Strongly Polynomial FPTASes for Monotone Dynamic Programs

In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone dynamic programs. While some ad-hoc SFPTASes for specific problems are already known, this is the first framework yielding such SFPTASes. In...

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Published in:	Algorithmica Vol. 84; no. 10; pp. 2785 - 2819
Main Authors:	Alon, Tzvi, Halman, Nir
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.10.2022 Springer Nature B.V
Subjects:	Agents (artificial intelligence) Algorithm Analysis and Problem Complexity Algorithms Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematical analysis Mathematics of Computing Polynomials Theory of Computation Strongly polynomial algorithms approximation sets and functions Dynamic programming
ISSN:	0178-4617, 1432-0541
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Abstract	In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone dynamic programs. While some ad-hoc SFPTASes for specific problems are already known, this is the first framework yielding such SFPTASes. In addition, it is possible to use our algorithm to get efficient (non strongly polynomial) FPTASes. Our results are derived by improving former (non strongly polynomial) FPTASes which were designed via the method of K -approximation sets and functions. We demonstrate our SFPTAS framework on five application problems, namely, 0/1 Knapsack, counting 0/1 Knapsack, Counting s - t paths, Mobile agent routing and Counting n -tuples, for the last problem we get the fastest SFPTAS known to date. In addition, we use our algorithm to get the fastest (non strongly polynomial) FPTASes for the following other three application problems: Stochastic ordered knapsack, Bi-criteria path problem with maximum survival probability and Minimizing the makespan of deteriorating jobs.
AbstractList	In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone dynamic programs. While some ad-hoc SFPTASes for specific problems are already known, this is the first framework yielding such SFPTASes. In addition, it is possible to use our algorithm to get efficient (non strongly polynomial) FPTASes. Our results are derived by improving former (non strongly polynomial) FPTASes which were designed via the method of K-approximation sets and functions. We demonstrate our SFPTAS framework on five application problems, namely, 0/1 Knapsack, counting 0/1 Knapsack, Counting s-t paths, Mobile agent routing and Counting n-tuples, for the last problem we get the fastest SFPTAS known to date. In addition, we use our algorithm to get the fastest (non strongly polynomial) FPTASes for the following other three application problems: Stochastic ordered knapsack, Bi-criteria path problem with maximum survival probability and Minimizing the makespan of deteriorating jobs. In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone dynamic programs. While some ad-hoc SFPTASes for specific problems are already known, this is the first framework yielding such SFPTASes. In addition, it is possible to use our algorithm to get efficient (non strongly polynomial) FPTASes. Our results are derived by improving former (non strongly polynomial) FPTASes which were designed via the method of K -approximation sets and functions. We demonstrate our SFPTAS framework on five application problems, namely, 0/1 Knapsack, counting 0/1 Knapsack, Counting s - t paths, Mobile agent routing and Counting n -tuples, for the last problem we get the fastest SFPTAS known to date. In addition, we use our algorithm to get the fastest (non strongly polynomial) FPTASes for the following other three application problems: Stochastic ordered knapsack, Bi-criteria path problem with maximum survival probability and Minimizing the makespan of deteriorating jobs.
Author	Halman, Nir Alon, Tzvi
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Cites_doi	10.1016/j.disopt.2017.11.001 10.1287/moor.1090.0391 10.1023/A:1009626427432 10.1145/2422.322418 10.1287/opre.1120.1093 10.1002/(SICI)1520-6750(199808)45:5<511::AID-NAV5>3.0.CO;2-6 10.1007/s11590-018-1251-0 10.1287/moor.1080.0330 10.1016/j.tcs.2016.06.015 10.1016/j.ic.2019.04.001 10.1137/130925153 10.1504/IJPS.2012.050127 10.1016/j.ejor.2018.04.013 10.1023/B:JOCO.0000021934.29833.6b 10.1287/ijoc.12.1.57.11901 10.1145/321906.321909 10.1007/s10951-019-00616-8 10.1007/s00291-018-0543-1 10.1002/nav.3800030106 10.1016/j.tcs.2007.03.006 10.1137/11083976X 10.1137/19M1308633 10.1137/18M1208423 10.1007/s10107-007-0189-2
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References	MittalSSchultzAA general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into oneOper. Res.201361386397304611710.1287/opre.1120.1093 HalmanNA technical note: fully polynomial time approximation schemes for minimizing makespan of deteriorating jobs with non-linear processing timesJ. Sched.2020236643648418200010.1007/s10951-019-00616-8 IbaraOHKimCEFast approximation algorithms for the knapsack and sum of subset problemsJ. ACM197522446346837846310.1145/321906.321909 KovalyovMYKubiakWA fully polynomial approximation scheme for minimizing makespan of deteriorating jobsJ. Heuristics19983428729710.1023/A:1009626427432 BertsekasDDynamic programming and optimal control1995MAAthena scientific Belmont0904.90170 OrlinJA faster strongly polynomial time algorithm for submodular function minimizationMath. Program.20091182237251247079010.1007/s10107-007-0189-2 ŠtefankovičDVempalaSVigodaEA deterministic polynomial-time approximation scheme for counting knapsack solutionsSIAM J. Comput.201241356366291433110.1137/11083976X DeanBGoemansMVondrákJApproximating the stochastic knapsack problem: the benefit of adaptivityMath. Oper. Res.2008334945964246465310.1287/moor.1080.0330 LevnerEElaloufAChengTCAn improved FPTAS for mobile agent routing with time constraintsJ. Univ. Comput. Sci.201117131854186228790181247.68021 HalmanNProvably near-optimal approximation schemes for implicit stochastic and for sample-based dynamic programsINFORMS J. Comput.202032411571181417783707303829 Gawrychowski, P., Markin, L., Weimann, O.: A faster FPTAS for # knapsack. arXiv:1802.05791 (2018) Halman, N., Kovalyov, M., Quilliot, A., Shabtay, D., Zofi, M.: Bi-criteria path problem with minimum length and maximum survival probability. OR Spectrum 41(2), 469–489 (2019) Díaz-NúñezFHalmanNVásquezÓThe TV advertisements scheduling problemOptim. Lett.20191318194390207710.1007/s11590-018-1251-0 SmithWVarious optimizers for single-stage productionNaval Res. Logist. Q.195631–259668910910.1002/nav.3800030106 GareyMRJohnsonDSComputers and Intractability: a Guide to the Theory of NP-Completeness1979New YorkW.H. Freeman0411.68039 HalmanNKellererHStrusevichVApproximation schemes for non-separable non-linear Boolean programming problems under nested knapsack constraintsEur. J. Oper. Res.20182702435447380708810.1016/j.ejor.2018.04.013 HalmanNNanniciniGToward breaking the curse of dimensionality: an FPTAS for stochastic dynamic programs with multidimensional actions and scalar statesSIAM J. Optim.201929211311163393933810.1137/18M1208423 KovalyovMvan de VeldeSScheduling deteriorating jobs to minimize makespanNav. Res. Logist.1998455511523163870510.1002/(SICI)1520-6750(199808)45:5<511::AID-NAV5>3.0.CO;2-6 HalmanNKlabjanDLiC-LOrlinJSimchi-LeviDFully polynomial time approximation schemes for stochastic dynamic programsSIAM J. Discret. Math.20142817251796326860410.1137/130925153 KovalyovMYKubiakWA Generic FPTAS For Partition Type Optimization ProblemsInt. J. Plann. Schedul.2012120923310.1504/IJPS.2012.050127 WoegingerGJWhen does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)?INFORMS J. Comput.2000125774176468610.1287/ijoc.12.1.57.11901 Alon, T.: Fully polynomial time approximation schemes (FPTAS) for some counting problems. Master’s thesis. arXiv preprintarXiv:1611.00992 (2016) HalmanNNanniciniGOrlinJOn the complexity of energy storage problemsDiscret. Optim.2018283153379903510.1016/j.disopt.2017.11.001 Hochbaum, D.: Various notions of approximations: good, better, best, and more. In: Hochbaum, D. (ed.) Approximation algorithms for NP-hard problems, PWS Publishing Company, Boston (1997) RizziRTomescuAFaster FPTASes for counting and random generation of knapsack solutionsInf. Comput.2019267135144395863010.1016/j.ic.2019.04.001 AlonTHalmanNAutomatic generation of FPTASes for stochastic monotone dynamic programs made easierSIAM J. Discret. Math.20213526792722433562410.1137/19M1308633 PruhsKWoegingerGJApproximation schemes for a class of subset selection problemsTheoret. Comput. Sci.2007382151156235211010.1016/j.tcs.2007.03.006 Chan, T.M.: Approximation schemes for 0–1 knapsack. In: Proceedings of the 1st Symposium of Simplicity in Algorithms (SOSA), pp. 5:1–5:12. (2018) Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer Science & Business Media (2012) HalmanNA deterministic fully polynomial time approximation scheme for counting integer knapsack solutions made easyTheor. Comput. Sci.20166454147354195610.1016/j.tcs.2016.06.015 CamponogaraEShimaRBMobile agent routing with time constraints: a resource constrained longest-path approachJ. Univ. Comput. Sci.20101633724011216.68302 MegiddoNLinear programming in linear time when the dimension is fixedJ. ACM (JACM)198431111412782138810.1145/2422.322418 HalmanNKlabjanDMostagirMOrlinJSimchi-LeviDA fully polynomial time approximation scheme for single-item stochastic inventory control with discrete demandMath. Oper. Res.200934674685255534210.1287/moor.1090.0391 KellererHPferschyUImproved dynamic programming in connection with an FPTAS for the knapsack problemJ. Comb. 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References_xml	– reference: GareyMRJohnsonDSComputers and Intractability: a Guide to the Theory of NP-Completeness1979New YorkW.H. Freeman0411.68039 – reference: Gawrychowski, P., Markin, L., Weimann, O.: A faster FPTAS for # knapsack. arXiv:1802.05791 (2018) – reference: HalmanNNanniciniGToward breaking the curse of dimensionality: an FPTAS for stochastic dynamic programs with multidimensional actions and scalar statesSIAM J. Optim.201929211311163393933810.1137/18M1208423 – reference: Hochbaum, D.: Various notions of approximations: good, better, best, and more. In: Hochbaum, D. (ed.) Approximation algorithms for NP-hard problems, PWS Publishing Company, Boston (1997) – reference: PruhsKWoegingerGJApproximation schemes for a class of subset selection problemsTheoret. Comput. Sci.2007382151156235211010.1016/j.tcs.2007.03.006 – reference: Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer Science & Business Media (2012) – reference: HalmanNA technical note: fully polynomial time approximation schemes for minimizing makespan of deteriorating jobs with non-linear processing timesJ. Sched.2020236643648418200010.1007/s10951-019-00616-8 – reference: Halman, N., Kovalyov, M., Quilliot, A., Shabtay, D., Zofi, M.: Bi-criteria path problem with minimum length and maximum survival probability. OR Spectrum 41(2), 469–489 (2019) – reference: RizziRTomescuAFaster FPTASes for counting and random generation of knapsack solutionsInf. Comput.2019267135144395863010.1016/j.ic.2019.04.001 – reference: HalmanNNanniciniGOrlinJOn the complexity of energy storage problemsDiscret. Optim.2018283153379903510.1016/j.disopt.2017.11.001 – reference: KovalyovMYKubiakWA fully polynomial approximation scheme for minimizing makespan of deteriorating jobsJ. Heuristics19983428729710.1023/A:1009626427432 – reference: KovalyovMvan de VeldeSScheduling deteriorating jobs to minimize makespanNav. Res. Logist.1998455511523163870510.1002/(SICI)1520-6750(199808)45:5<511::AID-NAV5>3.0.CO;2-6 – reference: Chan, T.M.: Approximation schemes for 0–1 knapsack. In: Proceedings of the 1st Symposium of Simplicity in Algorithms (SOSA), pp. 5:1–5:12. (2018) – reference: BertsekasDDynamic programming and optimal control1995MAAthena scientific Belmont0904.90170 – reference: MegiddoNLinear programming in linear time when the dimension is fixedJ. ACM (JACM)198431111412782138810.1145/2422.322418 – reference: Díaz-NúñezFHalmanNVásquezÓThe TV advertisements scheduling problemOptim. Lett.20191318194390207710.1007/s11590-018-1251-0 – reference: ŠtefankovičDVempalaSVigodaEA deterministic polynomial-time approximation scheme for counting knapsack solutionsSIAM J. Comput.201241356366291433110.1137/11083976X – reference: WoegingerGJWhen does a dynamic programming formulation guarantee the existence of a fully polynomial time approximation scheme (FPTAS)?INFORMS J. Comput.2000125774176468610.1287/ijoc.12.1.57.11901 – reference: MittalSSchultzAA general framework for designing approximation schemes for combinatorial optimization problems with many objectives combined into oneOper. Res.201361386397304611710.1287/opre.1120.1093 – reference: AlonTHalmanNAutomatic generation of FPTASes for stochastic monotone dynamic programs made easierSIAM J. Discret. Math.20213526792722433562410.1137/19M1308633 – reference: CamponogaraEShimaRBMobile agent routing with time constraints: a resource constrained longest-path approachJ. Univ. Comput. Sci.20101633724011216.68302 – reference: LevnerEElaloufAChengTCAn improved FPTAS for mobile agent routing with time constraintsJ. Univ. Comput. Sci.201117131854186228790181247.68021 – reference: HalmanNKellererHStrusevichVApproximation schemes for non-separable non-linear Boolean programming problems under nested knapsack constraintsEur. J. Oper. Res.20182702435447380708810.1016/j.ejor.2018.04.013 – reference: KellererHPferschyUImproved dynamic programming in connection with an FPTAS for the knapsack problemJ. Comb. Optim.20048511205812910.1023/B:JOCO.0000021934.29833.6b – reference: HalmanNKlabjanDLiC-LOrlinJSimchi-LeviDFully polynomial time approximation schemes for stochastic dynamic programsSIAM J. Discret. Math.20142817251796326860410.1137/130925153 – reference: SmithWVarious optimizers for single-stage productionNaval Res. Logist. Q.195631–259668910910.1002/nav.3800030106 – reference: DeanBGoemansMVondrákJApproximating the stochastic knapsack problem: the benefit of adaptivityMath. Oper. Res.2008334945964246465310.1287/moor.1080.0330 – reference: HalmanNProvably near-optimal approximation schemes for implicit stochastic and for sample-based dynamic programsINFORMS J. 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Snippet	In this paper we introduce a framework for the automatic generation of Strongly Polynomial Fully Polynomial Time Approximation Schemes (SFPTASes) for monotone...
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SubjectTerms	Agents (artificial intelligence) Algorithm Analysis and Problem Complexity Algorithms Approximation Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematical analysis Mathematics of Computing Polynomials Theory of Computation
Title	Strongly Polynomial FPTASes for Monotone Dynamic Programs
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