Tight Bounds for some Classical Problems Parameterized by Cutwidth

2025-02-21

AI Summary

This paper determines the exact exponential time complexity—under the structural parameter cutwidth (ctw)—of four classical problems: Hamiltonian Cycle, Triangle Packing, Max Cut, and Induced Matching. Methodologically, it combines ctw-specific path decomposition analysis, optimized dynamic programming, and fine-grained SETH-based reductions; a key innovation is the discovery of the “Z-cut” structure, which captures the intrinsic hardness bottleneck for Triangle Packing. The contributions are tight SETH-based bounds: $(1+sqrt{2})^{ ext{ctw}}$ for Hamiltonian Cycle; $3^{frac{1}{3}cdot ext{ctw}}$ for Triangle Packing—the first tight bound with a non-integer base; $2^{ ext{ctw}}$ for Max Cut; and $3^{ ext{ctw}}$ for Induced Matching. These results resolve longstanding open questions regarding the precise complexity of these problems parameterized by cutwidth, thereby completing the exact complexity landscape for cutwidth.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

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].

Problem

• Determine optimal Hamiltonian Cycle algorithm complexity by cutwidth.
• Provide tight bounds for Partition Into Triangles using cutwidth.
• Establish optimal running times for Max Cut and Induced Matching.

Innovation

• Optimal algorithm for Hamiltonian Cycle
• Tight bounds for Partition Into Triangles
• Matching lower bounds for Max Cut