Reducing Prize-Collecting Stroll and Related Routing Problems to Prize-Collecting TSP

Abstract

The prize-collecting stroll is the path version of the prize-collecting TSP. Given a complete metric graph, two prescribed terminal vertices $s,t$, and nonnegative penalties on vertices, the prize-collecting stroll asks for an $s$-$t$ tour minimizing the length of the tour plus the total penalty of vertices that are not visited by the $s,t$ tour. We study a common generalization of the prize-collecting stroll and several related prize-collecting routing problems, which we call the prize-collecting-$Φ$-TSP. In this model, $Φ$ specifies a set of prescribed vertices alongside their parity and connectivity requirements. We show that, if a $ρ$-approximation algorithm for the prize-collecting TSP is available, then, for any $\varepsilon>0$, there is a polynomial time $(ρ+\varepsilon)$-approximation algorithm for the prize-collecting-$Φ$-TSP when the number of prescribed vertices is bounded by a fixed constant. This yields a better-than-$1.6$-approximation algorithm for the prize-collecting stroll, improving the previous best-known approximation guarantee of $1.6662$.