An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality

Given a complete edge-weighted graph G, we present a polynomial time algorithm to compute a degree-four-bounded spanning Eulerian subgraph of 2G that has at most 1.5 times the weight of an optimal TSP solution of G. Based on this algorithm and a novel use of orientations in graphs, we obtain a ( 3 β / 4 + 3 β 2 / 4 ) -approximation algorithm for TSP with β-relaxed triangle inequality (β-TSP), where β ? 1 . A graph G is an instance of β-TSP, if it is a complete graph with edge weights c : E ( G ) ? Q ? 0 that are restricted as follows. For each triple of vertices u , v , w ? V ( G ) , c ( { u , v } ) ? β ( c ( { u , w } ) + c ( { w , v } ) ) . Improved approximation algorithm for TSP with relaxed triangle inequality.Approximate Min. weight degree-4-bounded Eulerian subgraph of doubled complete graph.Application of degree bounded matroids, b-matchings, and a new orientation technique.

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