A quadratic-order problem kernel for the traveling salesman problem parameterized by the vertex cover number

TL;DR

This paper introduces a quadratic-size problem kernel for the graphical traveling salesman problem parameterized by vertex cover number, enabling efficient approximation solutions.

Contribution

It presents a novel quadratic-order problem kernel for GTSP based on vertex cover number, improving preprocessing and approximation guarantees.

Findings

Kernel size is $ au^2+ au$ vertices, where $ au$ is the vertex cover number.

Approximate solutions on the kernel extend to the original problem with the same approximation ratio.

The approach enhances preprocessing efficiency for GTSP instances.

Abstract

The NP-hard graphical traveling salesman problem (GTSP) is to find a closed walk of total minimum weight that visits each vertex in an undirected edge-weighted and not necessarily complete graph. We present a problem kernel with $\tau^2+\tau$ vertices for GTSP, where $\tau$ is the vertex cover number of the input graph. Any $\alpha$-approximate solution for the problem kernel also gives an $\alpha$-approximate solution for the original instance, for any $\alpha\geq1$.