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

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.