Deadline TSP

We study the Deadline TSP problem. The input consists of a complete undirected graph G = (V,E), a metric c: E → Z₊, a reward function w: V → Z₊, a non-negative deadline function d: V → Z₊, and a starting node s ∈ V. A feasible solution is a path starting at s. Given such a path and a node v ∈ 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 (Formula presented)-approximation algorithm for every ɛ > 0 for the Deadline TSP, where a is the approximation ratio for Deadline TSP with a constant number of deadlines (currently α = 1/3 by [5]) and thus significantly improving the previously best known bi-criteria approximation for that problem (a bi-criteria (Formula presented))-approximation algorithm for every ɛ > 0 by Bansal et al. [1]). We also present improved bi-criteria (Formula presented)-approximation algorithms for the Deadline TSP on weighted trees.