Algorithmic Lower Bounds for Problems Parameterized by Clique-width

Abstract

Many NP-hard problems can be solved efficiently when the input is restricted to graphs of bounded tree-width or clique-width. In particular, by the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the tree-width of the input graph. On the other hand if we restrict ourselves to graphs of clique-width at most t, then there are many natural problems for which the running time of the best known algorithms is of the form nf(t), where n is the input length and f is some function. It was an open question whether natural problems like Graph Coloring, Max-Cut, Edge Dominating Set, and Hamiltonian Path are fixed parameter tractable when parameterized by the clique-width of the input graph. As a first step toward obtaining lower bounds for clique-width parameterizations, in [SODA 2009], we showed that unless FPT≠W[1], there is no algorithm with run time O(g(t) · nc), for some function g and a constant c not depending on t, for Graph Coloring, Edge Dominating Set and Hamiltonian Path. But the lower bounds obtained in [SODA 2009] are weak when compared to the upper bounds on the time complexity of the known algorithms for these problems when parameterized by the clique-width. In this paper, we obtain the asymptotically tight bounds for Max-Cut and Edge Dominating Set by showing that both problems cannot be solved in time f(t)no(t), unless Exponential Time Hypothesis (ETH) collapses; and can be solved in time nO(t), where f is an arbitrary function of t, on input of size n and clique-width at most t. We obtain our lower bounds by giving non-trivial structure-preserving “linear FPT reductions”.