Article

Random knapsack in expected polynomial time

Authors: Rene Beier, Berthold VöckingAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

Pages 232 - 241

https://doi.org/10.1145/780542.780578

Published: 09 June 2003 Publication History

Abstract

In this paper, we present the first average-case analysis proving an expected polynomial running time for an exact algorithm for the 0/1 knapsack problem. In particular, we prove, for various input distributions, that the number of dominating solutions (i.e., Pareto-optimal knapsack fillings) to this problem is polynomially bounded in the number of available items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of dominating solutions implies an algorithm with expected polynomial running time.The random input model underlying our analysis is very general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean, and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called strongly correlated instances are harder to solve than weakly correlated ones.

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Dey SDubey YMolinaro MMarx D(2021)Branch-and-bound solves random binary IPs in polytimeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458099(579-591)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458099
Erickson Jvan der Hoog IMiltzow T(2020)Smoothing the gap between NP and ER2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00099(1022-1033)Online publication date: Nov-2020
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Singh VChakraborty D(2020)Solving Knapsack Problem with Fuzzy Random Variable CoefficientsIntelligent Techniques and Applications in Science and Technology10.1007/978-3-030-42363-6_120(1037-1048)Online publication date: 3-Mar-2020
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Index Terms

Random knapsack in expected polynomial time
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Mixed discrete-continuous optimization
        Integer programming

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cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 09 June 2003

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STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

CA, San Diego, USA

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STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Dey SDubey YMolinaro MMarx D(2021)Branch-and-bound solves random binary IPs in polytimeProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458099(579-591)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458099
Erickson Jvan der Hoog IMiltzow T(2020)Smoothing the gap between NP and ER2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00099(1022-1033)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00099
Singh VChakraborty D(2020)Solving Knapsack Problem with Fuzzy Random Variable CoefficientsIntelligent Techniques and Applications in Science and Technology10.1007/978-3-030-42363-6_120(1037-1048)Online publication date: 3-Mar-2020
https://doi.org/10.1007/978-3-030-42363-6_120
Chen XPittel B(2019)On sparsity of the solution to a random quadratic optimization problemMathematical Programming10.1007/s10107-019-01456-2Online publication date: 9-Dec-2019
https://doi.org/10.1007/s10107-019-01456-2
Schulze BPaquete LKlamroth KFigueira J(2017)Bi-dimensional knapsack problems with one soft constraintComputers and Operations Research10.1016/j.cor.2016.07.01278:C(15-26)Online publication date: 1-Feb-2017
https://dl.acm.org/doi/10.1016/j.cor.2016.07.012
Chen XPeng J(2015)New Analysis on Sparse Solutions to Random Standard Quadratic Optimization Problems and ExtensionsMathematics of Operations Research10.1287/moor.2014.069240:3(725-738)Online publication date: 1-Mar-2015
https://dl.acm.org/doi/10.1287/moor.2014.0692
Veerapen NOchoa GHarman MBurke E(2015)An Integer Linear Programming approach to the single and bi-objective Next Release ProblemInformation and Software Technology10.1016/j.infsof.2015.03.00865:C(1-13)Online publication date: 1-Sep-2015
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Vaz DPaquete LFonseca CKlamroth KStiglmayr M(2015)Representation of the non-dominated set in biobjective discrete optimizationComputers and Operations Research10.1016/j.cor.2015.05.00363:C(172-186)Online publication date: 1-Nov-2015
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Mastin AJaillet P(2015)Average-Case Performance of Rollout Algorithms for Knapsack ProblemsJournal of Optimization Theory and Applications10.1007/s10957-014-0603-x165:3(964-984)Online publication date: 1-Jun-2015
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Chandrasekaran KVempala SNaor M(2014)Integer feasibility of random polytopesProceedings of the 5th conference on Innovations in theoretical computer science10.1145/2554797.2554838(449-458)Online publication date: 12-Jan-2014
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