ALGORITHMIC LOWER BOUNDS FOR PROBLEMS PARAMETERIZED BY CLIQUE-WIDTH

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 older 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 n(f(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 not equal 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. 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)n(o(t)), unless Exponential Time Hypothesis (ETH) collapses; and can be solved in time n(O(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".