# Online submodular maximization: beating 1/2 made simple

@article{Buchbinder2018OnlineSM,
title={Online submodular maximization: beating 1/2 made simple},
author={Niv Buchbinder and Moran Feldman and Mohit Garg},
journal={Mathematical Programming},
year={2018},
pages={1-21}
}
• Published 15 July 2018
• Computer Science
• Mathematical Programming
The Submodular Welfare Maximization problem (SWM) captures an important subclass of combinatorial auctions and has been studied extensively. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. For this setting, Korula et al. (SIAM J Comput 47(3):1056–1086, 2018) were able to show that the greedy algorithm is 0.5052-competitive when the items arrive in a uniformly random order. Unfortunately, however…
17 Citations
• Computer Science
Mathematical Programming
• 2020
An upper bound of 0.574 is proved on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of 1-1/e.
• Computer Science
ArXiv
• 2019
This work manages to adapt Sieve-Streaming to non-monotone objective functions by introducing a "just right" amount of randomness into it, and gets a semi-streaming polynomial time $4.282$-approximation algorithm for non- monotone objectives.
• Computer Science
Math. Oper. Res.
• 2022
Subsampling is proposed as a unified algorithmic technique for submodular maximization in centralized and online settings and it is shown that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods.
• Computer Science
Mathematics of Operations Research
• 2022
We study the problem of maximizing a nonmonotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi) streaming algorithm that
• Computer Science
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
• 2020
This paper presents the first online algorithm that breaks the long-standing 1/2 barrier and achieves a competitive ratio of at least 0.5086, and can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting.
• Computer Science
ITCS
• 2019
A polynomial time algorithm is given that substantially improves upon the previously best known approximation factor of $1/2 + 8 \times 10^{-14}$ [Norouzi-Fard et al. 2018] that used a memory buffer of size $O(k \log k)$.
• Computer Science
ArXiv
• 2022
This work provides the theoretical foundation for computing the weights approximately and shows that, with the proposed randomized data structures, the weights can be computed in sublinear time while still preserving the competitive ratio of the matching algorithm.
• Computer Science
ArXiv
• 2022
It is proved that the canonical perturbation function is the unique optimal perturbations function for vertex-weighted online bipartite matching, and the online budget-additive welfare maximization problem that is intermediate between AdWords and AdWords with unknown budgets is proposed.
• Computer Science
ALT
• 2020
A modified adaptive sampling procedure is given that obtains a better approximation ratio and is shown how to perform adaptive sampling when data has outliers, thus rendering distance-based sampling prone to picking the outliers.
• Computer Science, Mathematics
• 2018
This algorithm is the first deterministic algorithm known to improve over the 1/2approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.

## References

SHOWING 1-10 OF 38 REFERENCES

• Computer Science, Economics
SODA
• 2013
It is proved that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless NP = RP, which proves that Greedy provides the optimal competitive ratio.
• Mathematics, Computer Science
SIAM J. Comput.
• 2014
This work presents an optimal, combinatorial $1-1/e$ approximation algorithm for monotone submodular optimization over a matroid constraint, and generalizes to the case where the monot one sub modular function has restricted curvature.
• Computer Science
SODA
• 2014
A new variant of the continuous greedy algorithm, which interpolates between the classical greedy algorithm and a truly continuous algorithm, is developed, which can be implemented for matroid and knapsack constraints using O(n2) oracle calls to the objective function.
• Computer Science
Math. Oper. Res.
• 2017
A new algorithm is established for this general case of maximizing a monotone submodular function subject to a matroid independence constraint that establishes a surprising trade-off between two seemingly unrelated quantities.
• Computer Science
AAAI
• 2015
The first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint is developed, and it is shown that the randomized algorithm, STOCHASTIC-GREEDY, can achieve a (1 − 1/e − ε) approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data.
• Economics, Education
EC '09
• 2009
The problem of a search engine trying to assign a sequence of search keywords to a set of competing bidders, each with a daily spending limit, is considered, and the current literature on this problem is extended by considering the setting where the keywords arrive in a random order.
• Computer Science
2009 50th Annual IEEE Symposium on Foundations of Computer Science
• 2009
A novel application of the idea of the power of two choices from load balancing, which compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order to characterize an upper bound for the optimum in any scenario.
• Computer Science
ESA
• 2007
A (1 - 1/e)-competitive (optimal) algorithm is designed for the online ad-auctions problem, which is based on a clean primal-dual approach, matching the competitive factor obtained in Mehta et al.
• Computer Science
EC '08
• 2008
Tight information-theoretic lower bounds are provided for the welfare maximization problem in combinatorial auctions and the well-known value query model in which the permitted query to a valuation function is in the form of a subset of items, and the reply is the value assigned to that subset of Items by the valuation function.
• Computer Science
EC '11
• 2011
A new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time is introduced, and a 1-O(ε) approximation algorithm is given for the resource allocation problem.