Algorithmic Lower Bounds for Problems Parameterized with Clique-Width

Many NP-hard problems can be solved efficiently when the input is restricted to graphs of bounded tree-width or cliquewidth. 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 n, 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 6=W[1], there is no algorithm with run time O(g(t) ·n), 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