Online Stochastic Matching: New Algorithms and Bounds

Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1/e) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the “known I.I.D. model” where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to Haeupler, Mirrokni and Zadimoghaddam to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of Jaillet and Lu to 0.7299. We also consider an extension of stochastic rewards, a variant where each edge has an independent probability of being present. For the setting of stochastic rewards with non-integral arrival rates, we present a simple optimal non-adaptive algorithm with a ratio of 1 - 1/e. For the special case where each edge is unweighted and has a uniform constant probability of being present, we improve upon 1 - 1/e by proposing a strengthened LP benchmark.

💡 Research Summary

This paper studies online bipartite matching under the well‑known “known I.I.D. model,” where offline vertices (advertisers) are fixed and online vertices (impressions) arrive one by one according to a known probability distribution. The authors focus on the case of integral arrival rates and consider three variants: (i) edge‑weighted matching, (ii) vertex‑weighted matching, and (iii) stochastic‑reward matching where each edge exists with an independent probability.

For the edge‑weighted case, the state‑of‑the‑art competitive ratio was 0.667 (achieved by Haeupler, Mirrokni and Zadimoghaddam). The authors improve this to 0.705. Their approach begins with a linear program (LP) that captures the offline optimum and adds two novel constraints: (a) each vertex can be matched with probability at most 1‑1/e, and (b) any two edges incident to the same vertex can be simultaneously selected with probability at most 1‑1/e². Solving this strengthened LP yields a fractional solution f*. The key technical contribution is a “random move” procedure that converts f* into a collection of k matchings {M₁,…,M_k} whose edge‑incidence values belong to the discrete set {0,1/k,…,1}. By choosing k=2 and using the matchings in a correlated fashion (first try an edge from M₁, if unavailable try the corresponding edge from M₂), the probability that any edge is used is at least 0.705·f*_e. Lemma 2 shows that this edge‑wise lower bound directly translates into a global competitive ratio of 0.705.

In the vertex‑weighted setting, where each offline vertex u carries a weight w_u and all incident edges share that weight, the same LP framework is adapted. The authors again apply the random‑move rounding to obtain two (or more) matchings, but now the selection probability of a vertex is proportional to its weight. This yields a competitive ratio of 0.7299, improving the previous best of 0.7250 by Jaillet and Lu.

The third contribution concerns stochastic rewards. Each edge e appears with probability p_e when probed, and the algorithm must commit immediately upon a successful probe (query‑commit model). For non‑integral arrival rates, the authors present a simple non‑adaptive algorithm that follows a predetermined probing order derived from an LP with additional constraints of the form ∑_{e∈S} f_e ≤ (1‑exp(‑|S|p))/p for small sets S (|S| ≤ 2/p). This LP can be solved in polynomial time because the number of constraints is constant for fixed p. The resulting non‑adaptive policy achieves the optimal competitive ratio of 1‑1/e.

When all edges share a uniform existence probability p, the authors go further: by strengthening the LP benchmark they obtain a ratio strictly larger than 1‑1/e. The improvement is most pronounced for small p, where the uniform‑probability assumption allows tighter control over the probability mass assigned to small edge subsets.

Across all variants, the paper’s central methodological theme is the combination of (1) a strengthened LP that captures both matching feasibility and stochastic arrival constraints, (2) a randomized rounding that produces a k‑partition of the fractional solution into integral matchings with controlled overlap, and (3) a careful online decision rule that exploits the structure of these matchings to reduce edge‑collision probability. The analysis hinges on showing that each edge (or vertex) is selected with probability at least α·f*_e (or α·F*_u), which by linearity of expectation yields an overall competitive ratio of at least α.

The authors also discuss computational aspects: the LP can be solved in O(|E|²) time in the worst case, and the random‑move rounding runs in near‑linear time. The algorithms are non‑adaptive for the stochastic‑reward case, making them attractive for practical systems where online decisions must be pre‑specified.

In summary, the paper delivers four concrete advances: (i) a 0.705 competitive ratio for edge‑weighted online matching with integral arrivals, (ii) a 0.7299 ratio for vertex‑weighted matching, (iii) an optimal 1‑1/e ratio for stochastic‑reward matching with arbitrary arrival rates via a simple non‑adaptive policy, and (iv) a strictly better‑than‑1‑1/e ratio for the uniform‑probability stochastic‑reward setting through a strengthened LP benchmark. These results push the frontier of online matching theory and provide algorithmic tools that could be directly applied to real‑world ad allocation, ride‑sharing, and other real‑time resource allocation problems.