Clique-width: On the Price of Generality

Abstract

Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at most k. In particular, this implies that basic problems like Dominating Set, Graph Coloring, Clique, and Hamiltonian Cycle are solvable in linear time on graphs of bounded treewidth.A significant amount of research in graph algorithms has been devoted to extending this result to larger classes of graphs. It was shown that some of the algorithmic meta-theorems for treewidth can be carried over to graphs of bounded clique-width. Courcelle, Makowsky, and Rotics proved that the analogue of Courcelle's result holds for graphs of bounded clique-width when the logical formulas do not use edge set quantifications. Despite of its generality, this does not resolve the parameterized complexity of many basic problems concerning edge subsets (like Edge Dominating Set), vertex partitioning (like Graph Coloring), or global connectivity (like Hamiltonian Cycle). There are various algorithms solving some of these problems in polynomial time on graphs of clique-width at most k. However, these are not fixed parameter tractable algorithms and have typical running times O(n f (k) ), where n is the input length and f is some function.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, i.e. solvable in time O(g(k) · n c ), for some function g and a constant c not depending on k. In this paper we resolve this problem by showing that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]-hard parameterized by clique-width. This shows 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, namely F P T = W [1])-the price we pay for generality.

Algorithmic Lower Bounds for Problems Parameterized by Clique-width

“…Even though our results in [13] resolve 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 to compare the known algorithmic upper bounds. In this paper, we provide asymptotically tight optimal lower bounds for MAX-CUT and EDGE DOMINATING SET.…”

“…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 [15,22,23,24,17]. As the first step toward obtaining lower bounds for clique-width parameterizations, we have shown in [13] 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, for GRAPH COLORING, EDGE DOMINATING SET and HAMILTONIAN PATH.…”

Open Problems on Graph Coloring for Special Graph Classes

“…Hence, it is natural to research whether one can improve the FPT result for Coloring from treewidth to clique-width. However, Fomin et al [24] showed that Coloring is W[1]-hard when parameterized by the clique-width of the input graph. We also note that Precoloring Extension is NP-complete for distance-hereditary graphs [4], which have clique-width at most 3 [29].…”

Better Polynomial Algorithms on Graphs of Bounded Rank-Width

“…[5]. On the other hand, [9] recently proved that various problems, such as the chromatic number or hamiltonicity, can not be solved by FPT algorithms parameterized by clique-/ rank-width.…”