The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3 2, even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only 4 3. Very recently, significant progress has been made for the important special case of graphic metrics, first by Oveis , and then by Momke and Svensson. In this paper, we provide an improved analysis for the approach introduced by Momke and Svensson yielding a bound of 13 9 on the approximation factor, as well as a bound of 19 12+epsilon for any epsilon>0 for a more general Travelling Salesman Path Problem in graphic metrics.
Abstract In their 1971 paper on the travelling salesman problem and minimum spanning trees, Held and Karp showed that finding an optimally weighted 1-tree is equivalent to solving a linear program for the traveling salesman problem (TSP) with only node-degree constraints and subtour elimination constraints. In this paper we show that the Held-Karp 1-trees have a certain monotonicity property: given a particular instance of the symmetric TSP with triangle inequality, the cost of the minimum weighted 1-tree is monotonic with respect to the set of nodes included. As a consequence, we obtain an alternate proof of a result of Wolsey and show that linear programs with node-degree and subtour elimination constraints must have a cost at least 2 3 OPT where OPT is the cost of the optimum solution to the TSP instance. Abstract : An O(n sup 3) heuristic algorithm is described for solving n-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition. The algorithm involves as substeps the computation of a shortest spanning tree of the graph G defining the TSP, and the finding of a minimum cost perfect matching of a certain induced subgraph of G. A worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3 2. This represents a 50 reduction over the value 2 which was the previously best known such ratio for the performance of other polynomial-growth algorithms for the TSP.