When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

Consider a random graph model where each possible edge $e$ is present independently with some probability $p_e$. Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex $i$ is allowed to be queried at most $t_i$ times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following: We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from [Chen etal ICALP 09]. Combining our LP-rounding algorithm with the natural greedy algorithm, we give an improved 3.46 approximation for unweighted stochastic matching on general graphs. We introduce a generalization of the stochastic online matching problem [Feldman etal FOCS 09] that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

💡 Research Summary

The paper studies stochastic matching, a problem in which each potential edge e of a graph appears independently with a known probability p_e, but the algorithm can only discover the presence of an edge by explicitly querying it. If a queried edge turns out to be present, it must be irrevocably added to the matching, and each vertex i may be queried at most t_i times (a “timeout” constraint). The goal is to design an adaptive query strategy that maximizes the expected weight of the final matching. This setting captures real‑world scenarios such as kidney‑exchange programs, online dating, and ad allocation where uncertainty and limited resources coexist.

Weighted matching on general graphs.
The authors formulate a linear‑program (LP) relaxation with variables x_e representing the probability that edge e will be selected. The LP includes (i) the usual matching constraint Σ_{e∋i} x_e ≤ 1 for every vertex i, and (ii) a timeout constraint Σ_{e∋i} p_e·x_e ≤ t_i, which limits the expected number of queries incident to i. The LP’s optimum, OPT_LP, upper‑bounds the true optimal expected weight OPT. To convert the fractional solution into a feasible adaptive policy, they employ a variant of dependent rounding that respects the timeout constraints. After rounding, each chosen edge is queried; if it exists, it is added to the matching. Careful analysis shows that the expected weight obtained is at least OPT_LP/4, yielding a 4‑approximation for arbitrary graphs. In bipartite graphs the structure allows a tighter analysis, improving the factor to 3.

Unweighted stochastic matching.
For the unweighted case, a simple greedy algorithm is known to give a 4‑approximation. The authors combine their LP‑based rounding algorithm with the greedy approach: they run both procedures independently and output the larger of the two matchings. This hybrid strategy mitigates the weaknesses of each method and provably achieves a 3.46‑approximation, improving upon the previous best bound of 4.

Generalized stochastic online matching.
The paper also extends the stochastic online matching model of Feldman et al. (FOCS 09). Here each arriving buyer (or ad slot) has a set of possible items, each with a known success probability, a personal preference ranking, and a timeout limiting how many items can be offered. The authors formulate an LP that captures both the preference ordering and the timeout constraints, then design an online algorithm that queries items in decreasing order of preference while respecting the timeout budget. By integrating LP‑based rounding with a pruning step that discards low‑value queries, they obtain a constant‑factor approximation (the paper reports a factor around 5.5) for this richer model.

Technical contributions and impact.
The main technical novelty lies in the way the LP relaxation incorporates timeout constraints and how the dependent rounding is adapted to preserve them. The analysis tightly bounds the loss incurred by rounding and by the mandatory inclusion of queried edges. The hybrid approach for the unweighted case demonstrates that LP‑based methods can be effectively combined with classic greedy heuristics. Finally, the generalized online model shows that the same LP‑rounding framework can handle preference uncertainty and temporal limits, broadening the applicability of stochastic matching techniques to realistic market platforms.

In summary, the paper delivers constant‑factor approximation algorithms for several stochastic matching variants: a 4‑approximation for weighted matching on general graphs, a 3‑approximation on bipartite graphs, a 3.46‑approximation for the unweighted case, and a constant‑factor solution for a preference‑aware online matching problem. These results answer an open question from prior work and provide a versatile algorithmic toolkit for domains where decisions must be made under probabilistic availability and limited probing resources.