A Primal-Dual Approximation Algorithm for the Asymmetric Prize Collecting TSP
Résumé
We present a primal-dual $\lceil\log{n}\rceil$-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous work on the problem [9] is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.'s heuristic [6] or the recent Asadpour et al.'s heuristic for the ATSP [2]. Depending on which of the two heuristics is used, it gives respectively $1+\lceil\log{n}\rceil$ and $3+8\frac{\log{n}}{\log{\log{n}}}$ as an approximation ratio. Our approximation ratio $\lceil\log{n}\rceil$ outperforms the first in theory and the second in practice. Moreover, unlike the method in [9], our algorithm is combinatorial.