An Optimal Approximation for Submodular Maximization under a Matroid Constraint in the Adaptive Complexity Model Eric Balkanski Aviad Rubinstein Yaron Singer [email protected] [email protected] [email protected] Harvard University Stanford University Harvard University School of Engineering and Applied Department of Computer Science School of Engineering and Applied Sciences USA Sciences USA USA ABSTRACT KEYWORDS In this paper we study submodular maximization under a matroid Adaptivity, submodular optimization, matroids, parallel algorithms constraint in the adaptive complexity model. This model was re- cently introduced in the context of submodular optimization to ACM Reference Format: quantify the information theoretic complexity of black-box opti- Eric Balkanski, Aviad Rubinstein, and Yaron Singer. 2019. An Optimal Ap- proximation for Submodular Maximization under a Matroid Constraint in mization in a parallel computation model. Informally, the adaptivity the Adaptive Complexity Model. In Proceedings of the 51st Annual ACM of an algorithm is the number of sequential rounds it makes when SIGACT Symposium on the Theory of Computing (STOC ’19), June 23–26, each round can execute polynomially-many function evaluations in 2019, Phoenix, AZ, USA. ACM, New York, NY, USA, 12 pages. https://doi. parallel. Since submodular optimization is regularly applied on large org/10.1145/3313276.3316304 datasets we seek algorithms with low adaptivity to enable speedups via parallelization. Consequently, a recent line of work has been devoted to designing constant factor approximation algorithms for 1 INTRODUCTION maximizing submodular functions under various constraints in the In this paper we study submodular maximization under matroid adaptive complexity model. constraints in the adaptive complexity model. The adaptive com- Despite the burst in work on submodular maximization in the plexity model was recently introduced in the context of submodular adaptive complexity model, the fundamental problem of maximiz- optimization in [BS18a] to quantify the information theoretic com- ing a monotone submodular function under a matroid constraint plexity of black-box optimization in a parallel computation model. has remained elusive. In particular, all known techniques fail for Informally, the adaptivity of an algorithm is the number of sequen- this problem and there are no known constant factor approximation tial rounds it makes when each round can execute polynomially- algorithms whose adaptivity is sublinear in the rank of the matroid many function evaluations in parallel. The concept of adaptivity is k or in the worst case sublinear in the size of the ground set n. heavily studied in computer science and optimization as it provides In this paper we present an approximation algorithm for the a measure of efficiency of parallel computation. problem of maximizing a monotone submodular function under a Since submodular optimization is regularly applied on very large matroid constraint in the adaptive complexity model. The approxi- datasets, we seek algorithms with low adaptivity to enable speedups mation guarantee of the algorithm is arbitrarily close to the optimal via parallelization. For the basic problem of maximizing a monotone 1 − 1/e and it has near optimal adaptivity of O¹log¹nº log¹kºº. This submodular function under a cardinality constraint k the celebrated result is obtained using a novel technique of adaptive sequencing greedy algorithm which iteratively adds to the solution the element which departs from previous techniques for submodular maximiza- with largest marginal contribution is Ω¹kº adaptive. Until very tion in the adaptive complexity model. In addition to our main result recently, even for this basic problem, there was no known constant- we show how to use this technique to design other approximation factor approximation algorithm whose adaptivity is sublinear in algorithms with strong approximation guarantees and polyloga- k. In the worst case k 2 Ω¹nº and hence greedy and all other rithmic adaptivity. algorithms had adaptivity that is linear in the size of the ground set. CCS CONCEPTS The main result in [BS18a] is an adaptive sampling algorithm for • Theory of computation → Approximation algorithms anal- maximizing a monotone submodular function under a cardinality ysis. constraint that achieves a constant factor approximation arbitrarily close to 1/3 in O¹lognº adaptive rounds as well as a lower bound Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed that shows that no algorithm can achieve a constant factor approx- for profit or commercial advantage and that copies bear this notice and the full citation imation in o˜¹lognº rounds. Consequently, this algorithm provided on the first page. Copyrights for components of this work owned by others than ACM a constant factor approximation with an exponential speedup in must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a parallel runtime for monotone submodular maximization under a fee. Request permissions from [email protected]. cardinality constraint. STOC ’19, June 23–26, 2019, Phoenix, AZ, USA In [BRS19, EN19], the adaptive sampling technique was extended © 2019 Association for Computing Machinery. ACM ISBN 978-1-4503-6705-9/19/06...$15.00 to achieve an approximation guarantee arbitrarily close to the op- https://doi.org/10.1145/3313276.3316304 timal 1 − 1/e in O¹lognº adaptive rounds. This result was then STOC ’19, June 23–26, 2019, Phoenix, AZ, USA Eric Balkanski, Aviad Rubinstein, and Yaron Singer obtained with a linear number of queries [FMZ19], which is op- Our main result is largely powered by a new technique developed timal. Functions with bounded curvature have also been studied in this paper which we call adaptive sequencing. This technique using adaptive sampling under a cardinality constraint [BS18b]. proves to be extremely powerful and is a departure from all previous The more general family of packing constraints, which includes techniques for submodular maximization in the adaptive complexity partition and laminar matroids, has been considered in [CQ19]. In model. In addition to our main result we show that this technique particular, under m packing constraints, a 1−1/e −ϵ approximation gives us a set of other strong results that include: 2 was obtained in O¹log m lognº rounds using a combination of con- • An O¹log¹nº log¹kºº adaptive combinatorial algorithm that tinuous optimization and multiplicative weight update techniques. 1 obtains a 2 − ϵ approximation for monotone submodular maximization under a matroid constraint (Theorem1); 1.1 Submodular Maximization under a Matroid • An O¹log¹nº log¹kºº adaptive combinatorial algorithm that 1 − Constraint obtains a P+1 ϵ approximation for monotone submodular For the fundamental problem of maximizing a monotone submod- maximization under intersection of P matroids (Theorem7); ular function under a general matroid constraint it is well known • An O¹log¹nº log¹kºº adaptive algorithm that obtains an ap- since the late 70s that the greedy algorithm achieves a 1/2 approxi- proximation of 1 − 1/e − ϵ for monotone submodular max- mation [NWF78] and that even for the special case of cardinality imization under a partition matroid constraint that can be constraint no algorithm can obtain an approximation guarantee implemented in the PRAM model with polylogarithmic depth better than 1−1/e using polynomially-many value queries [NW78]. (AppendixA). Thirty years later, in seminal work, Vondrák introduced the contin- In addition to these results the adaptive sequencing technique uous greedy algorithm which approximately maximizes the multi- can be used to design algorithms that achieve the same results linear extension of the submodular function [CCPV07] and showed as those for cardinality constraint in [BRS19, EN19, FMZ19] and it obtains the optimal 1 − 1/e approximation guarantee [Von08]. for non-monotone submodular maximization under cardinality Despite the surge of interest in adaptivity of submodular max- constraint as in [BBS18] (AppendixA). imization, the problem of maximizing a monotone submodular function under a matroid constraint in the adaptive complexity 1.3 Technical Overview model has remained elusive. As we discuss in Section 1.4, when The standard approach to obtain an approximation guarantee arbi- it comes to matroid constraints there are fundamental limitations trarily close to 1 − 1/e for maximizing a submodular function under of the techniques developed in this line of work. The best known a matroid constraint M is by the continuous greedy algorithm due adaptivity for obtaining a constant factor approximation guarantee to Vondrák [Von08]. This algorithm approximately maximizes the for maximizing a monotone submodular function under a matroid multilinear extension F of the submodular function [CCPV07] in constraint is achieved by the greedy algorithm and is linear in the O¹nº adaptive steps. In each step the algorithm updates a continu- rank of the matroid. The best known adaptivity for obtaining the ous solution x 2 »0; 1¼ in the direction of 1 , where S is chosen by optimal 1 − 1/e guarantee is achieved by the continuous greedy S maximizing an additive function under a matroid constraint. and is linear in the size of the ground set. In this