Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

Abstract

Abstract. We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dominating Set problems on graphs of bounded clique-width. We show that on an n-vertex graph of clique-width t both problems (1) cannot be solved in time f (t)n o(t) for any function f of t unless exponential time hypothesis fails, and (2) can be solved in time n O(t) .Key words. exponential time hypothesis, clique-width, max-cut, edge dominating set AMS subject classifications. 05C85, 68R10, 68Q17, 68Q25, 68W40 DOI. 10.1137/1309109321. Introduction. Tree-width is one of the most fundamental parameters in graph algorithms. Graphs of bounded tree-width enjoy good algorithmic properties similar to trees, and this is why many problems which are hard on general graphs can be solved efficiently when the input is restricted to graphs of bounded tree-width. On the other hand, many hard problems also become tractable when restricted to graphs "similar to complete graphs." Courcelle and Olariu [6] introduced the notion of clique-width which captures nice algorithmic properties of both extremes.Since 2000, the research on algorithmic and structural aspects of clique-width is an active direction in graph algorithms, logic, and complexity. Corneil et al. [4] show that graphs of clique-width at most 3 can be recognized in polynomial time. By the meta-theorem of Courcelle, Makowsky, and Rotics [5], all problems expressible in M S 1 -logic (monadic second-order logic on graphs with quantification over subsets of vertices but not of edges) are fixed parameter tractable when parameterized by the clique-width of a graph and the expression size. For many other problems not expressible in this logic including problems like Max-Cut, Edge Dominating Set, Graph Coloring, or Hamiltonian Cycle, there is a significant amount of the literature devoted to algorithms for these problems and their generalizations on graphs of bounded clique-width

Cliquewidth III: The Odd Case of Graph Coloring Parameterized by Cliquewidth

“…A subset of the authors of the current paper, together with Fomin [20], showed that the EDS, HP, and GC problems parameterized by cliquewidth are all W[1]-hard. In particular, this implies that these problems do not admit algorithms with running times of the form O(g(k) · n c ), for any function g and constant c independent of k, unless FPT=W [1].…”

“…In particular, this implies that these problems do not admit algorithms with running times of the form O(g(k) · n c ), for any function g and constant c independent of k, unless FPT=W [1]. However, the lower bounds of Fomin et al [20] did not rule out non-trivial improvements to the exponent f (k) of n in the running times.…”

An Optimal XP Algorithm for Hamiltonian Cycle on Graphs of Bounded Clique-Width

“…In this paper, we prove that Hamiltonian Cycle can be solved in time n O(k) , thereby resolving the open problem in [9]. A Hamiltonian cycle in a graph is a cycle containing all vertices of the graph.…”

“…This question has been carefully answered by Fomin et al [8,9]. In particular, they showed that for Max-Cut and Edge Dominating Set, there is no f (k) · n o(k) -time algorithm unless the Exponential Time Hypothesis (ETH) fails, and proposed for both problems algorithms with running time n O(k) .…”

$${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

“…As a consequence, we obtain an analogous result to Corollary 3 for rows using the following result. Fomin et al [16] showed that the MaxCut problem can be solved in time O(n 2t+O (1) ) where t is clique-width of the input graph. By the combination of their result and our upper-bounds on clique-width (Theorem 21 in Sect.…”