An improved analysis of the M\"omke-Svensson algorithm for graph-TSP on subquartic graphs

TL;DR

This paper refines the analysis of the M"omke-Svensson algorithm for graph-TSP, establishing a tighter approximation ratio of 25/18 specifically for subquartic graphs, improving understanding of its performance.

Contribution

The paper proves that the M"omke-Svensson algorithm has an approximation guarantee of at most 25/18 on subquartic graphs, refining previous bounds and analyzing circulation costs with new methods.

Findings

M"omke-Svensson algorithm has a 25/18 approximation ratio on subquartic graphs.

New methods to upper bound circulation costs improve analysis accuracy.

The result applies to graphs with subquartic support in the LP relaxation.

Abstract

M\"omke 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 M\"omke 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 M\"omke-Svensson algorithm is tight. We show that this is not the case and that M\"omke 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.