Guess Free Maximization of Submodular and Linear Sums

We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (Math Oper Res 42(4):1197–1218, 2017) described an algorithm for this problem whose approximation guarantee is optimal...

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Vydáno v:	Algorithmica Ročník 83; číslo 3; s. 853 - 878
Hlavní autor:	Feldman, Moran
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.03.2021 Springer Nature B.V
Témata:	Algorithm Analysis and Problem Complexity Algorithms Algorithms and Data Structures (WADS 2019) Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Linear functions Mathematical analysis Mathematics of Computing Maximization Optimization Polytopes Theory of Computation 68Q25 Continuous greedy 68W25 90C27 Submodular maximization Curvature
ISSN:	0178-4617, 1432-0541
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Abstract	We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (Math Oper Res 42(4):1197–1218, 2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al. (2017). We also show that the guarantee of our algorithm becomes slightly better when the polytope is down-monotone, and that this better guarantee is tight for such polytopes.
AbstractList	We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (Math Oper Res 42(4):1197–1218, 2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al. (2017). We also show that the guarantee of our algorithm becomes slightly better when the polytope is down-monotone, and that this better guarantee is tight for such polytopes. We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (Math Oper Res 42(4):1197–1218, 2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al. (2017). We also show that the guarantee of our algorithm becomes slightly better when the polytope is down-monotone, and that this better guarantee is tight for such polytopes.
Author	Feldman, Moran
Author_xml	– sequence: 1 givenname: Moran orcidid: 0000-0002-1535-2979 surname: Feldman fullname: Feldman, Moran email: moranfe3@gmail.com organization: The Open University of Israel (Currently Affiliated with the University of Haifa, Israel)
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Cites_doi	10.1002/0471722154 10.1109/FOCS.2010.60 10.1287/moor.2018.0955 10.1137/1.9781611974331.ch72 10.1007/BF01588971 10.1137/1.9781611973402.110 10.1007/978-3-642-36694-9_18 10.1145/3070685 10.1287/moor.3.3.177 10.1137/110839655 10.1137/1.9781611973402.106 10.1137/080733991 10.1109/FOCS.2018.00080 10.1609/aaai.v31i1.10653 10.1287/moor.2016.0809 10.1287/moor.2016.0842 10.1109/FOCS.2016.34 10.1109/FOCS.2011.46
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References	CălinescuGChekuriCPálMVondrákJMaximizing a monotone submodular function subject to a matroid constraintSIAM J. Comput.201140617401766286319310.1137/080733991 Ene, A., Nguyen, H.L.: Constrained submodular maximization: Beyond 1/e. In: FOCS, pp. 248–257 (2016) BuchbinderNFeldmanMConstrained submodular maximization via a non-symmetric techniqueMath. Oper. Res.20194439881005399665510.1287/moor.2018.0955 BuchbinderNFeldmanMSchwartzRComparing apples and oranges: query trade-off in submodular maximizationMath. Oper. Res.2017422308329365199310.1287/moor.2016.0809 Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: SODA, pp. 1497–1514 (2014) Gupta, A., Nagarajan, V.: A stochastic probing problem with applications. In: IPCO, pp. 205–216 (2013) NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functions-IMath. Program.19781426529450386610.1007/BF01588971 Soma, T., Yoshida, Y.: A new approximation guarantee for monotone submodular function maximization via discrete convexity. In: ICALP, pp. 99:1–99:14 (2018) Feldman, M., Naor, J., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: FOCS, pp. 570–579 (2011) Harshaw, C., Feldman, M., Ward, J., Karbasi, A.: Submodular maximization beyond non-negativity: Guarantees, fast algorithms, and applications. In: ICML, pp. 2634–2643 (2019) NemhauserGLWolseyLABest algorithms for approximating the maximum of a submodular set functionMath. Oper. Res.19783317718850665610.1287/moor.3.3.177 Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: Submodular maximization with cardinality constraints. In: SODA, pp. 1433–1452 (2014) Feldman, M., Svensson, O., Zenklusen, R.: Online contention resolution schemes. In: SODA, pp. 1014–1033 (2016) Adamczyk, M., Wlodarczyk, M.: Random order contention resolution schemes. In: FOCS, pp. 790–801 (2018) Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: FOCS, pp. 575–584 (2010) ChekuriCVondrákJZenklusenRSubmodular function maximization via the multilinear relaxation and contention resolution schemesSIAM J. Comput.201443618311879328128710.1137/110839655 Feldman, M.: Maximizing symmetric submodular functions. ACM Trans. Algorithms 13(3), 39:1–39:36 (2017) SviridenkoMVondrákJWardJOptimal approximation for submodular and supermodular optimization with bounded curvatureMath. Oper. Res.201742411971218372243210.1287/moor.2016.0842 AlonNSpencerJHThe Probabilistic Method20002LondonWiley10.1002/0471722154 N Buchbinder (757_CR6) 2017; 42 757_CR18 757_CR3 C Chekuri (757_CR9) 2014; 43 757_CR1 757_CR15 757_CR14 M Sviridenko (757_CR19) 2017; 42 757_CR13 757_CR12 GL Nemhauser (757_CR16) 1978; 3 757_CR11 757_CR10 G Călinescu (757_CR7) 2011; 40 GL Nemhauser (757_CR17) 1978; 14 N Buchbinder (757_CR4) 2019; 44 757_CR8 757_CR5 N Alon (757_CR2) 2000
References_xml	– reference: Chekuri, C., Vondrák, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: FOCS, pp. 575–584 (2010) – reference: Adamczyk, M., Wlodarczyk, M.: Random order contention resolution schemes. In: FOCS, pp. 790–801 (2018) – reference: Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: Submodular maximization with cardinality constraints. In: SODA, pp. 1433–1452 (2014) – reference: Gupta, A., Nagarajan, V.: A stochastic probing problem with applications. In: IPCO, pp. 205–216 (2013) – reference: NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functions-IMath. Program.19781426529450386610.1007/BF01588971 – reference: Feldman, M., Svensson, O., Zenklusen, R.: Online contention resolution schemes. In: SODA, pp. 1014–1033 (2016) – reference: SviridenkoMVondrákJWardJOptimal approximation for submodular and supermodular optimization with bounded curvatureMath. Oper. Res.201742411971218372243210.1287/moor.2016.0842 – reference: BuchbinderNFeldmanMConstrained submodular maximization via a non-symmetric techniqueMath. Oper. Res.20194439881005399665510.1287/moor.2018.0955 – reference: Soma, T., Yoshida, Y.: A new approximation guarantee for monotone submodular function maximization via discrete convexity. In: ICALP, pp. 99:1–99:14 (2018) – reference: ChekuriCVondrákJZenklusenRSubmodular function maximization via the multilinear relaxation and contention resolution schemesSIAM J. Comput.201443618311879328128710.1137/110839655 – reference: BuchbinderNFeldmanMSchwartzRComparing apples and oranges: query trade-off in submodular maximizationMath. Oper. Res.2017422308329365199310.1287/moor.2016.0809 – reference: CălinescuGChekuriCPálMVondrákJMaximizing a monotone submodular function subject to a matroid constraintSIAM J. Comput.201140617401766286319310.1137/080733991 – reference: NemhauserGLWolseyLABest algorithms for approximating the maximum of a submodular set functionMath. Oper. Res.19783317718850665610.1287/moor.3.3.177 – reference: Harshaw, C., Feldman, M., Ward, J., Karbasi, A.: Submodular maximization beyond non-negativity: Guarantees, fast algorithms, and applications. In: ICML, pp. 2634–2643 (2019) – reference: Feldman, M.: Maximizing symmetric submodular functions. ACM Trans. Algorithms 13(3), 39:1–39:36 (2017) – reference: Ene, A., Nguyen, H.L.: Constrained submodular maximization: Beyond 1/e. In: FOCS, pp. 248–257 (2016) – reference: Feldman, M., Naor, J., Schwartz, R.: A unified continuous greedy algorithm for submodular maximization. In: FOCS, pp. 570–579 (2011) – reference: AlonNSpencerJHThe Probabilistic Method20002LondonWiley10.1002/0471722154 – reference: Badanidiyuru, A., Vondrák, J.: Fast algorithms for maximizing submodular functions. In: SODA, pp. 1497–1514 (2014) – volume-title: The Probabilistic Method year: 2000 ident: 757_CR2 doi: 10.1002/0471722154 – ident: 757_CR8 doi: 10.1109/FOCS.2010.60 – ident: 757_CR15 – volume: 44 start-page: 988 issue: 3 year: 2019 ident: 757_CR4 publication-title: Math. Oper. Res. doi: 10.1287/moor.2018.0955 – ident: 757_CR13 doi: 10.1137/1.9781611974331.ch72 – volume: 14 start-page: 265 year: 1978 ident: 757_CR17 publication-title: Math. Program. doi: 10.1007/BF01588971 – ident: 757_CR3 doi: 10.1137/1.9781611973402.110 – ident: 757_CR14 doi: 10.1007/978-3-642-36694-9_18 – ident: 757_CR11 doi: 10.1145/3070685 – volume: 3 start-page: 177 issue: 3 year: 1978 ident: 757_CR16 publication-title: Math. Oper. Res. doi: 10.1287/moor.3.3.177 – volume: 43 start-page: 1831 issue: 6 year: 2014 ident: 757_CR9 publication-title: SIAM J. Comput. doi: 10.1137/110839655 – ident: 757_CR5 doi: 10.1137/1.9781611973402.106 – volume: 40 start-page: 1740 issue: 6 year: 2011 ident: 757_CR7 publication-title: SIAM J. Comput. doi: 10.1137/080733991 – ident: 757_CR1 doi: 10.1109/FOCS.2018.00080 – ident: 757_CR18 doi: 10.1609/aaai.v31i1.10653 – volume: 42 start-page: 308 issue: 2 year: 2017 ident: 757_CR6 publication-title: Math. Oper. Res. doi: 10.1287/moor.2016.0809 – volume: 42 start-page: 1197 issue: 4 year: 2017 ident: 757_CR19 publication-title: Math. Oper. Res. doi: 10.1287/moor.2016.0842 – ident: 757_CR10 doi: 10.1109/FOCS.2016.34 – ident: 757_CR12 doi: 10.1109/FOCS.2011.46
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Algorithms and Data Structures (WADS 2019) Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Linear functions Mathematical analysis Mathematics of Computing Maximization Optimization Polytopes Theory of Computation
Title	Guess Free Maximization of Submodular and Linear Sums
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