Tight Bounds for some W[1]-hard Problems Parameterized by Multi-clique-width

2026-04-28

AI Summary

This work investigates the fine-grained complexity of Max Cut, Hamiltonian Cycle, and Edge Dominating Set parameterized by modular treewidth. Assuming the Exponential Time Hypothesis (ETH), it establishes that Max Cut admits no algorithm running in time \(n^{2^{o(k)}} \cdot f(k)\), highlighting a fundamental distinction from its behavior under standard treewidth. In contrast, the study presents \(n^{O(k)}\)-time algorithms for both Hamiltonian Cycle and Edge Dominating Set, matching known conditional lower bounds and thereby resolving three long-standing open problems. By integrating conditional lower bounds, parameterized algorithm design, and structural analysis of graphs of bounded modular treewidth, this research provides a complete characterization of the complexity landscape for these problems under this parameterization.

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 $n^{2^{o(k)}} \cdot f(k)$ on graphs of multi-clique-width $k$ for any computable function $f$. For clique-width $k$ an $n^{\mathcal{O}(k)}$ 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 $n^{2^{o(k)}} \cdot f(k)$. As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time $n^{\mathcal{O}(k)}$ 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].

Problem

• multi-clique-width
• fine-grained complexity
• W[1]-hard
• parameterized complexity
• tight bounds

Innovation

• multi-clique-width
• fine-grained complexity
• Max Cut
• ETH lower bound
• parameterized algorithms