Parameterized TSP: Beating the Average

TL;DR

This paper extends Vizing's classic result by providing a fixed-parameter tractable algorithm that determines whether a Hamilton cycle exists with weight significantly below the average, for any edge weighting of the complete graph.

Contribution

It introduces a polynomial-time algorithm for the generalized Vizing problem, deciding if a Hamilton cycle of weight at most the average minus a fixed parameter exists.

Findings

Algorithm runs in polynomial time for fixed k

Decides existence of Hamilton cycle below average minus k

Generalizes Vizing's original result

Abstract

In the Travelling Salesman Problem (TSP), we are given a complete graph $K_n$ together with an integer weighting $w$ on the edges of $K_n$, and we are asked to find a Hamilton cycle of $K_n$ of minimum weight. Let $h(w)$ denote the average weight of a Hamilton cycle of $K_n$ for the weighting $w$. Vizing (1973) asked whether there is a polynomial-time algorithm which always finds a Hamilton cycle of weight at most $h(w)$. He answered this question in the affirmative and subsequently Rublineckii (1973) and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalisation of Vizing's result: for each fixed $k$, we give an algorithm that decides whether, for any input edge weighting $w$ of $K_n$, there is a Hamilton cycle of $K_n$ of weight at most $h(w)-k$ (and constructs such a cycle if it exists). For $k$ fixed, the running time of the algorithm is polynomial in $n$, where the degree of the polynomial does not depend on $k$ (i.e., the generalised Vizing problem is fixed-parameter tractable with respect to the parameter $k$).