2014/07/09 by Alantha Newman, Newman, Alantha
Computer Science · #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #cs.DS
paper · pdf · doi:10.48550/arxiv.1407.2524
Journal version
arxiv created 2020/03/03 · arxiv updated 2020/03/04
Mömke and Svensson presented a beautiful new approach for the traveling salesman problem on a graph metric (graph-TSP), which yields 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 25/18 on subquartic graphs. To prove this, we present different methods to upper bound the minimum cost of a circulation on the network C(G,T). Our approximation guarantee holds for all graphs that have an optimal solution to a standard linear programming relaxation of graph-TSP with subquartic support.