An Improved Analysis of the Mömke-Svensson Algorithm for Graph-TSP on Subquartic Graphs

Abstract

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.

A 97-approximation algorithm for Graphic TSP in cubic bipartite graphs

“…Currently, the best approximation algorithm for Graphic TSP in cubic graphs is due to Correa, Larré, and Soto [5], whose algorithm achieves an approximation factor of ( 43 − 1 61236 ) for 2-edge-connected cubic graphs. Similarly, a 4 3 -approximation was recently obtained for instances of sub-quartic graphs [14]. Progress in approximating Graphic TSP in cubic graphs also relates to traditional graph theory, as Barnette's conjecture [2] states that all bipartite, planar, 3-connected, cubic graphs are Hamiltonian.…”

On approximating a new generalization of traveling salesman problem

“…Moreover, further improvements have been achieved for certain special cases of graphic TSP. Newman [30] provided a new analysis to show that Mömke-Svensson algorithm [27] can achieve a performance guarantee of for subquartic graphs. Combining with the polyhedral idea, Boyd et al [31] obtained a performance ratio of 1.4 and for the subcubic and cubic graphs, respectively.…”