Tight Bounds for some Classical Problems Parameterized by Cutwidth

TL;DR

This paper establishes tight bounds for classical graph problems parameterized by cutwidth, providing optimal algorithms and matching lower bounds, and introduces new technical insights into the complexity based on specific cut structures.

Contribution

It presents the first tight bounds for problems parameterized by cutwidth, including Hamiltonian Cycle, Partition Into Triangles, Max Cut, and Induced Matching, with novel technical methods.

Findings

Matching upper and lower bounds for Hamiltonian Cycle parameterized by cutwidth.

Tight bounds for Partition Into Triangles and Triangle Packing, involving non-integral bases.

Optimal algorithms for Max Cut and Induced Matching with proven lower bounds.

Abstract

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].