A faster polynomial-space algorithm for Hamiltonian cycle parameterized by treedepth

2026-03-25

AI Summary

This work proposes a randomized algorithm for the Hamiltonian cycle and path problems on graphs of tree-depth τ, achieving a running time of 4^τ·n^{O(1)} while using only polynomial space. The key innovation lies in introducing a novel representation termed “ordered consistent matching pairs,” which replaces the conventional use of perfect matchings in auxiliary graphs within dynamic programming frameworks. This new formulation enables more efficient state transitions in the dynamic programming process, thereby improving the time complexity from the previous best-known 5^τ to 4^τ for polynomial-space algorithms parameterized by tree-depth. To the best of our knowledge, this constitutes the fastest known polynomial-space algorithm for these problems under the tree-depth parameterization.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$. Moreover, assuming the Strong Exponential-Time Hypothesis (SETH) we have essentially matching lower bounds for many such problems. They main drawback of these results is that the corresponding dynamic programming algorithms use exponential space, which makes them infeasible for larger $w$, and there is some evidence that this cannot be avoided. This motivates using somewhat more restrictive structure/decompositions of the graph to also get good (exponential) dependence on the corresponding parameter but use only polynomial space. A number of papers have contributed to this quest by studying problems relative to treedepth, and have obtained fast polynomial space algorithms, often matching the dependence on treewidth in the time bound. E.g., a number of connectivity problems could be solved by adapting the cut-and-count technique of Cygan et al. (FOCS 2011, TALG 2022) to treedepth, but this excluded well-known path and cycle problems such as Hamiltonian Cycle (Hegerfeld and Kratsch, STACS 2020). Recently, Nederlof et al. (SIDMA 2023) showed how to solve Hamiltonian Cycle, and several related problems, in $5^τn^{O(1)}$ randomized time and polynomial space when provided with an elimination forest of depth $τ$. We present a faster (also randomized) algorithm, running in $4^τn^{O(1)}$ time and polynomial space, for the same set of problems. We use ordered pairs of what we call consistent matchings, rather than perfect matchings in an auxiliary graph, to get the improved time bound.

Problem

• Hamiltonian Cycle
• treedepth
• polynomial space
• parameterized algorithm
• NP-hard graph problems

Innovation

• Hamiltonian Cycle
• treedepth
• polynomial space
• consistent matchings
• randomized algorithm