Intractability of Clique-Width Parameterizations

Abstract

Abstract. We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]-hard parameterized by clique-width. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, that is, solvable in time g(k) · n O(1) on n-vertex graphs of clique-width k, where g is some function of k only. Our results imply that the running time O(n f (k) ) of many clique-width based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namelyKey words. Parameterized complexity, clique-width, tree-width, chromatic number, edge domination, hamiltonian cycle AMS subject classifications. 68Q17, 68Q25, 68W401. Introduction. One of the most frequent approaches for solving graph problems is based on decomposition methods. Tree decomposition, and the corresponding parameter, the tree-width of a graph, are among the most commonly used concepts. We refer to the surveys of Bodlaender [3] and Hlinený et al. [22] for further references on tree-width and related parameters. In the quest for alternative graph decompositions that can be applied to broader classes than to those of bounded tree-width and still enjoy good algorithmic properties, Courcelle and Olariu [10] introduced the clique-width of a graph. Clique-width can be seen as a generalization of tree-width, in a sense that every graph class of bounded tree-width also have bounded clique-width [5].In recent years, clique-width has received much attention. Corneil, Habib, Lanlignel, Reed, and Rotics [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. 3 ) and computing (2 k+1 − 1)-expressions for a graph G of clique-width at most k. It is also worth to mention here the related graph parameters NLC-width introduced by Wanke [30] and rank-width introduced by Oum and Seymour [27], which are equivalent to clique-width in the sense that the same classes of graph have bounded clique-width, NLC-width ans rank-width.By the seminal result of Courcelle [6, 9] (see also [1]), every decision problem on graphs expressible in monadic second order logic is fixed parameter tractable when parameterized by the tree-width of the input graph. For problems expressible in monadic second order logic with logical formulas that do not use edge set quantifications (so-called M S 1 -logic), it is possible to extend the meta theorem of Courcelle to graphs of bounded clique-width. As it was shown by Courcelle, Makowsky, and

Solving Hamiltonian Cycle by an EPT algorithm for a non-sparse parameter

“…The classical structural graph parameter bounded also on some non-sparse graphs is clique-width. Unfortunately, it is unlikely that such a result exists for clique-width, as Hamiltonian Cycle has been shown to be W-hard when parameterized by clique-width [5]. So, we must focus on a non-sparse parameter which is less general than clique-width.…”

“…When parameterized by the more general parameter cliquewidth, Hamiltonian Cycle becomes W[1]-hard, as shown by Fomin et al [5]. Saether and Telle address this problem in their paper [13] by introducing a new parameter, split-matching-width, which lies between treewidth and clique-width in terms of generality.…”

On the Complexity of Some Colorful Problems Parameterized by Treewidth

“…It is also somewhat surprising that this good news does not extend to List Coloring, Precoloring Extension or Equitable Coloring, all of which turn out to be hard for W [1]. Results of the preliminary version of this paper [13] have led to thorough investigations of structural parameterizations like treewidth or clique-width [9,16,17,27].…”

Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

“…Even the existence of fixed parameter tractable algorithms (with clique-width being the parameter) for all these problems (or their generalizations) was open until very recently [16,24,25,27,18]. As the first step toward obtaining lower bounds for clique-width parameterizations, we have shown in [15] that unless FPT = W [1], there is no function g such that Graph Coloring, Edge Dominating Set, and Hamiltonian Path are solvable in time g(t) · n O (1) . While [15] resolves the parameterized complexity of these problems, the conclusion that unless FPT = W [1] there is no algorithm with run time O(g(t) · n c ) for some function g and a constant c not depending on t is weak compared to the known algorithmic upper bounds.…”

“…As the first step toward obtaining lower bounds for clique-width parameterizations, we have shown in [15] that unless FPT = W [1], there is no function g such that Graph Coloring, Edge Dominating Set, and Hamiltonian Path are solvable in time g(t) · n O (1) . While [15] resolves the parameterized complexity of these problems, the conclusion that unless FPT = W [1] there is no algorithm with run time O(g(t) · n c ) for some function g and a constant c not depending on t is weak compared to the known algorithmic upper bounds. For example, it does not rule out an algorithm of running time n O( √ t) 2 t .…”