Improved Competitive Ratio for Edge-Weighted Online Stochastic Matching

We consider the edge-weighted online stochastic matching problem, in which an edge-weighted bipartite graph G=(I\cup J, E) with offline vertices J and online vertex types I is given. The online vertices have types sampled from I with probability proportional to the arrival rates of online vertex types. The online algorithm must make immediate and irrevocable matching decisions with the objective of maximizing the total weight of the matching. For the problem with general arrival rates, Feldman et al. (FOCS 2009) proposed the Suggested Matching algorithm and showed that it achieves a competitive ratio of 1-1/e \approx 0.632. The ratio has recently been improved to 0.645 by Yan (2022), who proposed the Multistage Suggested Matching (MSM) algorithm. In this paper, we propose the Evolving Suggested Matching (ESM) algorithm, and show that it achieves a competitive ratio of 0.650.Comment: To appear in WINE202

On the Perturbation Function of Ranking and Balance for Weighted Online Bipartite Matching

Ranking and Balance are arguably the two most important algorithms in the online matching literature. They achieve the same optimal competitive ratio of 1-1/e for the integral version and fractional version of online bipartite matching by Karp, Vazirani, and Vazirani (STOC 1990) respectively. The two algorithms have been generalized to weighted online bipartite matching problems, including vertex-weighted online bipartite matching and AdWords, by utilizing a perturbation function. The canonical choice of the perturbation function is f(x) = 1-e^{x-1} as it leads to the optimal competitive ratio of 1-1/e in both settings. We advance the understanding of the weighted generalizations of Ranking and Balance in this paper, with a focus on studying the effect of different perturbation functions. First, we prove that the canonical perturbation function is the unique optimal perturbation function for vertex-weighted online bipartite matching. In stark contrast, all perturbation functions achieve the optimal competitive ratio of 1-1/e in the unweighted setting. Second, we prove that the generalization of Ranking to AdWords with unknown budgets using the canonical perturbation function is at most 0.624 competitive, refuting a conjecture of Vazirani (2021). More generally, as an application of the first result, we prove that no perturbation function leads to the prominent competitive ratio of 1-1/e by establishing an upper bound of 1-1/e-0.0003. Finally, we propose the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets, and we design an optimal 1-1/e competitive algorithm by generalizing Balance

Edge-weighted Online Stochastic Matching: Beating 1−1e1-\frac1e

We study the edge-weighted online stochastic matching problem. Since Feldman, Mehta, Mirrokni, and Muthukrishnan proposed the $(1-\frac1e)$-competitive Suggested Matching algorithm, there has been no improvement for the general edge-weighted online stochastic matching problem. In this paper, we introduce the first algorithm beating the $1-\frac1e$ barrier in this setting, achieving a competitive ratio of $0.645$. Under the LP proposed by Jaillet and Lu, we design an algorithmic preprocessing, dividing all edges into two classes. Then based on the Suggested Matching algorithm, we adjust the matching strategy to improve the performance on one class in the early stage and on another class in the late stage, while keeping the matching events of different edges highly independent. By balancing them, we finally guarantee the matched probability of every single edge

Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of $0.567$ for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being $1/2$ were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is $(1-1/e)$ competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic $0.596$ competitive algorithm for the previously well studied case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based analysis. This allows us to compare against the natural benchmarks of adaptive offline algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches.Comment: Preliminary version in EC 202