The smoothed number of Pareto-optimal solutions in bicriteria integer optimization
A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which is expon...
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| Published in | Mathematical programming Vol. 200; no. 1; pp. 319 - 355 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2023
Springer |
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| Online Access | Get full text |
| ISSN | 0025-5610 1436-4646 1436-4646 |
| DOI | 10.1007/s10107-022-01885-6 |
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| Abstract | A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which is exponential in the worst case. We study bicriteria integer optimization problems in the model of smoothed analysis, in which inputs are subject to a small amount of random noise, and we prove an almost tight polynomial bound on the expected number of Pareto-optimal solutions. Our results give rise to tight polynomial bounds for the expected running time of the Nemhauser-Ullmann algorithm for the knapsack problem and they improve known results on the running times of heuristics for the bounded knapsack problem and the bicriteria shortest path problem. |
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| AbstractList | A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this principle are often successful in practice. Their running time, however, depends on the number of enumerated solutions, which is exponential in the worst case. We study bicriteria integer optimization problems in the model of smoothed analysis, in which inputs are subject to a small amount of random noise, and we prove an almost tight polynomial bound on the expected number of Pareto-optimal solutions. Our results give rise to tight polynomial bounds for the expected running time of the Nemhauser-Ullmann algorithm for the knapsack problem and they improve known results on the running times of heuristics for the bounded knapsack problem and the bicriteria shortest path problem. |
| Audience | Academic |
| Author | Beier, René Röglin, Heiko Rösner, Clemens Vöcking, Berthold |
| Author_xml | – sequence: 1 givenname: René surname: Beier fullname: Beier, René organization: Max-Planck-Institut für Informatik – sequence: 2 givenname: Heiko surname: Röglin fullname: Röglin, Heiko email: roeglin@cs.uni-bonn.de organization: Department of Computer Science, University of Bonn – sequence: 3 givenname: Clemens surname: Rösner fullname: Rösner, Clemens organization: Department of Computer Science, University of Bonn – sequence: 4 givenname: Berthold surname: Vöcking fullname: Vöcking, Berthold organization: Department of Computer Science, RWTH Aachen University |
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| Cites_doi | 10.1016/j.tcs.2005.09.022 10.1017/CBO9780511813603 10.1016/j.jcss.2004.04.004 10.1002/(SICI)1520-6750(200002)47:1<57::AID-NAV4>3.0.CO;2-4 10.1007/978-3-540-74208-1_6 10.1145/1562764.1562785 10.1007/978-3-540-24777-7 10.1007/b101915 10.1145/2699445 10.1007/978-3-540-72792-7_5 10.1109/FOCS.2009.21 10.1145/990308.990310 10.1287/mnsc.15.9.494 10.1007/BF00938761 10.1007/978-3-319-55911-7_39 10.1016/S0305-0548(97)00027-0 10.1007/3-540-44688-5_15 10.1137/110851833 10.4086/toc.2014.v010a010 10.1016/S0305-0548(99)00037-4 |
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| Keywords | Smoothed analysis Bicriteria optimization Pareto-optimal solutions Integer optimization |
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Res.2000276507524175520510.1016/S0305-0548(99)00037-40955.90144 – reference: Beier, René, Röglin, Heiko, Vöcking, Berthold: the smoothed number of pareto optimal solutions in bicriteria integer optimization. In: Proceedings of the 12th international conference on integer programming and combinatorial optimization (IPCO), pp. 53–67 (2007) – reference: SpielmanDaniel ATengShang-HuaSmoothed analysis: an attempt to explain the behavior of algorithms in practiceCommun. ACM20095210768410.1145/1562764.1562785 – reference: BrunschTobiasRöglinHeikoImproved smoothed analysis of multiobjective optimizationJ. ACM20156214:14:58332377010.1145/26994451321.90118 – reference: Müller-Hannemann, Matthias, Weihe, Karsten: Pareto shortest paths is often feasible in practice. In: Proceedings of the 5th international workshop on algorithm engineering (WAE), pages 185–198 (2001) – reference: NemhauserGeorge LUllmannZevDiscrete dynamic programming and capital allocationManag. Sci.196915949450538168010.1287/mnsc.15.9.4941231.90339 – reference: CorleyH WilliamMoonIDouglasShortest paths in networks with vector weightsJ. Optimiz. Theor. Appl.1985461798679259510.1007/BF009387610542.90099 – reference: Röglin, Heiko, Teng, Shang-Hua: Smoothed analysis of multiobjective optimization. In: Proceedings of the 50th Annual IEEE symposium on foundations of computer science (FOCS), pp. 681–690 (2009) – reference: RöglinHeikoRösner, Clemens: the smoothed number of Pareto-optimal solutions in non-integer bicriteria optimizationTheory and applications of models of computation2017ChamSpringer54355510.1007/978-3-319-55911-7_391462.90122 – reference: BeierRenéVöckingBertholdRandom knapsack in expected polynomial timeJ. Comput. Syst. Sci.2004693306329208793810.1016/j.jcss.2004.04.0041062.90037 – reference: MustafaAdliGohMarkFinding integer efficient solutions for bicriteria and tricriteria network flow problems using dinasComput. Oper. Res.1998252139157160544310.1016/S0305-0548(97)00027-00907.90133 – reference: MantheyBodoRöglinHeikoSmoothed analysis: analysis of algorithms beyond worst caseInf. Technol.2011536280286 – reference: MoitraAnkurO’DonnellRyanPareto optimal solutions for smoothed analystsSIAM J. Comput.201241512661284302379310.1137/1108518331263.90089 – reference: MitzenmacherMichaelUpfalEliProbability and Computing2005Cambridge University Press10.1017/CBO97805118136031092.60001 – reference: Beier, R. : Probabilistic analysis of discrete optimization problems. PhD thesis, Universität des Saarlandes (2004) – reference: VassilvitskiiSergeiYannakakisMihalisEfficiently computing succinct trade-off curvesTheor. Comput. Sci.20053482334356218138610.1016/j.tcs.2005.09.0221080.90069 – reference: SpielmanDaniel ATengShang-HuaSmoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial timeJ. 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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Pareto efficiency Theoretical |
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| Title | The smoothed number of Pareto-optimal solutions in bicriteria integer optimization |
| URI | https://link.springer.com/article/10.1007/s10107-022-01885-6 https://link.springer.com/content/pdf/10.1007/s10107-022-01885-6.pdf |
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