TY - JOUR

T1 - APPROXIMABILITY of MONOTONE SUBMODULAR FUNCTION MAXIMIZATION under CARDINALITY and MATROID CONSTRAINTS in the STREAMING MODEL

AU - Huang, Chien Chung

AU - Kakimura, Naonori

AU - Mauras, Simon

AU - Yoshida, Yuichi

N1 - Funding Information:
∗Received by the editors August 3, 2020; accepted for publication (in revised form) September 10, 2021; published electronically February 8, 2022. https://doi.org/10.1137/20M1357317 Funding: The first author was supported by the French National Research Agency (ANR), with grants ANR-19-CE48-0016 and ANR-18-CE40-0025-01. The second author was supported by JSPS KAKENHI grants JP17K00028, JP18H05291, 20H05795, and 21H03397. The fourth author was supported by JSPS KAKENHI grant 20H05965. †CNRS, DI ENS, PSL, Paris, 75005, France (villars@gmail.com). ‡Keio University, Yokohama, 223-8522, Japan (kakimura@math.keio.ac.jp). §Université de Paris, IRIF, CNRS, Paris, 75013, France (simon.mauras@irif.fr). ¶National Institute of Informatics, Chiyoda-ku, Tokyo, 101-8430, Japan (yyoshida@nii.ac.jp).
Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics

PY - 2022

Y1 - 2022

N2 - Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is large gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat 1− 1e in the single-pass streaming model. Let n be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio 2−√2+ε requires Ω(Kn2 ) space for any ε > 0, where K is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio 2KK−1 + ε requires Ω(Kn2 ) space for any ε > 0, where K is the rank of the given matroid. In addition, we give streaming algorithms that assume access to the objective function via a weak oracle that can only be used to evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios 2KK−1 and 21, respectively, whose space complexity is exponential in K but is independent of n. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of 2KK−1 for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of n.

AB - Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is large gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat 1− 1e in the single-pass streaming model. Let n be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio 2−√2+ε requires Ω(Kn2 ) space for any ε > 0, where K is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio 2KK−1 + ε requires Ω(Kn2 ) space for any ε > 0, where K is the rank of the given matroid. In addition, we give streaming algorithms that assume access to the objective function via a weak oracle that can only be used to evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios 2KK−1 and 21, respectively, whose space complexity is exponential in K but is independent of n. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of 2KK−1 for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of n.

KW - approximation algorithm

KW - inapproximability

KW - streaming model

KW - submodular function maximization

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U2 - 10.1137/20M1357317

DO - 10.1137/20M1357317

M3 - Article

AN - SCOPUS:85118644525

SN - 0895-4801

VL - 36

SP - 355

EP - 382

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -