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 n f (t) , where n is the input length and f is some function. 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; andwhere 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".

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

“…While ETH is now a widely believed assumption, and has been used as a starting point to prove running time lower bounds for numerous problems [5,4,11,18,17], SETH remains largely untouched (with one exception [21]). The reason for this is two-fold.…”

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].…”

What’s Next? Future Directions in Parameterized Complexity

“…One should examine it separately for each problem whether the lower bound obtained this way is tight or not. Many of the more involved reductions from Clique use edge selection gadgets (see e.g., [48,51,79]). As a clique of size k has Θ(k 2 ) edges, this means that the reduction typically increases the parameter to Θ(k 2 ) at least and, similarly to Theorem 2.8, what we can conclude is that there is no f (k)n o( √ k) time algorithm for the target problem (unless ETH fails).…”