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].

翻译：本文致力于研究由多重团宽度参数化问题的细粒度复杂性，该方向由Fürer [ITCS 2017] 开创，并由Chekan与Kratsch [MFCS 2023] 进一步推进。多重团宽度是一种与团宽度类似定义的参数，但允许每个顶点同时持有多个标签。该参数受团宽度和树宽（加上常数）的上界约束，因此无需指数级增长即可推广两者。反之，多重团宽度为$k$的图其团宽度至多为$2^k$，且存在团宽度至少为$2^{Ω(k)}$的图。因此，尽管这两个参数在功能上等价，但问题的细粒度复杂性可能因参数不同而异。作为首要结果，我们证明在ETH假设下，对于多重团宽度为$k$的图，Max Cut问题无法在时间$n^{2^{o(k)}} \cdot f(k)$内求解（其中$f$为任意可计算函数）。对于团宽度为$k$的情况，Fomin等人 [SIAM J. Comput. 2014] 提出的$n^{\mathcal{O}(k)}$算法在ETH下是紧的。这使得Max Cut成为首个已知的、针对团宽度与多重团宽度参数化时紧运行时间不同的问题，并丰富了形式为$n^{2^{o(k)}} \cdot f(k)$的下界短列表。作为第二个贡献，我们证明多重团宽度为$k$的图上的哈密顿回路问题与边支配集问题可在时间$n^{\mathcal{O}(k)}$内求解，这与团宽度的紧运行时间相匹配。这些结果解答了Chekan与Kratsch [MFCS 2023] 遗留的三个开放问题。