Parameterized Local Search for Vertex Cover: When only the Search Radius is Crucial

2026-05-08

AI Summary

This work addresses the local search optimization of the Vertex Cover problem: given a vertex cover, it investigates whether there exists an exchange operation of size at most $k$ that yields a strictly smaller (or lower-weight) cover. The authors propose a novel algorithmic framework whose running time exhibits mild dependence on structural graph parameters—such as treewidth, h-index, and modular width—and exponential dependence only on the exchange radius $k$. They introduce the maximum degree of the quotient graph as a new structural parameter. The approach applies uniformly to both unweighted and weighted settings, yielding fixed-parameter algorithms with running time $\ell^{f(k)} \cdot n^{O(1)}$, effectively balancing solution quality and computational efficiency. The method is further successfully extended to the weighted $d$-exchange scenario.

Abstract

A vertex set $W$ in a graph $G$ is a valid $k$-swap for a vertex cover $S$ of $G$ if $W$ has size at most $k$ and $S'=(S \setminus W) \cup (W \setminus S)$, the symmetric difference of $S$ and $W$, is a vertex cover of $G$. If $|S'| < |S|$, then $W$ is improving. In LS Vertex Cover, one is given a vertex cover $S$ of a graph $G$ and wants to know if there is a valid improving $k$-swap for $S$ in $G$. In applications of LS Vertex Cover, $k$ is a very small parameter that can be set by a user to determine the trade-off between running time and solution quality. Consequently, $k$ can be considered to be a constant. Motivated by this and the fact that LS Vertex Cover is W[1]-hard with respect to $k$, we aim for algorithms with running time $\ell^{f(k)}\cdot n^{\mathcal{O}(1)}$ where $\ell$ is a structural graph parameter upper-bounded by $n$. We say that such a running time grows mildly with respect to $\ell$ and strongly with respect to $k$. We obtain algorithms with such a running time for $\ell$ being the $h$-index of $G$, the treewidth of $G$, or the modular-width of $G$. In addition, we consider a novel parameter, the maximum degree over all quotient graphs in a modular decomposition of $G$. Moreover, we adapt these algorithms to the more general problem where each vertex is assigned a weight and where we want to find a valid $d$-improving $k$-swap, that is, a valid $k$-swap which decreases the weight of the vertex cover by at least $d$.

Problem

• Vertex Cover
• Local Search
• Parameterized Complexity
• k-swap
• Graph Parameter

Innovation

• parameterized local search
• vertex cover
• structural graph parameters
• modular decomposition
• fixed-parameter tractability