由簇顶点删除数参数化的诱导子图同构与最大公共诱导子图的复杂度
Complexity of induced subgraph isomorphism and maximum common induced subgraph parameterized by cluster vertex deletion number
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中文总结 AI 辅助
该研究以簇顶点删除数$k$为参数，证明ISI可在随机化$O^*(k^{O(k)})$时间内求解，MCIS可在随机化$O^*(2^{O(k^2)})$时间内求解且更难，还证明3-MCIS在$k=2$时为NP难，解决了开放问题并给出相关下界。
中文摘要 AI 辅助
我们研究以簇顶点删除数$k$为参数的诱导子图同构（Induced Subgraph Isomorphism, ISI）和最大公共诱导子图（Maximum Common Induced Subgraph, MCIS）的参数化复杂度。对于ISI，我们给出了一个随机化的$O^*(k^{O(k)})$时间算法，证明ISI在该参数下是固定参数可处理的，解决了Hanaka等人[WALCOM 2026]提出的开放问题。该算法在指数时间假设（Exponential Time Hypothesis, ETH）下是最优的，其核心是将问题归约为可通过代数技术求解的精确多色匹配问题。对于MCIS，我们通过归约为精确多色匹配的加权变体，给出了一个随机化的$O^*(2^{O(k^2)})$时间算法；同时，我们证明了与之匹配的基于ETH的下界，具体是证明了具有行列列表约束的$k$阶二元矩阵可行性问题不存在$O^*(2^{o(k^2)})$时间算法，该问题可能具有独立研究价值。这些结果表明，在该参数设置下，MCIS比ISI更难。最后，对于三图变体3-MCIS，我们证明当每个输入图的簇顶点删除数为2时，它已成为NP难问题。
英文摘要
We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.
