Parameterized Approximation Algorithms for TSP

Abstract

We study the Traveling Salesman problem (TSP), where given a complete undirected graph G = (V,E) with n vertices and an edge cost function c:E↦R_{⩾0}, the goal is to find a minimum-cost cycle visiting every vertex exactly once. It is well-known that unless P = NP, TSP cannot be approximated in polynomial time within a factor of ρ(n) for any computable function ρ, while the metric case of TSP, that the edge cost function satisfies the △-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized ρ-approximation algorithm returns a ρ-approximation solution in f(k)⋅|I|^O(1) time, where f is a computable function and k is a parameter of the input I. We introduce two parameters, which measure the distance of a given TSP-instance from the metric case, and achieve the following two results:  
- A 3-approximation algorithm for TSP in O((3k₁)! 8^k₁⋅ n²+n³) time, where k₁ is the number of triangles in which the edge costs violate the △-inequality. 
- A 3-approximation algorithm for TSP in O(n^O(k₂)) time and a (6k₂+9)-approximation algorithm for TSP in O(k₂^O(k₂)⋅n³) time, where k₂ is the minimum number of vertices, whose removal results in a metric graph.
To our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs.