A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

We present a primal-dual ⌈log(n)⌉-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 algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294–301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.’s heuristic (Frieze et al. in Networks 12:23–39, 1982) or the recent Asadpour et al.’s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+⌈log(n)⌉ and $3+ 8\frac{\log(n)}{\log(\log(n))}$ as an approximation ratio. Our algorithm achieves an approximation ratio of ⌈log(n)⌉ which is weaker than $3+ 8\frac{\log(n)}{\log(\log(n))}$ but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.