# Online submodular maximization: beating 1/2 made simple

@article{Buchbinder2019OnlineSM,
title={Online submodular maximization: beating 1/2 made simple},
author={Niv Buchbinder and Moran Feldman and Mohit Garg},
journal={Mathematical Programming},
year={2019},
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…
16 Citations

### Making a Sieve Random: Improved Semi-Streaming Algorithm for Submodular Maximization under a Cardinality Constraint

• 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.

### Streaming Maximization of Submodular Functions Subject to Cardinality Constraint

This work studies the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model, and presents a single-pass streaming algorithm that uses roughly O(k/ε2) memory and enjoys a fast update time of O( log k+log(1/α) ε2 ) per element.

### The Power of Subsampling in Submodular Maximization

• 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.

### An Optimal Streaming Algorithm for Submodular Maximization with a Cardinality Constraint

• 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

### Edge-Weighted Online Bipartite Matching

• 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.

### Submodular Secretary Problem with Shortlists

• 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 $O(k \log k)$.

### Online Assortment Optimization for Two-sided Matching Platforms

• Computer Science
EC
• 2021
This work considers the online assortment optimization problem faced by a two-sided matching platform that hosts a set of suppliers waiting to match with a customer, and develops specialized balancing algorithms, which are preference-aware, that leverage general information about the suppliers' choice models.

### Sublinear Time Algorithm for Online Weighted Bipartite Matching

• 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.

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

• Computer Science
• 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.

### Robust Algorithms for Online k-means Clustering

• 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.

## References

SHOWING 1-10 OF 38 REFERENCES

### Online Submodular Welfare Maximization: Greedy is Optimal

• 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.

### Monotone Submodular Maximization over a Matroid via Non-Oblivious Local Search

• 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.

### Fast algorithms for maximizing submodular functions

• 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.

### The Submodular Welfare Problem with Demand Queries

• Economics
Theory Comput.
• 2010
It is shown that the Submodular Welfare Problem is NP -hard to approximate within a ratio better than some r < 1, and an incentive compatible mechanism based on fair division queries that achieves an optimal solution is presented.

### Comparing Apples and Oranges: Query Trade-off in Submodular Maximization

• 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.

### A Unified Continuous Greedy Algorithm for Submodular Maximization

• Computer Science, Mathematics
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
• 2011
This work presents a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications.

### Online budgeted matching in random input models with applications to Adwords

• Computer Science, Economics
SODA '08
• 2008
An online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints is studied, with a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 - 1/e for maximizing the value of the assignment.

### Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs

• Computer Science
STOC '11
• 2011
This paper studies the ranking algorithm in the random arrivals model, and shows that it has a competitive ratio of at least 0.696, beating the 1-1/e ≈ 0.632 barrier in the adversarial model.

### Lazier Than Lazy Greedy

• 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.

### The adwords problem: online keyword matching with budgeted bidders under random permutations

• 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.