🤖 AI Summary
This paper studies the online minimum-cost matching problem on a line: requests arrive sequentially and must be irrevocably matched to unused servers upon arrival, aiming to minimize total matching distance. To address the gap between the empirically strong performance of the greedy algorithm and its lack of theoretical guarantees, we analyze its competitive ratio under three stochastic input models. First, we establish— for the first time—a constant competitive ratio for the greedy algorithm under the two-sided uniform random model. Second, we improve the best-known competitive ratio in the server-surplus setting from $O(log^3 n)$ to a constant. Third, we prove that one-sided randomness only ensures a $Theta(log n)$ competitive ratio and provide a tight lower bound. Through probabilistic analysis, semi-random modeling, and adversarial lower-bound constructions, our work delivers the first rigorous theoretical justification for the empirical superiority of the greedy algorithm, significantly advancing beyond prior state-of-the-art results.
📝 Abstract
In the online minimum cost matching problem, there are n servers and, at each of n time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched, with the goal of minimizing the sum of the distances between the matched pairs. Online minimum cost matching is a central problem in applications such as ride-hailing platforms and food delivery services. Despite achieving a worst-case competitive ratio that is exponential in n even on the line, the simple greedy algorithm, which matches each request to its nearest available server, performs well in practice and has a number of attractive features such as strategyproofness. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy on the line over instances that are at least partially random. When both the requests and the servers are drawn uniformly and independently from [0, 1], we obtain a constant competitive ratio for greedy, which improves over the previously best-known bound of [EQUATION] for greedy in this setting. We also show that this constant competitive ratio also holds in the excess supply setting where there is a linear excess of servers, which improves over the previously best-known bound of O(log3 n) for greedy in this setting. We moreover show that in the semi-random model where the requests are still drawn uniformly and independently but where the servers are chosen adversarially, greedy achieves an O(log n) competitive ratio. Even though this one-sided randomness allows a large improvement in greedy's competitive ratio compared to the model where the requests are fully adversarial or arrive in a random order, we show that it is not sufficient to obtain a constant competitive ratio by giving a tight Ω(log n) lower bound. These results invite further investigation about how much randomness is necessary and sufficient to obtain strong theoretical guarantees for the greedy algorithm for online minimum cost matching, on the line and beyond. A full version of this paper can be found at https://arxiv.org/abs/2210.03166.
