Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees

2025-03-19

AI Summary

This paper studies the Degree-Constrained Minimum Spanning Tree (DC-MST) problem, supporting three types of vertex-degree constraints: list-based, upper-bound, and exact-degree constraints. Methodologically, it integrates structural graph decompositions, conditional lower-bound techniques (under SETH and ETH), high-dimensional convolution, and state-compression dynamic programming. The main contributions are fine-grained complexity bounds parameterized by structural graph parameters: an $O^*(2^{mathrm{pw}})$ algorithm and matching SETH-tight lower bound for pathwidth (pw) and cutwidth; an $O^*(2^{mathrm{cw}})$ ETH-tight algorithm for cliquewidth (cw); and an $O^*(4^{mathrm{tw}})$ algorithm for treewidth (tw), which is nearly SETH-optimal. These results establish the first comprehensive and tight fine-grained complexity landscape for DC-MST with respect to major structural parameters, achieving theoretical optimality in both algorithm design and hardness proofs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial 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
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth.

Problem

• Computing minimum-cost spanning trees with vertex degree constraints.
• Analyzing fine-grained complexity using graph parameters like pathwidth and treewidth.
• Providing SETH-tight bounds and ETH-tight algorithms for constrained spanning trees.

Innovation

• Novel techniques for degree-constrained spanning trees
• SETH-tight bounds for pathwidth and cutwidth
• ETH-tight algorithm for cliquewidth parameterization