Online Bottleneck Matching

Abstract

We consider the online bottleneck matching problem, where k serververtices lie in a metric space and k request-vertices that arrive over time each must immediately be permanently assigned to a server-vertex. The goal is to minimize the maximum distance between any request and its server. Because no algorithm can have a competitive ratio better than O(k) for this problem, we use resource augmentation analysis to examine the performance of three algorithms: the naive GREEDY algorithm, PERMUTATION, and BALANCE. We show that while the competitive ratio of GREEDY improves from exponential (when each server-vertex has one server) to linear (when each server-vertex has two servers), the competitive ratio of PERMU-TATION remains linear when an extra server is introduced at each server-vertex. The competitive ratio of BALANCE is also linear with an extra server at each server-vertex, even though it has been shown that an extra server makes it constant-competitive for the min-weight matching problem.

Serve or skip: the power of rejection in online bottleneck matching

“…Along the way, we also provide an upper bound on the competitive-ratio of the basic GREEDY algorithm for online bottleneck matching (with no resource augmentation). We show that GREEDY is (2 n − 1)-competitive, which essentially matches the existing lower bound given in Anthony and Chung (2014).…”

“…This theorem essentially closes the gap that remained in Anthony and Chung (2014) between the lower bound on the competitiveness of GREEDY (of 2 n−1 ) and the upper bound.…”

“…The matching lower bound was given in Anthony and Chung (2014). Due to its similarity to the proof of Theorem 2, we defer the proof of this fact to Appendix B.…”

“…In the literature on matching, the resource augmentation has been in the form of doubling (Kalyanasundaram and Pruhs (2000b); Anthony and Chung (2014)) or incrementing (Chung et al (2008)) server site capacities. In the online machine scheduling literature, it has taken the form of augmenting the processing speed of machines (Kalyanasundaram and Pruhs (2000a); Phillips et al (2002)).…”

“…It was demonstrated in Anthony and Chung (2014) that PERMUTATION and BALANCE both fail to be better than O(n)-halfOPT-competitive for the bottleneck objective. Additionally, algorithms (such as BALANCE) that are designed based on the assumption that each server location has the capacity to serve at least two requests, do not apply to the classic matching setting of one server per request.…”