Deadline TSP

Abstract

We study the Deadline TSP problem. The input consists of a complete undirected graph $G=(V,E)$, a metric $c:E \rightarrow \mathbf {Z}_+$, a reward function $w:V\rightarrow \mathbf {Z}_+$, a non-negative deadline function $d:V\rightarrow \mathbf {Z}_+$, and a starting node $s \in V$. A feasible solution is a path starting at s. Given such a path and a node $v\in V$, we say that the path visits v by its deadline if the length of the prefix of the path starting at s until the first time it traverses v is at most d(v) (in particular, it means that the path traverses v). If a path visits v by its deadline, it gains the reward w(v). The objective is to find a path P starting at s that maximizes the total reward. In our work we present a bi-criteria ($1+\varepsilon ,\frac{\alpha }{1+\varepsilon }$)-approximation algorithm for every $\varepsilon >0$ for the Deadline TSP, where $\alpha$ is the approximation ratio for Deadline TSP with a constant number of deadlines (currently $\alpha =\frac{1}{3}$ by [5]) and thus significantly improving the previously best known bi-criteria approximation for that problem (a bi-criteria $(1+\varepsilon ,\frac{1}{O(\log (1/\varepsilon ))})$-approximation algorithm for every $\varepsilon >0$ by Bansal et al. [1]). We also present improved bi-criteria $(1+\varepsilon ,\frac{1}{1+\varepsilon })$-approximation algorithms for the Deadline TSP on weighted trees.

This research was supported by a grant from the GIF, the German-Israeli Foundation for Scientific Research and Development (grant number I-1366-407.6/2016).