 skew cutsets, joins, etc., can be formulated as list M-partition problems [8]. Moreover, the study of list M-partition problems can lead to efficient solutions of some of these problems [4]. In [8] we have given polynomial time algorithms for many list M-partition problems, and quasi-polynomial (O(n log n )) time algorithms for certain others. In [6] we have shown that all list M-partition problems are solvable in quasi-polynomial time, or are NP-complete. (We call such a result a quasidichotomy.) Many of our quasi-polynomial time algorithms from [8] were improved to polynomial time algorithms in [2, 4], but it is not known whether all list M-partition problems are polynomial time solvable or NP-complete; this is known as the Dichotomy Problem for list M-partitions. In this paper, we consider the restrictions of both the M-partition and the list M-partition problems to instances G that are chordal graphs. The two corresponding problems will be called the chordal M-partition problem and chordal list M-partition problem. There are several classical examples to suggest that M-partitions of chordal graphs can be found in polynomial time. For instance, k-colorability of chordal graphs (M is the k k matrix with 0 on the diagonal and everywhere else) can be decided efficiently using a perfect elimination ordering [9]; in fact, the algorithm either produces a k-coloring of the input graph or produces the unique forbidden subgraph K k+1. A similar result is known about clique covering (M is the l l matrix with 1 on the diagonal and elsewhere). In [10] we have shown more generally that there is a polynomial time recognition algorithm, and a forbidden subgraph characterization, of graphs that can be partitioned into k independent sets and l cliques (M has k zeros and l ones on the diagonal, everywhere else). We further extend these results to the list M-partition problem. We also extend the class of matrices M for which we can give polynomial time algorithms, and forbidden subgraph characterizations. However, we also find M-partition problems that remain NP-complete for chordal graphs, even in the absence of lists. Certain dichotomy and quasi-dichotomy results will also be claimed. 2 Matrices M with 0, 1 diagonal If the diagonal of the matrix M contains no, we have several large classes of polynomially solvable list M-partition problems, including the list versions of the above problem of partitioning G into k independent sets and l cliques. Consider first the case where the k k matrix M has zero diagonal. 3

4 Theorem 2.1 If all diagonal entries of M are zero, then the chordal list M-partition problem can be solved in polynomial time. Proof. A chordal graph G which admits an M-partition with such a matrix M cannot have a clique with k + 1 vertices; hence it must have treewidth at most k 1. The existence of a list M-partition of a graph of bounded treewidth can be tested by standard dynamic programming techniques [1, 5, 11]. Recall that a tree decomposition of a graph G is a pair (X,U) where U is a tree and X = (X i ) i V (U) is a collection of subsets of V (G) whose union equals V (G), such that each edge xy of G is included in some X i, and such that for each vertex x of G, the set of all X i containing x forms a subtree of G. The treewidth of a decomposition is the maximum value of X i 1, and the treewidth of a graph is the minimum treewidth of a decomposition. A tree decomposition in which U has a fixed root r is called nice [1] if each node of the rooted tree U has at most two children, and the following conditions are satisfied: If i has two children (a join node), say j and h, then X i = X j = X h ; if i has one child j then X i is obtained from X j by adding (an introduce node) or deleting (a forget node) a single vertex of G, and if X i = 1 for each leaf (start node) i of U. It is known that a nice tree decomposition of a chordal graph of bounded treewidth can be obtained in linear time [1]. Given a nice tree decomposition (X, U) of G with root r, we denote by G i the subgraph of G induced by the union of X i and all X j where j is a descendant of i. Let F(i) be the set of all pairs (Π,S), where Π is an assignment of the vertices in X i to parts, obtained by restricting a list M- partition Σ of G i, and S is the set of those parts in the partition Σ which contain vertices of G i X i. Note that each F(i) has at most k k 2 k elements. We can compute the set F(i) for any node, once all its descendants j have had their values F(j) calculated. This is not hard to see, considered separately the start, introduce, forget, and join nodes. For instance, suppose i is a forget node, with the unique child j, and X i = X j x. For each (Π,S) F(j) we add to F(i) the pair (Π,S ), where Π is Π restricted to X i and S equals either S, if the part a that x was assigned in Π was already present in S, or equals S a. On the other hand, if i is an introduce node, with the unique child j and X j = X i x, then for each (Π,S) F(j) we consider all possible values x can take with the current assignment Π, because of the adjacencies of x in X j, and also because of the non-adjacencies of x in G i X i ; it is for this purpose that we keep track of the set S. The above proof yields an algorithm for the list M-partition problem 4

5 restricted to graphs of treewidth at most k 1 (and hence for all chordal graphs), of complexity O(n(2k) k ); the complexity analysis is easily adapted from that of [5]. We next consider the case where the l l matrix M has all diagonal entries one. Let G with lists L(x) be an instance of the chordal list M- partition problem. A rectangle in G is a collection of sublists L x L(x),x V (G), such that any choice of parts from L x for each x constitutes a solution. Theorem 2.2 If all diagonal entries of M are one, then the chordal list M-partition problem can be solved in time polynomial in n l. The set of solutions to an instance is the union of at most n 2l rectangles, and can be found in polynomial time. Proof. Consider a perfect elimination ordering of the graph. If there are l parts, then choose l pairs (x i,y i ) of vertices in the input graph G, where x i is the first vertex in the perfect elimination ordering to go to part i, and y i is the last vertex in the perfect elimination ordering to go to part i. This involves n 2l possible choices. For each choice, remove part i from the list of any vertex that occurs either before x i or after y i in the elimination ordering. Remove from all lists of vertices z forbidden parts j given their adjacency or non-adjacency to the vertices x i,y i that go to part i according to what M requires. That is, vertex z cannot go to part j if there is an edge zx i or an edge zy i in G and the entry M(i,j) = 0. Similarly, vertex z cannot go to part j if there is no edge zx i or no edge zy i in G and the entry M(i,j) = 1. Finally, assign parts to vertices from their resulting reduced lists L x arbitrarily. Suppose z i,z j end up in parts i,j respectively and are adjacent, but M(i,j) = 0. Say z i was eliminated before z j. Then z i is adjacent to y i, since M(i,i) = 1. Thus y i and z j are both neighbors of z i, and both were eliminated after z i, so y i is adjacent to z j by the definition of a perfect elimination ordering. Since M(i,j) = 0, part j would have been removed from the list of z j. In the other case, suppose z i,z j end up in parts i,j respectively and are not adjacent, but M(i,j) = 1. Say z i was eliminated before z j. Then x i is adjacent to z i since M(i,i) = 1. Also x i is adjacent to z j since M(i,j) = 1. Thus x i is adjacent to both z i and z j, and both were eliminated after x i, so z i is adjacent to z j by the definition of a perfect elimination ordering, a contradiction. Thus we end up with n 2l families of solutions, each family given only by restrictions on possible parts for each element, so that each family is a rectangle. 5

6 In the rest of the paper we often focus on (k + l) (k + l) matrices M which consist of a k k diagonal matrix A and an l l diagonal matrix B, with an off-diagonal k l matrix C (and its l k transpose). We shall call such matrices A, B, C-block matrices. Assume now that all diagonal entries of A are zero, and all diagonal entries of B are one. We shall also consider restrictions on C. Feder, Hell, Klein, and Motwani [8] showed the following. Let A and B be two classes of graphs that are closed under taking induced subgraphs, and for which membership can be tested in polynomial time. Suppose further that there exists a constant c such that any graph both in A and B has at most c vertices. They consider the question of partitioning the vertices of a graph G into two sets S A and S B so that the subgraph G A induced by S A is in A, and the subgraph G B induced by S B is in B. They show that there are at most n 2c such partitions, and that all such partitions can be found in polynomial time. In our application, we let A be the class of graphs without a clique with k + 1 vertices, and B the class of graphs without an independent set of l+1 vertices. A chordal graph without a clique with k + 1 vertices and without an independent set with l + 1 vertices has at most c = kl vertices, since it is k-colorable, and thus a union of k independent sets. Given an instance G, let S A be the set of vertices that are placed in the parts corresponding to the k k matrix A, and let S B be the set of vertices that go to the parts corresponding to the l l matrix B. It follows that S A A,S B B. Suppose first the k by l matrix C is all. Then for each of the n 2c valid partitions into two graphs G A and G B, we can restrict the lists for G A to parts in A and solve the problem for matrix A on G A with the algorithm of Theorem 2.1. Similarly, we can restrict the lists for G B to parts in B and solve the problem for matrix B on G B with the algorithm of Theorem 2.2. More generally, we call a matrix C horizontal if all entries of C corresponding to a part i in A are the same, and vertical if all entries of C corresponding to a part j in B are the same. Finally, we call matrix C crossed if the entries of C are all 0 or (or all 1 or ) and every zero (respectively every one) belongs to either a row or a column of all zeros (respectively all ones). Theorem 2.3 Suppose M is an A, B, C-block matrix. If all diagonal entries of A are zero, all diagonal entries of B are one, and if C is either horizontal, vertical, or crossed, then the chordal list M- partition can be solved in time polynomial in n kl. 6

7 Proof. For each choice of G A and G B, if all entries of C corresponding to a part i in A are zero (respectively one) then it suffices to remove the part i from the list of any vertex v of G A that has a neighbor in G B (respectively for which some vertex of G B is not a neighbor). Similarly, if all entries of C corresponding to a part j in B are zero (respectively one) then it suffices to remove the part j from the list of any vertex v of G B that has a neighbor in G A (respectively for which some vertex of G A is not a neighbor). Once the conditions given by C are met, we can replace C by an all matrix and solve the problem for G A and G B using Theorems 2.1 and 2.2. We can generalize this result to matrices A which consist of diagonal blocks A i with zero diagonals, and matrices B which consist of diagonal blocks B i with all diagonal entries one, as long as all entries of A not in the diagonal blocks are one, all entries of B not in the diagonal blocks are zero, and all block matrices C i,j of C corresponding to A i,b j are either horizontal, vertical, or crossed. 3 NP-complete problems Consider a fixed bipartite graph H. The list H-coloring problem is defined as follows: An instance is a bipartite graph G with lists (white vertices of G have lists consisting of white vertices of H and similarly for black vertices), and a solution is a mapping of vertices of G to vertices of H so that adjacency is preserved and each vertex of G is mapped to a member of its list. (Such a mapping is called a list H-coloring of G.) Feder, Hell and Huang [7] showed that the list H-coloring problem is polynomial time solvable if the bipartite graph H is the complement of a circular arc graph (a cocircular graph), and is NP-complete otherwise. Based on this result, it will be possible to find NP-complete chordal list M-partition problems. Given a bipartite graph H with k white vertices (forming the set V A ) and l black vertices (forming the set V B ), the matrix corresponding to H is the k l matrix C with C(i,j) = if ij is an edge in H (with i V A,j V B ), and with C(i,j) = 0 otherwise. Theorem 3.1 Let M be an A,B,C-block matrix. Suppose A does not contain any 1 s, and B does not contain any 0 s. If C is the matrix corresponding to a bipartite graph H that is not a cocircular graph, then the chordal list M-partition problem is NP-complete. Proof. Consider an instance G of the list H-coloring problem, and define the graph G to be obtained from G by adding all edges between pairs of 7

8 black vertices. (The lists of G remain the same as in G.) It is easy to see that G has a list H-coloring if and only if G has a list M-partition. Since G is a split graph (it can be partitioned into a clique and an independent set), it is also chordal [9]. The same result holds if C is obtained from the matrix corresponding to a bipartite graph H by replacing each 0 with a 1. (This follows by replacing the bipartite graph G by the bipartite complement G of G.) The proof implies that the list M-partition problems corresponding to graphs that are not cocircular are NP-complete even for split graphs. It is easy to see that, in the special case when A is an all zero matrix and B is an all one matrix, we in fact obtain the following dichotomy (again, valid also for matrices C obtained by replacing all zeros by ones): Theorem 3.2 Let M be an A,B,C-block matrix. If A = 0, B = 1, and C is the matrix corresponding to a bipartite graph H, then the chordal list M-partition problem is polynomial if H is a cocircular graph and is NP-complete otherwise. A similar quasi-dichotomy result can be derived from the theorem of Feder and Hell [6], who showed that on general instances, all list M-partition problems are quasi-polynomial or NP-complete. In particular, if A and B are as above (all-zero and all-one matrices), and C is an arbitrary matrix (not necessarily corresponding to a graph H), then it can be shown that in fact the chordal list M-partition problem is quasi-polynomial or NP-complete. Several generalizations of these dichotomy and quasi-dichotomy results can be proved: It is enough to assume, for instance, that B (instead of being an all one matrix) has ones on the diagonal and no zeros. The quasidichotomy also applies if B is only assumed to have ones on the diagonal, as long as A has zeros on the diagonal and no s. In this case, if additionally C has no zeros (or no ones), we have dichotomy. These results will be proved elsewhere. We now focus on constructing NP-complete M-partition problems (without lists). Let H again be a bipartite graph. The H-retraction problem is the restriction of the list H-coloring problem to instances G containing H as a subgraph, and with lists either L(g) = g, if g V (H), or L(g) = V (H), otherwise. A list H-coloring of G is called an H-retraction of G, in this situation. Many bipartite graphs H are known to yield NP-complete H- retraction problems, although a complete classification of complexity is not 8

9 known, and dichotomy has not been proved, for H-retractions. In particular, it is known that if H is an even cycle of length greater than four, the H-retraction problem is NP-complete [7]. Theorem 3.3 For every bipartite graph H such that the H-retraction problem is NP-complete, there exists a matrix M H such that the M H -partition problem (without lists) is also NP-complete. Proof. Let H be a bipartite graph such that the H-retraction problem is NP-complete. We first extend the graph H to a larger bipartite graph H, by attaching to each white vertex of H a path of length five and to each black vertex of H a path of length four. Note that all the leaves (vertices of degree one) of H are black. We now introduce an auxiliary problem, which we shall call the weak H -retraction problem. Suppose that the bipartite graph H has k black vertices, forming the set V B, and let L denote the set of all black leaves of H. An instance of the weak H-retraction problem is a bipartite graph G with a specified set X of k black vertices, such that each vertex of G not in X has at most one neighbour in X. A solution to the instance is an edge-preserving and color-preserving mapping of the vertices of G to the vertices of H such that X is mapped bijectively to V B. We now show that the H-retraction problem reduces to the weak H -retraction problem. Suppose G is an instance of the H-retraction problem, i.e., a bipartite graph containing H. We transform G to an instance G (with a set X) of the weak H -retraction problem as follows: Let X be another copy of the set V B, disjoint from G. Consider the union of G and X, and identify each vertex of L in X with the corresponding vertex of L in G. Finally, add internally disjoint paths of length four joining all pairs of vertices of X which correspond to vertices in V B of distance two or four in H. Call the resulting graph G. We now argue that G admits an H-retraction if and only if G admits a weak H -retraction. On the one hand, suppose f is an H-retraction of G. Then f, extended by taking each vertex of X L to the corresponding vertex of V B is a weak H -retraction of G. For the other direction, we note that any bijection between X and V B has to map vertices of L to vertices of L, since leaves in H have exactly two vertices in H at distance two or four, while black vertices of H that are not leaves have at least three vertices in H at distance two or four. Therefore, any weak H -retraction of G which maps the vertices of X bijectively to the vertices of V B must map the copy of H in G isomorphically to H. It follows that G admits an H -retraction, which can easily 9

10 be modified to an H-retraction by mapping all the added paths of H into H. Next, we define a matrix M H such that the chordal M H -partition problem (without lists) is NP-complete, as claimed in the theorem. The matrix M H will be an A,B,C-block matrix in which the diagonal matrix A is an all zero matrix; the diagonal matrix B has all diagonal entries one and all other entries ; and finally, the matrix C will be the matrix corresponding to the bipartite graph H. We now reduce the weak H -retraction problem to the M H -partition problem. Given an instance G for the weak H -retraction problem, we construct an instance G of the M H -partition problem as follows. We replace each white vertex a of G by a set I(a) of k + 1 independent vertices (where k = V B ), and each black vertex b of G by a clique K(b) of two vertices. Whenever a and b are adjacent in G, all vertices of I a are adjacent to all vertices of K b in G. Finally, we add all edges between K b and K b unless both b and b are in X. Note that each vertex every I(a) is adjacent to at most one K(b) with b X. We claim that G admits a weak H -retraction if and only if G admits an M H -partition. Indeed, if f is a weak H -retraction of G, all vertices of a set I(a) can be placed in the part f(a) and all vertices of a set K(b) can be placed in the part f(b). Conversely, each M H -partition of G must place at least one of the two vertices in any K(b) to a part in B, since A is an all-zero matrix. Also, if b,b are both in X, these vertices must be placed in distinct parts of B. By a similar argument, at least one vertex of each I(a) must be placed in a part in A, since the vertices placed to parts in B are covered by k cliques. This way we deduce an H -retraction of G. It remains to argue that the instance G is a chordal graph. We first note that each vertex of every I(a) is only adjacent to vertices in K(b) with b X expept possibly in one K(b) with b X. According to the definition of G, these vertices are all mutually adjacent, i.e., a clique. Thus we can repeatedly remove simplicial vertices (vertices whose neighbours form a clique) from the sets I(a), until G is reduced to the union of the K(b), which is clearly chordal. 4 Conclusions We also have some forbidden subgraph charaterizations of M-partitionable chordal graphs. It is well-known, for example, that a chordal graph G is k- colorable if and only if it does not contain a K k+1. In [8] we have extended 10

11 this as follows: A chordal graph G can be partitioned into k cliques and l independent sets if and only if it does not contain an induced subgraph isomorphic to (l + 1)K k+1. For many other matrices M it is possible to characterize non-m-partitionable chordal graphs G by a finite number of forbidden subgraphs. At the same time, it follows from our results that, unless P=NP, this is not the case for all matrices M. Once again, we shall consider only A,B,C-block matrices M, with A having zero diagonal and B having a diagonal of ones. Moreover, we shall assume that all off-diagonal entries of A are the same, say a, all off-diagonal entries of B are the same, say b, and all entries of C are the same, say c. Note that we may assume a 0 and b 1, otherwise we may replace M by a matrix with k = 1 or l = 1 respectively. The result of [10] states that when a = b = c =, a chordal graph is non-m-partitionable if and only if it contains in induced subgraph isomorphic to (l + 1)K k+1. We can show that if c, the non-m-partitionable chordal graphs can always be characterized by a finite number of forbidden subgraphs, all with at most (k + 1)(l + 1) vertices. This bound does not always apply: if c =, we know that in the particular case of k = 1 and b = 0, the largest minimal forbidden subgraph has on the order of l 2 vertices. Nevertheless, even in the case c = we can prove that there is always only a finite number of obstructions. If a = 1, the best bounds we currently have for the size of minimal obstructions are t k, where t = O(l) if b =, and t = O(l 2 ) if b = 0. If the remaining case a =,b = 0, we have the bound 2(k + 1) (k+2)l+1. We will return to these results in a future paper. References [1] H. L. Bodlaender and T. Kloks, Efficient and constructive algorithms for the pathwidth and treewidth of graphs, J. Algorithms 21 (1996) [2] K. Cameron, E. M. Eschen, C. T. Hoang, and R. Sritharan, The list partition problem for graphs, SODA [3] M. Chudnovsky, N. Robertson, P. Seymour, and R. Thomas, The strong perfect graph theorem, manuscript,

12 [4] C. M. H. de Figueiredo, S. Klein, Y. Kohayakawa, and B. A. Reed, Finding skew partitions efficiently, Journal of Algorithms 37 (2000) [5] J. Diaz, M. Serna, and D. M. Thilikos, The complexity of parametrized H-colorings: a survey, Dimacs Series in Discrete Mathematics, [6] T. Feder and P. Hell, List constraint satisfaction and list partition, submitted to SIAM J. Comput. (2003). [7] T. Feder, P. Hell, and J. Huang, List homomorphisms and circular arc graphs, Combinatorica 19 (1999) [8] T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of list partitions, Proc. 31st Ann. ACM Symp. on Theory of Computing (1999) SIAM J. Comput., in press. [9] M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, [10] P. Hell, S. Klein, L.T. Nogueira, F. Protti, Partitioning chordal graphs into independent sets and cliques, Discrete Applied Math., to appear. [11] A. Proskurowski and S. Arnborg, Linear time algorithms for NP-hard problems restricted to partial k-trees, Discrete Applied Math (23)

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