Parameterized TSP: Beating the Average

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 $k$)

A domination algorithm for {0,1}\{0,1\}-instances of the travelling salesman problem

We present an approximation algorithm for $\{0,1\}$-instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial-time algorithm which has domination ratio $1-n^{-1/29}$. In other words, given a $\{0,1\}$-edge-weighting of the complete graph $K_n$ on $n$ vertices, our algorithm outputs a Hamilton cycle $H^*$ of $K_n$ with the following property: the proportion of Hamilton cycles of $K_n$ whose weight is smaller than that of $H^*$ is at most $n^{-1/29}$. Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial-time algorithm with domination ratio $1/2-o(1)$ for arbitrary edge-weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant $C$ such that $n^{-1/29}$ cannot be replaced by $\exp(-(\log n)^C)$ in the result above.Comment: 29 pages (final version to appear in Random Structures and Algorithms

Parameterized Traveling Salesman Problem:Beating the Average

In the traveling salesman problem (TSP), we are given a complete graph Kn together with an integer weighting w on the edges of Kn, and we are asked to find a Hamilton cycle of Kn of minimum weight. Let h(w) denote the average weight of a Hamilton cycle of Kn for the weighting w. Vizing in 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, also in 1973, and others described several other TSP heuristics satisfying this property. In this paper, we prove a considerable generalization of Vizing’s result: for each fixed k, we give an algorithm that decides whether, for any input edge weighting w of Kn, there is a Hamilton cycle of Kn 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 generalized Vizing problem is fixed-parameter tractable with respect to the parameter k)