On Polynomial Kernels for Traveling Salesperson Problem and its Generalizations

TL;DR

This paper investigates polynomial kernels for the Traveling Salesperson Problem (TSP) and its generalizations, providing positive results for certain structural parameters and negative results for others, advancing kernelization understanding in TSP.

Contribution

It introduces polynomial kernels for TSP based on structural parameters and establishes limitations for kernelization with respect to other parameters.

Findings

Polynomial kernels exist for TSP with respect to feedback edge set number.

Kernelization results extend to TSP generalizations with parameters like vertex cover.

Negative results show unlikely polynomial kernels for certain combined parameters.

Abstract

For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today's computation, we employ one of the most successful models of such precomputation -- the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations. We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of Subset-TSP, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.