Real-time solving of computationally hard problems using optimal algorithm portfolios

Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T , e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time App...

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Published in:	Annals of mathematics and artificial intelligence Vol. 89; no. 7; pp. 693 - 710
Main Authors:	Nof, Yair, Strichman, Ofer
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.07.2021 Springer Springer Nature B.V
Subjects:	Algorithms Artificial Intelligence Combinatorial analysis Complex Systems Computation Computer Science Investment analysis Mathematics Optimization Polynomials Real time Search algorithms Algorithm portfolios NP-optimization 68T20 Real-time
ISSN:	1012-2443, 1573-7470
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Abstract	Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T , e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T . What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n , for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T . We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout.
AbstractList	Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T, e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T. What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n, for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T. We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout. Keywords Algorithm portfolios * NP-optimization * Real-time Mathematics Subject Classification (2010) 68T20 Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T, e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T. What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n, for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T. We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout. Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short and fixed amount of time T , e.g., T = 0.5 seconds. For such a task, the exact, exponential algorithms, as well as various Polynomial-Time Approximation Schemes, are irrelevant because they can exceed T . What is left in practice is to combine various anytime algorithms in a parallel portfolio. The question is how to build such an optimal portfolio, given a budget of K computing cores. It is certainly not as simple as choosing the K best performing algorithms, because their results are possibly correlated (e.g., there is no point in choosing two good algorithm for the portfolio if they win on a similar set of instances). We prove that the decision variant of this problem is NP-complete, and furthermore that the optimization problem is approximable. On the practical side, our main contribution is a solution of the optimization problem of choosing K algorithms out of n , for a machine with K computing cores, and the related problem of detecting the minimum number of required cores to achieve an optimal portfolio, with respect to a given training set of instances. As a benchmark, we took instances of a hard optimization problem that is prevalent in the real-time industry, in which the challenge is to decide on the best action within time T . We include the results of numerous experiments that compare the various methods. Hence, a side effect of our tests is that it gives the first systematic empirical evaluation of the relative success of various known stochastic-search algorithms in coping with a hard combinatorial optimization problems under a very short and fixed timeout.
Audience	Academic
Author	Strichman, Ofer Nof, Yair
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References	CohenEDahlweidMHillebrandMLeinenbachDMoskalMSantenTSchulteWTobiesSVCC: a Practical System for Verifying Concurrent C2009BerlinSpringer2342https://doi.org/10.1007/978-3-642-03359-9_2 Nof, Y.: Real time solving of discrete optimization problems. Master’s thesis, Technion, Israel Institute of Technology. Available online in https://ie.technion.ac.il/~ofers/publications/theses/yair_nof.pdf (2017) KirkpatrickSGelattCDJrVecchiMPOptimization by simulated annealingScience198322067168070248510.1126/science.220.4598.671 SinzCTowards an optimal cnf encoding of boolean cardinality constraintsCP200537098278311153.68488 Di GasperoLSchaerfAEasylocal++: an object-oriented framework for flexible design of local search algorithmsSoftware — Practice & Experience200333873376510.1002/spe.524 HentenryckPVMichelLConstraint-Based Local Search2005CambridgeMIT Press1160.68556 Hoos, H., Leyton-Brown, K., Schaub, T., Schneider, M.: Algorithm configuration for portfolio-based parallel sat-solving. In: Workshop on Combining Constraint Solving with Mining and Learning (2012) Crescenzi, P., Kann, V.: A compendium of NP optimization problems. In: WWW Spring 1994 (1994) Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations. Springer, US (1972) Dutertre, B., de Moura, L.M.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) Computer Aided Verification, 18th International Conference, CAV 2006, Seattle, WA, USA, August 17-20, 2006, Proceedings, Lecture Notes in Computer Science, vol. 4144, pp 81–94. Springer (2006). https://doi.org/10.1007/11817963_11 HubermanBALukoseRMHoggTAn economics approach to hard computational problemsScience19972755296515410.1126/science.275.5296.51 Leino, K.R.M.: Dafny: an automatic program verifier for functional correctness. In: International Conference on Logic for Programming Artificial Intelligence and Reasoning, pp 348–370. 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Springer (2011) YangXSNature-Inspired Metaheuristic Algorithms20102nd edn.United KingdomLuniver Press1213 Wei, Y., Pei, Y., Furia, C.A., Silva, L.S., Buchholz, S., Meyer, B., Zeller, A.: Automated fixing of programs with contracts. In: Proceedings of the 19th International Symposium on Software Testing and Analysis, pp 61–72. ACM (2010) GomesCPSelmanBAlgorithm portfoliosArtif. Intell.20011261-24362181548910.1016/S0004-3702(00)00081-3 Ansótegui, C., Sellmann, M., Tierney, K.: A gender-based genetic algorithm for the automatic configuration of algorithms. In: International Conference on Principles and Practice of Constraint Programming, pp 142–157. Springer, Berlin (2009) HoosHHStutzleTStochastic Local Search: Foundations and Applications2004BurlingtonMorgan Kaufmann Massachusetts1126.68032 Malitsky, Y., Sellmann, M.: Instance-specific algorithm configuration as a method for non-model-based portfolio generation. 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Springer, Berlin (1997) López-IbáñezMDubois-LacosteJCáceresLPBirattariMStützleTThe irace package: iterated racing for automatic algorithm configurationOperations Research Perspectives201634358357917510.1016/j.orp.2016.09.002 ForrestSMitchellMRelative Building-Block fitness and the Building-Block hypothesisFoundations of Genetic Algorithms19932109126 KroeningDStrichmanODecision Procedures: an Algorithmic Point of View. Texts in Theoretical Computer Science. 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References_xml	– reference: Leino, K.R.M.: Dafny: an automatic program verifier for functional correctness. In: International Conference on Logic for Programming Artificial Intelligence and Reasoning, pp 348–370. Springer (2010) – reference: López-IbáñezMDubois-LacosteJCáceresLPBirattariMStützleTThe irace package: iterated racing for automatic algorithm configurationOperations Research Perspectives201634358357917510.1016/j.orp.2016.09.002 – reference: Michel, L., Van Hentenryck, P.: Localizer a modeling language for local search. In: Smolka, G. (ed.) Principles and Practice of Constraint Programming-CP97, pp 237–251. Springer, Berlin (1997) – reference: KirkpatrickSGelattCDJrVecchiMPOptimization by simulated annealingScience198322067168070248510.1126/science.220.4598.671 – reference: Crescenzi, P., Kann, V.: A compendium of NP optimization problems. In: WWW Spring 1994 (1994) – reference: Di GasperoLSchaerfAEasylocal++: an object-oriented framework for flexible design of local search algorithmsSoftware — Practice & Experience200333873376510.1002/spe.524 – reference: ForrestSMitchellMRelative Building-Block fitness and the Building-Block hypothesisFoundations of Genetic Algorithms19932109126 – reference: GomesCPSelmanBAlgorithm portfoliosArtif. Intell.20011261-24362181548910.1016/S0004-3702(00)00081-3 – reference: Wei, Y., Pei, Y., Furia, C.A., Silva, L.S., Buchholz, S., Meyer, B., Zeller, A.: Automated fixing of programs with contracts. In: Proceedings of the 19th International Symposium on Software Testing and Analysis, pp 61–72. ACM (2010) – reference: Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations. Springer, US (1972) – reference: PetrikMZilbersteinSLearning parallel portfolios of algorithmsAnn. Math. Artif. Intell.20064818510623371521121.68095 – reference: KroeningDStrichmanODecision Procedures: an Algorithmic Point of View. Texts in Theoretical Computer Science. An EATCS Series2010BerlinSpringerhttp://www.decision-procedures.org – reference: RubinsteinRKroeseDThe Cross-Entropy Method: a Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning2004New YorkSpringer10.1007/978-1-4757-4321-0 – reference: YangXSNature-Inspired Metaheuristic Algorithms20102nd edn.United KingdomLuniver Press1213 – reference: Ansótegui, C., Sellmann, M., Tierney, K.: A gender-based genetic algorithm for the automatic configuration of algorithms. In: International Conference on Principles and Practice of Constraint Programming, pp 142–157. Springer, Berlin (2009) – reference: de MouraLBjørnerNZ3: an Efficient SMT Solver2008BerlinSpringer337340https://doi.org/10.1007/978-3-540-78800-3_24 – reference: Hoos, H., Leyton-Brown, K., Schaub, T., Schneider, M.: Algorithm configuration for portfolio-based parallel sat-solving. In: Workshop on Combining Constraint Solving with Mining and Learning (2012) – reference: NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functions - IMath. Program.197814126529450386610.1007/BF01588971 – reference: SinzCTowards an optimal cnf encoding of boolean cardinality constraintsCP200537098278311153.68488 – reference: HubermanBALukoseRMHoggTAn economics approach to hard computational problemsScience19972755296515410.1126/science.275.5296.51 – reference: HentenryckPVMichelLConstraint-Based Local Search2005CambridgeMIT Press1160.68556 – reference: GloverFFuture paths for integer programming and links to artificial intelligenceComput. Oper. 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Snippet	Various hard real-time systems have a desired requirement which is impossible to fulfill: to solve a computationally hard optimization problem within a short...
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SubjectTerms	Algorithms Artificial Intelligence Combinatorial analysis Complex Systems Computation Computer Science Investment analysis Mathematics Optimization Polynomials Real time Search algorithms
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