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Abstract-The online matching problem has received significant attention in recent years because of its connections to allocation problems in Internet advertising, crowd-sourcing, etc. In these real-world applications, the typical goal is not to maximize the number of allocations; rather it is to maximize the number of "successful" allocations, where success of an allocation is governed by a stochastic process which follows the allocation. To address such applications, we propose and study the online matching problem with stochastic rewards (called the ONLINE STOCHASTIC MATCHING problem) in this paper. Our problem also has close connections to the existing literature on stochastic packing problems; in fact, our work initiates the study of online stochastic packing problems.We give a deterministic algorithm for the ONLINE STOCHAS-TIC MATCHING problem whose competitive ratio converges to (approximately) 0.567 for uniform and vanishing probabilities. We also give a randomized algorithm which outperforms the deterministic algorithm for higher probabilities. Finally, we complement our algorithms by giving an upper bound on the competitive ratio of any algorithm for this problem. This result shows that the best achievable competitive ratio for the ONLINE STOCHASTIC MATCHING problem is provably worse than that for the (non-stochastic) online matching problem.

We give new upper and lower bounds for the dynamic set cover problem. First, we give a (1 + ε)f -approximation for fully dynamic set cover in O(f 2 log n/ε 5 ) (amortized) update time, for any ǫ > 0, where f is the maximum number of sets that an element belongs to. In the decremental setting, the update time can be improved to O(f 2 /ε 5 ), while still obtaining an (1 + ε)f -approximation. These are the first algorithms that obtain an approximation factor linear in f for dynamic set cover, thereby almost matching the best bounds known in the offline setting and improving upon the previous best approximation of O(f 2 ) in the dynamic setting.To complement our upper bounds, we also show that a linear dependence of the update time on f is necessary unless we can tolerate much worse approximation factors. Using the recent distributed PCP-framework, we show that any dynamic set cover algorithm that has an amortized update time of O(f 1−ε ) must have an approximation factor that is Ω(n δ ) for some constant δ > 0 under the Strong Exponential Time Hypothesis.

In this paper, we study the set cover problem in the fully dynamic model. In this model, the set of active elements, i.e., those that must be covered at any given time, can change due to element arrivals and departures. The goal is to maintain an algorithmic solution that is competitive with respect to the current optimal solution. This model is popular in both the dynamic algorithms and online algorithms communities. The difference is in the restriction placed on the algorithm: in dynamic algorithms, the running time of the algorithm making updates (called update time) is bounded, while in online algorithms, the number of updates made to the solution (called recourse) is limited.We give new results in both settings (all recourse and update time bounds are amortized):• In the update time setting, we obtain O(log n)-competitiveness with O(f log n) update time, and O(f 3 )-competitiveness with O(f 2 ) update time. The O(log n)-competitive algorithm is the first one to achieve a competitive ratio independent of f in this setting. The second result improves on previous work by removing an O(log n) factor in the update time bound. This has an important consequence: we obtain the first deterministic constant-competitive, constant update time algorithm for fully-dynamic vertex cover.• In the recourse setting, we show a competitive ratio of O(min{log n, f }) with constant recourse. The most relevant previous result is the O(log m log n) bound for online set cover in the insertion-only model with no recourse. Note that we can match the best offline bounds with O(1) recourse, something that is impossible in the classical online model.These results also yield, as corollaries, new results for the maximum k-coverage problem and the non-metric facility location problem in the fully dynamic model. Our results are based on two algorithmic frameworks in the fully-dynamic model that are inspired by the classic greedy and primal-dual algorithms for offline set cover. We show that both frameworks can be used for obtaining both recourse and update time bounds, thereby demonstrating algorithmic techniques common to these strands of research.

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