Cutwidth is a widely studied parameter that quantifies how well a graph can be decomposed along small edge-cuts. It complements pathwidth, which captures decomposition by small vertex separators, and it is well-known that cutwidth upper-bounds pathwidth. The SETH-tight parameterized complexity of problems on graphs of bounded pathwidth (and treewidth) has been actively studied over the past decade while for cutwidth the complexity of many classical problems remained open. For Hamiltonian Cycle, it is known that a $(2+\sqrt{2})^{\operatorname{pw}} n^{O(1)}$ algorithm is optimal for pathwidth under SETH~[Cygan et al.\ JACM 2022]. Van Geffen et al.~[J.\ Graph Algorithms Appl.\ 2020] and Bojikian et al.~[STACS 2023] asked which running time is optimal for this problem parameterized by cutwidth. We answer this question with $(1+\sqrt{2})^{\operatorname{ctw}} n^{O(1)}$ by providing matching upper and lower bounds. Second, as our main technical contribution, we close the gap left by van Heck~[2018] for Partition Into Triangles (and Triangle Packing) by improving both upper and lower bound and getting a tight bound of $\sqrt[3]{3}^{\operatorname{ctw}} n^{O(1)}$, which to our knowledge exhibits the only known tight non-integral basis apart from Hamiltonian Cycle. We show that cuts inducing a disjoint union of paths of length three (unions of so-called $Z$-cuts) lie at the core of the complexity of the problem -- usually lower-bound constructions use simpler cuts inducing either a matching or a disjoint union of bicliques. Finally, we determine the optimal running times for Max Cut ($2^{\operatorname{ctw}} n^{O(1)}$) and Induced Matching ($3^{\operatorname{ctw}} n^{O(1)}$) by providing matching lower bounds for the existing algorithms -- the latter result also answers an open question for treewidth by Chaudhary and Zehavi~[WG 2023].

翻译：割宽是一种被广泛研究的参数，用于量化图沿小边割的分解能力。它补充了路径宽（后者通过小顶点分离器刻画分解），且众所周知割宽是路径宽的上界。在过去十年中，关于有界路径宽（及树宽）图上问题的SETH-紧参数化复杂性得到了积极研究，而对于割宽，许多经典问题的复杂性仍然悬而未决。对于哈密顿环问题，已知在SETH下，针对路径宽的$(2+\sqrt{2})^{\operatorname{pw}} n^{O(1)}$算法是最优的~[Cygan et al.\ JACM 2022]。Van Geffen等人~[J.\ Graph Algorithms Appl.\ 2020]和Bojikian等人~[STACS 2023]曾提出，对于以割宽参数化的该问题，何种运行时间是最优的。我们通过提供匹配的上界和下界，以$(1+\sqrt{2})^{\operatorname{ctw}} n^{O(1)}$回答了这个问题。其次，作为我们的主要技术贡献，我们通过改进上界和下界，得到了$\sqrt[3]{3}^{\operatorname{ctw}} n^{O(1)}$的紧界，从而填补了van Heck~[2018]在划分为三角形（及三角形填充）问题上留下的空白；据我们所知，这是除哈密顿环外唯一已知的紧非整数基。我们证明了诱导长度为三的路径无交并（即所谓的$Z$-割的并）的割位于该问题复杂性的核心——通常下界构造使用更简单的割，这些割要么诱导一个匹配，要么诱导一个完全二分图的无交并。最后，我们通过为现有算法提供匹配的下界，确定了最大割（$2^{\operatorname{ctw}} n^{O(1)}$）和诱导匹配（$3^{\operatorname{ctw}} n^{O(1)}$）的最优运行时间——后一结果也回答了Chaudhary和Zehavi~[WG 2023]关于树宽的一个开放性问题。