Tight Bounds for Some W[1]-Hard Problems Parameterized by Multi-Clique-Width

Abstract

In this work we contribute to the study of the fine-grained complexity of problems parameterized by multi-clique-width, which was initiated by Fürer [ITCS 2017] and pursued further by Chekan and Kratsch [MFCS 2023]. Multi-clique-width is a parameter defined analogously to clique-width but every vertex is allowed to hold multiple labels simultaneously. This parameter is upper-bounded by both clique-width and treewidth (plus a constant), hence it generalizes both of them without an exponential blow-up. Conversely, graphs of multi-clique-width k have clique-width at most 2^k, and there exist graphs with clique-width at least 2^{Ω(k)}. Thus, while the two parameters are functionally equivalent, the fine-grained complexity of problems may differ relative to them.
As our first and main result we show that under ETH the Max Cut problem cannot be solved in time n^{2^{o(k)}} ⋅ f(k) on graphs of multi-clique-width k for any computable function f. For clique-width k an n^{𝒪(k)} algorithm by Fomin et al. [SIAM J. Comput. 2014] is tight under ETH. This makes Max Cut the first known problem for which the tight running times differ for parameterization by clique-width and multi-clique-width and it contributes to the short list of known lower bounds of form n^{2^{o(k)}} ⋅ f(k). As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time n^{𝒪(k)} on graphs of multi-clique-width k matching the tight running time for clique-width. These results answer three questions left open by Chekan and Kratsch [MFCS 2023].