Clique-width III: Hamiltonian Cycle and the Odd Case of Graph Coloring

Abstract

MAX-CUT, EDGE DOMINATING SET, GRAPH COLORING, and HAMILTONIAN CYCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO1 (which would have yielded linear-time algorithms on bounded clique-width graphs by a well-known theorem of Courcelle, Makowsky, and Rotics). Each of these problems can be solved in time g(k)nf(k) on graphs of clique-width k. Fomin et al. (2010) showed that the running times cannot be improved to g(k)nO(1) assuming W[1]≠FPT. However, this does not rule out non-trivial improvements to the exponent f(k) in the running times. In a follow-up paper, Fomin et al. (2014) improved the running times for EDGE DOMINATING SET and MAX-CUT to nO(k), and proved that these problems cannot be solved in time g(k)no(k) unless ETH fails. Thus, prior to this work, EDGE DOMINATING SET and MAX-CUT were known to have tight nΘ (k) algorithmic upper and lower bounds. In this article, we provide lower bounds for HAMILTONIAN CYCLE and GRAPH COLORING. For HAMILTONIAN CYCLE, our lower bound g(k)no(k) matches asymptotically the recent upper bound nO(k) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight nΘ(k) bounds for EDGE DOMINATING SET, MAX-CUT, and HAMILTONIAN CYCLE, the GRAPH COLORING problem has an upper bound of nO(2k) and a lower bound of merely no(√ [4]k) (implicit from the W[1]-hardness proof). In this article, we close the gap for GRAPH COLORING by proving a lower bound of n2o(k). This shows that GRAPH COLORING behaves qualitatively different from the other three problems. To the best of our knowledge, GRAPH COLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n.