We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multiplayer model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two-player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the connections mentioned previously, have multiple implications in the data stream and robust setting.Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets (i.e., sets respecting the cardinality constraint) are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the link to the (non-streaming) robust setting mentioned previously, both of these algorithms improve on the current state of the art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2+I hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best-known approximation algorithm.
Bibliographical noteFunding Information:
M. Feldman was supported in part by the Israel Science Foundation (ISF) under grant 1357/16. O. Svensson was supported by the Swiss National Science Foundation under grant 200021-184656, “Randomness in Problem Instances and Randomized Algorithms.” R. Zenklusen was supported by Swiss National Science Foundation grants 200021_184622 and 200021_165866, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 817750) .
M. Feldman was supported in part by the Israel Science Foundation (ISF) under grant 1357/16. O. Svensson was supported by the Swiss National Science Foundation under grant 200021-184656, Randomness in Problem Instances and Randomized Algorithms. R. Zenklusen was supported by Swiss National Science Foundation grants 200021_184622 and 200021_165866, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 817750)
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- Additional Key Words and PhrasesCommunication complexity
- approximation algorithms
- submodular maximization
ASJC Scopus subject areas
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence