Clique-width III

Abstract

M AX -C UT , E DGE D OMINATING S ET , G RAPH C OLORING , and H AMILTONIAN C YCLE on graphs of bounded clique-width have received significant attention as they can be formulated in MSO 2 (and, therefore, have linear-time algorithms on bounded treewidth graphs by the celebrated Courcelle’s theorem), but cannot be formulated in MSO 1 (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 ) n f ( k ) on graphs of clique-width k . Fomin et al. (2010) showed that the running times cannot be improved to g ( k ) n O (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 E DGE D OMINATING S ET and M AX -C UT to n O ( k ) , and proved that these problems cannot be solved in time g ( k ) n o ( k ) unless ETH fails. Thus, prior to this work, E DGE D OMINATING S ET and M AX -C UT were known to have tight n Θ ( k ) algorithmic upper and lower bounds. In this article, we provide lower bounds for H AMILTONIAN C YCLE and G RAPH C OLORING . For H AMILTONIAN C YCLE , our lower bound g ( k ) n o ( k ) matches asymptotically the recent upper bound n O ( k ) due to Bergougnoux, Kanté, and Kwon (2017). As opposed to the asymptotically tight n Θ( k ) bounds for E DGE D OMINATING S ET , M AX -C UT , and H AMILTONIAN C YCLE , the G RAPH C OLORING problem has an upper bound of n O (2 k ) and a lower bound of merely n o (√ [4] k ) (implicit from the W[1]-hardness proof). In this article, we close the gap for G RAPH C OLORING by proving a lower bound of n 2 o ( k ) . This shows that G RAPH C OLORING behaves qualitatively different from the other three problems. To the best of our knowledge, G RAPH C OLORING is the first natural problem known to require exponential dependence on the parameter in the exponent of n .