Online Matching with Convex Delay Costs

TL;DR

This paper introduces convex-MPMD, a new online matching problem with convex delay costs, and presents an optimal deterministic algorithm with competitive ratios depending on the metric space size and aspect ratio.

Contribution

It formulates convex-MPMD to better model impatient requests and devises a new deterministic algorithm with proven optimal competitive ratios for this problem.

Findings

The algorithm achieves an $O(k)$ competitive ratio on $k$-point uniform metrics.

The algorithm achieves an $O(k riangle)$ ratio on metrics with aspect ratio $ riangle$.

Lower bounds show the algorithm's competitive ratio is optimal.

Abstract

We investigate the problem of Min-cost Perfect Matching with Delays (MPMD) in which requests are pairwise matched in an online fashion with the objective to minimize the sum of space cost and time cost. Though linear-MPMD (i.e., time cost is linear in delay) has been thoroughly studied in the literature, it does not well model impatient requests that are common in practice. Thus, we propose convex-MPMD where time cost functions are convex, capturing the situation where time cost increases faster and faster. Since the existing algorithms for linear-MPMD are not competitive any more, we devise a new deterministic algorithm for convex-MPMD problems. For a large class of convex time cost functions, our algorithm achieves a competitive ratio of $O(k)$ on any $k$-point uniform metric space, or $O(k\Delta)$ when the metric space has aspect ratio $\Delta$. Moreover, we prove lower bounds for the competitive ratio of deterministic and randomized algorithms, indicating that our deterministic algorithm is optimal. This optimality uncover a substantial difference between convex-MPMD and linear-MPMD, since linear-MPMD allows a deterministic algorithm with constant competitive ratio on any uniform metric space.