[論文レビュー] A faster polynomial-space algorithm for Hamiltonian cycle parameterized by treedepth

Stefan Kratsch|arXiv (Cornell University)|Mar 25, 2026

Complexity and Algorithms in Graphs被引用数 0

ひとこと要約

tldr: The paper presents a randomized polynomial-space algorithm that solves Partial Cycle Cover and Hamiltonian Cycle in 4^{τ} n^{O(1)} time given a graph with elimination forest depth τ, improving over prior 5^{τ} bounds by using consistent matchings and an inclusion-exclusion framework.

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.

研究の動機と目的

• Motivate the search for fast, space-efficient algorithms for NP-hard graph problems under structural parameters like treedepth.
• Show that polynomial-space algorithms with single-exponential dependence on treedepth are possible for Hamiltonian Cycle and related problems.
• Develop a novel counting approach based on consistent matchings to achieve improved time bounds.
• Provide a Monte-Carlo algorithm that avoids false positives while maintaining polynomial space.

提案手法

• Define and use consistent matchings M1 and M2 (M1 ∩ M2 = ∅ and V(M1) = V(M2)) to represent partial cycle covers.
• Reduce Partial Cycle Cover to a bipartite setting to ensure even-length cycles.
• Develop an inclusion-exclusion formula to count ordered pairs of consistent matchings with given cardinality and weight (|Mw,ℓ|).
• Introduce two polynomial constructs P_(v)(J) and Q_[v](J′) to manage contributions along the elimination forest.
• Recursively compute the counting polynomials along the elimination tree of depth τ to obtain |˜M_{w,ℓ}| in 4^{τ} n^{O(1)} time using polynomial space.
• Apply the isolation lemma to reduce to a unique minimum-weight partial cycle cover with high probability, enabling detection via modulo 2^{k+1} counting.

実験結果

リサーチクエスチョン

• RQ1 Can Hamiltonian Cycle and related path/cycle problems be solved in polynomial space with single-exponential dependence on treedepth?
• RQ2 What is the exact base of the exponential dependence when using polynomial-space methods parameterized by treedepth?
• RQ3 How can consistent matchings be leveraged to replace perfect matchings in auxiliary constructions for cycle covers?
• RQ4 Does the inclusion-exclusion framework extend to partial cycle covers and weighted variants within polynomial space?
• RQ5 Can randomized isolation techniques yield correct decisions with controllable error for these problems under treedepth parameterization?

主な発見

• There exists a 4^{τ} n^{O(1)} time and polynomial-space Monte-Carlo algorithm for Partial Cycle Cover and Hamiltonian Cycle given a graph and an elimination forest of depth τ.
• The algorithm has no false positives and at most 1/2 probability of false negatives.
• Two key polynomials, P_(v)(J) and Q_[v](J′), enable recursive combination of child subproblems along the elimination forest to compute the count of ordered pairs of consistent matchings.
• A reduction to bipartite structures ensures that partial cycle covers consist of even-length cycles, aiding the counting method.
• The method generalizes to several path and cycle problems, yielding the same time/space bounds for Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover.