FPT Approximation Algorithms for TSP on Non-Metric Graphs

The Traveling Salesman Problem (TSP) is a classic and extensively studied problem with numerous real-world applications in artificial intelligence and operations research. It is well-known that TSP admits a constant approximation ratio on metric graphs but becomes NP-hard to approximate within any computable function f (n) on general graphs. This disparity highlights a significant gap between the results on metric graphs and general graphs. Recent research has introduced some parameters to measure the "distance" of general graphs from being metric and explored Fixed-Parameter Tractable (FPT) approximation algorithms parameterized by these parameters. Two commonly studied parameters are p, the number of vertices in triangles violating the triangle inequality, and q, the minimum number of vertices whose removal results in a metric graph. In this paper, we present improved FPT approximation algorithms with respect to these two parameters. For p, we propose an FPT algorithm with a 1.5-approximation ratio, improving upon the previous ratio of 2.5. For q, we significantly enhance the approximation ratio from 11 to 3, advancing the state of the art in both cases. In addition, when p (or q) is a constant, we obtain a better approximation ratio.