Relaxing the Irrevocability Requirement for Online Graph Algorithms

Online graph problems are considered in models where the irrevocability requirement is relaxed. Motivated by practical examples where, for example, there is a cost associated with building a facility and no extra cost associated with doing it later, we consider the Late Accept model, where a request can be accepted at a later point, but any acceptance is irrevocable. Similarly, we also consider a Late Reject model, where an accepted request can later be rejected, but any rejection is irrevocable (this is sometimes called preemption). Finally, we consider the Late Accept/Reject model, where late accepts and rejects are both allowed, but any late reject is irrevocable. For Independent Set, the Late Accept/Reject model is necessary to obtain a constant competitive ratio, but for Vertex Cover the Late Accept model is sufficient and for Minimum Spanning Forest the Late Reject model is sufficient. The Matching problem has a competitive ratio of 2, but in the Late Accept/Reject model, its competitive ratio is 3/2

Improved Approximation Algorithms for PRIZE-COLLECTING STEINER TREE and TSP

Abstract — We study the prize-collecting versions of the Steiner tree, traveling salesman, and stroll (a.k.a. PATH-TSP) problems (PCST, PCTSP, and PCS, respectively): given a graph (V, E) with costs on each edge and a penalty (a.k.a. prize) on each node, the goal is to find a tree (for PCST), cycle (for PCTSP), or stroll (for PCS) that minimizes the sum of the edge costs in the tree/cycle/stroll and the penalties of the nodes not spanned by it. In addition to being a useful theoretical tool for helping to solve other optimization problems, PCST has been applied fruitfully by AT&T to the optimization of real-world telecommunications networks. The most recent improvements for the first two problems, giving a 2-approximation algorithm for each, appeared first in 1992. (A 2-approximation for PCS appeared in 2003.) The natural linear programming (LP) relaxation of PCST has an integrality gap of 2, which has been a barrier to further improvements for this problem. We present (2 − ɛ)-approximation algorithms for all three problems, connected by a unified technique for improving prizecollecting algorithms that allows us to circumvent the integrality gap barrier. 1