New approximation algorithms for (1,2)-TSP

We give faster and simpler approximation algorithms for the (1, 2)-TSP problem, a well-studied variant of the traveling salesperson problem where all distances between cities are either 1 or 2. Our main results are two approximation algorithms for (1, 2)-TSP, one with approximation factor 8/7 and run time O(n3) and the other having an approximation guarantee of 7/6 and run time O(n2.5). The 8/7-approximation matches the best known approximation factor for (1, 2)-TSP, due to Berman and Karpinski (SODA 2006), but considerably improves the previous best run time of O(n9). Thus, ours is the first improvement for the (1, 2)-TSP problem in more than 10 years. The algorithm is based on combining three copies of a minimum-cost cycle cover of the input graph together with a relaxed version of a minimum weight matching, which allows using “half-edges”. The resulting multigraph is then edge-colored with four colors so that each color class yields a collection of vertex-disjoint paths. The paths from one color class can then be extended to an 8/7-approximate traveling salesperson tour. Our algorithm, and in particular its analysis, is simpler than the previously best 8/7-approximation. The 7/6-approximation algorithm is similar and even simpler, and has the advantage of not using Hartvigsen's complicated algorithm for computing a minimum-cost triangle-free cycle cover.