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 (1+ε,[Formula presented])-approximation algorithm for every ε>0 for the Deadline TSP, where α is the approximation ratio for Deadline TSP with a constant number of deadlines (currently α=[Formula presented] by [5]) and thus significantly improving the previously best known bi-criteria approximation for that problem (a bi-criteria (1+ε,[Formula presented])-approximation algorithm for every ε>0 by Bansal et al. [1]). We also present improved bi-criteria (1+ε,[Formula presented])-approximation algorithms for the Deadline TSP on weighted trees.