Faster Algorithms on Branch and Clique Decompositions Hans L. Bodlaender1, Erik Jan van Leeuwen2, Johan M.M. van Rooij1, and Martin Vatshelle2 1 Department of Information and Computing Sciences, Utrecht University P.O. Box 80.089, NL-3508 TB Utrecht, The Netherlands {hansb,jmmrooij}@cs.uu.nl 2 Department of Informatics, University of Bergen P.O. Box 7803, N-5020 Bergen, Norway {E.J.van.Leeuwen,martin.vatshelle}@ii.uib.no Abstract. We combine two techniques recently introduced to obtain faster dynamic programming algorithms for optimization problems on graph decompositions. The unification of generalized fast subset convo- lution and fast matrix multiplication yields significant improvements to the running time of previous algorithms for several optimization prob- ∗ ω k lems. As an example, we give an O (3 2 ) time algorithm for Minimum Dominating Set on graphs of branchwidth k, improving on the previous O∗(4k) algorithm. Here ω is the exponent in the running time of the best matrix multiplication algorithm (currently ω<2.376). For graphs of cliquewidth k,weimprovefromO∗(8k)toO∗(4k). We also obtain an algorithm for counting the number of perfect matchings of a graph, ∗ ω k given a branch decomposition of width k, that runs in time O (2 2 ). Generalizing these approaches, we obtain faster algorithms for all so- called [ρ, σ]-domination problems on branch decompositions if ρ and σ are finite or cofinite. The algorithms presented in this paper either attain or are very close to natural lower bounds for these problems. 1 Introduction Graph decompositions have over the last few years shown their worth in attack- ing NP-hard graph optimization problems. Most of this success is due to tree decompositions, which form the basis of results in many areas, from approxi- mation algorithms to exact algorithms and have become part of any algorith- mic toolbox. This success has motivated researchers to define and study other types of graph decompositions that are ‘better’ than tree decompositions. In this paper, we investigate algorithms for optimization problems on two such de- compositions, namely branch decompositions and clique decompositions. By for the first time combining two recent techniques used in designing algorithms on graph decompositions, generalized fast subset convolution and fast matrix mul- tiplication, in conjunction with the use of asymmetric vertex states, we obtain significant improvements on previous results. P. Hlinˇen´yandA.Kuˇcera (Eds.): MFCS 2010, LNCS 6281, pp. 174–185, 2010. c Springer-Verlag Berlin Heidelberg 2010 Faster Algorithms on Branch and Clique Decompositions 175 Algorithmic Techniques. Fast subset convolutions were introduced by Bj¨orklund et al. [1] to improve the running time of algorithms for optimization problems admitting a convolution-like recursive definition. These ideas were re- cently applied and generalized to show that a whole range of problems has faster algorithms on tree decompositions [16]. In particular, Van Rooij et al. [16] showed that Minimum Dominating Set has an O(n tw2(G)3tw(G)) time algorithm. Us- ing a generalized form of fast subset convolutions, they were able to obtain the fastest algorithms for a large class of so-called [ρ, σ]-domination problems. Matrix multiplication has been used for a much longer time as a basic tool for solving combinatorial problems. The best possible exponent in the running time of an algorithm performing multiplication of two n × n matrices is denoted by ω, i.e. the running time is O(nω). Currently we know that ω<2.376, due to an algorithm by Coppersmith and Winograd [4], but it is frequently hypothesized that ω = 2. Dorn [8] recently showed that matrix multiplication can also be used as a tool in solving many optimization problems on branch decompositions. One of the main results of this paper is that fast subset convolutions and fast matrix multiplication can be combined to obtain faster algorithms on branch decompositions for many optimization problems. Graph Decompositions. The notion of a branch decomposition was proposed by Robertson and Seymour as part of their graph minors project [12]. All of the recent results aimed at obtaining faster exact or fixed-parameter algorithms for Minimum Dominating Set on planar graphs and graphs excluding a fixed minor rely on branch decompositions [9,10,8]. The branchwidth and the treewidth of a graph are very closely related; the branchwidth of a graph is always less than its treewidth, but never by more than a factor 2/3. The notion of cliquewidth was first studied by Courcelle et al. [5]. Whereas the treewidth of the n-vertex clique is equal to n − 1, its cliquewidth is equal to 2. Moreover, the cliquewidth of a graph is always bounded by a function of its treewidth [6]. This makes cliquewidth an interesting graph parameter to consider on graphs where the tree- or branchwidth is too high for efficient algorithms. Our Results. In this paper, we improve on the currently best algorithms for Minimum Dominating Set on branch and clique decompositions. Dorn [8] showed that Minimum Dominating Set has an O(m 4k) time algorithm on branch de- compositions of width k. By combining fast subset convolution and fast matrix ω multiplication, we improve on this algorithm and obtain an O(mk2 3 2 k)time algorithm. A further innovation is the use of asymmetric vertex states.When combining two tables in this kind of dynamic programming algorithms, the set of vertex states used by these tables is always the same. We however use different states to obtain further speed-ups. This result extends to counting the number of dominating sets of each size. Another counting problem where we can apply this technique is counting the number of perfect matchings of a graph. This problem generalizes the problem of computing the permanent of a matrix and is a well-known #P-hard problem [15]. 176 H.L. Bodlaender et al. We give an algorithm for this problem on branch decompositions, running in time 2 ω k O(mk 2 2 i×(n)), where i×(n) is the time to multiply two n-bit numbers. Using the ideas of these algorithms, we solve existence, minimization, maxi- mization and counting variations of all [ρ, σ]-domination problems with finite or ∗ ω cofinite σ and ρ in O (s 2 k) time, where s is the number of vertex states used. Examples of such problems are Strong Stable Set, Independent Dominating Set, Perfect Code, Induced Bounded Degree Subgraph, and p-Dominating Set. On clique decompositions, we report an even bigger improvement for Mini- mum Dominating Set. We present an O∗(4cw(G)) time algorithm improving the current best O∗(8cw(G)) time algorithm obtained in [3]. 2 Preliminaries A branch decomposition (T,l) of a graph G is a ternary tree T and a bijection l between the edges of G and the leaves of T . Associated with every edge e ∈ E(T ) is the middle set Xe of e, defined as the set of vertices in V(G) which have incident edges e1,e2 such that the leaves l(e1)andl(e2) are in different components of T − e.Thewidth of a branch decomposition is the size of the largest set Xe.The branchwidth bw(G) is the minimum width of a branch decomposition of G. Create a root of T as follows. Choose an edge e =(t, t) ∈ E(T ), subdivide it, and add a new vertex r to T adjacent to the vertex created in the subdivision. The middle set of each edge of the subdivision is set to Xe and the middle set of the edge incident to r is set to ∅.RootT at r.Givenanyedge(t, t)=e ∈ E(T ), we can now speak of Te, which is the subtree of T induced by t, t ,andtheir descendants. The root of Te is t or t , whichever is closer to r in T . Consider an arbitrary internal vertex v ∈ V (T )andlete, f, g be its incident edges, such that e connects to the parent of v (and f,g to the two children of v). We call f the left child of e and g the right child. Consider the middle sets Xe, Xf , Xg. Note that any vertex in at least one of these sets is in at least two of them. We can then partition Xe ∪ Xf ∪ Xg into four sets: I = Xe ∩ Xf ∩ Xg L =(Xe ∩ Xf ) − I R =(Xe ∩ Xg) − IF=(Xf ∩ Xg) − I Observe that I, L,andR partition Xe, I, L,andF partition Xf ,andI, R,and F partition Xg. We can now prove the following lemma. | ∪ ∪ | | | | | | | | |≤ 3 · Lemma 1. Xe Xf Xg = I + L + R + F 2 bw(G). The notion of cliquewidth is defined as follows. A k-expression combines any number of the following four operations: – create a new labelled graph with one vertex labelled i ∈{1,...,k}, – relabel all vertices with label i to j (i = j), – connect all vertices with label i to all vertices labelled j (i = j), – take the disjoint union of two labelled graphs. Faster Algorithms on Branch and Clique Decompositions 177 The cliquewidth cw(G) of a graph G is the minimum k for which there is a k- expression that evaluates to a graph isomorphic to G. This definition can also be turned into a decomposition based on a rooted tree of degree at most three, labeling leaves of the tree by vertices of G and internal vertices by one of the above operations. We call this a clique decomposition. However, the definition of k-expressions is more useful in this paper. Given sets ρ, σ ⊆ N,a[ρ, σ]-dominating set is a subset D ⊆ V such that |N(v) ∩ D|∈ρ for every v ∈ V \ D and |N(v) ∩ D|∈σ for every v ∈ D.The [ρ, σ]-domination problems were introduced by Telle in [13,14] and form a large class of graph covering problems.