Calum MacRury

Brian Brubach, Nathaniel Grammel, Will Ma et al.

Matching is one of the most fundamental and broadly applicable problems across many domains. In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models. Here, each edge in the graph has a known, independent probability of existing derived from some prediction. Algorithms must probe edges to determine existence and match them irrevocably if they exist. Further, each vertex may have a patience constraint denoting how many of its neighboring edges can be probed. We present new ordered contention resolution schemes yielding improved approximation guarantees for some of the foundational problems studied in this area. For stochastic matching with patience constraints in general graphs, we provide a 0.382-approximate algorithm assuming each vertex has patience at least $2$. Under this assumption, we improve upon the previous best 0.31-approximation of Baveja et al. (2018). When the vertices do not have patience constraints, we describe a 0.432-approximate random order probing algorithm with several corollaries such as an improved guarantee for the Prophet Secretary problem under Edge Arrivals. Finally, for the special case of bipartite graphs with unit patience constraints on one of the partitions, we show a 0.632-approximate algorithm that improves on the recent $1/3$-guarantee of Hikima et al. (2021).

Mahsa Derakhshan, Andisheh Ghasemi, Calum MacRury

Matching problems under uncertainty arise in applications such as kidney exchange, hiring, and online marketplaces. A decision-maker must sequentially explore potential matches under local exploration constraints, while committing irrevocably to successful matches as they are revealed. The query-commit matching problem captures these challenges by modeling edges that succeed independently with known probabilities and must be accepted upon success, subject to vertex patience (time-out) constraints limiting the number of incident queries. In this work, we introduce the action-reward query-commit matching problem, a strict generalization of query-commit matching in which each query selects an action from a known action space, determining both the success probability and the reward of the queried edge. If an edge is queried using a chosen action and succeeds, it is irrevocably added to the matching, and the corresponding reward is obtained; otherwise, the edge is permanently discarded. We study the design of approximation algorithms for this problem on bipartite graphs. This model captures a broad class of stochastic matching problems, including the sequential pricing problem introduced by Pollner, Roghani, Saberi, and Wajc (EC~2022). On the positive side, Pollner et al. designed a polynomial-time approximation algorithm achieving a ratio of $0.426$ in the one-sided patience setting, which degrades to $0.395$ when both sides have bounded patience. In this work, we design computationally efficient algorithms for the action-reward query-commit in one-sided and two-sided patience settings, achieving approximation ratios of $1-1/e \approx 0.63$ and $\frac{1}{27}\!\left(19-67/e^3\right) \approx 0.58$ respectively. These results improve the state of the art for the sequential pricing problem, surpassing the previous guarantees of $0.426$ and $0.395$.