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An improved analysis of the Mömke-Svensson algorithm for graph-TSP on subquartic graphs

Alantha Newman 1, 2
1 G-SCOP_OC - Optimisation Combinatoire
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : Recently, Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yielded a 4 3 -approximation guarantee on subcubic graphs as well as a substantial improvement over the 3 2 -approximation guarantee of Christofides' algorithm on general graphs. The crux of their approach is to compute an upper bound on the minimum cost of a circulation in a particular network, C(G, T), where G is the input graph and T is a carefully chosen spanning tree. The cost of this circulation is directly related to the number of edges in a tour output by their algorithm. Mucha subsequently improved the analysis of the circulation cost, proving that Mömke and Svensson's algorithm for graph-TSP has an approximation ratio of at most 13 9 on general graphs. This analysis of the circulation is local, and vertices with degree four and five can contribute the most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph with degree at most four at each vertex) for which Mucha's analysis of the Mömke-Svensson algorithm is tight. We show that this is not the case and that Mömke and Svensson's algorithm for graph-TSP has an approximation guarantee of at most 46 33 on subquartic graphs. To prove this, we present a different method to upper bound the minimum cost of a circulation on the network C(G, T). Our approximation guarantee actually holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.
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Alantha Newman. An improved analysis of the Mömke-Svensson algorithm for graph-TSP on subquartic graphs. European Symposium on Algorithms, Sep 2014, Wroclaw, Poland. pp.737 - 749. ⟨hal-01119635⟩

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