FIXED RATIO POLYNOMIAL TIME APPROXIMATION ALGORITHM FOR THE PRIZE-COLLECTING ASYMMETRIC TRAVELING SALESMAN PROBLEM

We develop the first fixed-ratio approximation algorithm for the well-known Prize-Collecting Asymmetric Traveling Salesman Problem,  which has numerous valuable applications in operations research. An instance of this problem is given by a complete node- and edge-weighted digraph \(G\). Each node of the graph \(G\) can either be visited by the resulting route or skipped, for some penalty, while the arcs of \(G\)  are weighted by non-negative transportation costs that fulfill the triangle inequality constraint. The goal is to find a closed walk that minimizes the total transportation costs augmented by the accumulated penalties. We show that an arbitrary \(\alpha\)-approximation algorithm for the Asymmetric Traveling Salesman Problem induces an \((\alpha+1)\)-approximation for the problem in question. In particular, using the recent \((22+\varepsilon)\)-approximation algorithm of V. Traub and J. Vygen that improves the seminal result of  O. Svensson, J. Tarnavski, and L. Végh,  we obtain \((23+\varepsilon)\)-approximate solutions for the problem.