Tight complexity bounds for FPT subgraph problems parameterized by the clique-width
We give tight algorithmic lower and upper bounds for some double-parameterized subgraph problems when the clique-width of the input graph is one of the parameters. Let G be an arbitrary input graph on n vertices with clique-width at most w. We prove the following results. The DENSE (SPARSE) k-SUBGRAPH problem, which asks whether there exists an induced subgraph of G with k vertices and at least q edges (at most q edges, respectively), can be solved in time k(O(w)) . n, but it cannot be solved in time 2(0(w log k)) . n(0(1)) unless the Exponential Time Hypothesis (ETH) fails. The d-REGULAR INDUCED SUBGRAPH problem, which asks whether there exists a d-regular induced subgraph of G, and the MINIMUM SUBGRAPH OF MINIMUM DEGREE AT LEAST d problem, which asks whether there exists a subgraph of G with k vertices and minimum degree at least d, can be solved in time d(0(w)) . n, but they cannot be solved in time 2(0(w log d)) . n(0(1)) unless ETH fails. (C) 2013 Elsevier B.V. All rights reserved.
clique-width exponential time hypothesis parameterized complexity