research-article

Combinatorial dominance guarantees for problems with infeasible solutions

Authors:

Steven S. Skiena,

Yochai TwittoAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 5, Issue 1

Article No.: 8, Pages 1 - 29

https://doi.org/10.1145/1435375.1435383

Published: 12 December 2008 Publication History

Abstract

The design and analysis of approximation algorithms for NP-hard problems is perhaps the most active research area in the theory of combinatorial algorithms. In this article, we study the notion of a combinatorial dominance guarantee as a way for assessing the performance of a given approximation algorithm. An f(n) dominance bound is a guarantee that the heuristic always returns a solution not worse than at least f(n) solutions. We give tight analysis of many heuristics, and establish novel and interesting dominance guarantees even for certain inapproximable problems and heuristic search algorithms. For example, we show that the maximal matching heuristic of VERTEX COVER offers a combinatorial dominance guarantee of 2ⁿ − (1.839 + o(1))ⁿ. We also give inapproximability results for most of the problems we discuss.

References

[1]

Alon, N., Gutin, G., and Krivelevich, M. 2004. Algorithms with large domination ratio. J. Algor. 50, 118--131.

Digital Library

[2]

Balas, E., and Simonetti, N. 2001. Linear time dynamic programming algorithms for some new classes of restricted TSPs: A computational study. INFORMS J. Comput. 13, 56--75.

Digital Library

[3]

Bellare, M., Goldreich, O., and Sudan, M. 1995. Free bits, PCPs, and non-approximability—towards tight results. In Proceedings of the IEEE 36th Annual Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA, 422--431.

Digital Library

[4]

Berend, D., Skiena, S., and Twitto, Y. 2006. Dominance certificates for combinatorial optimization problems. submitted.

[5]

Deineko, V., and Woeginger, G. 2000. A study of exponential neighborhoods for the traveling salesman problem and the quadratic assignment problem. Math. Prog., Ser. A 87, 519--542.

[6]

Glover, F., and Punnen, A. 1997. The travelling salesman problem: New solvable cases and linkages with the development of new approximation algorithms. J. Oper. Res. Soc. 48, 502--510.

[7]

Gutin, G., Vainshtein, A., and Yeo, A. 2003. Domination analysis of combinatorial optimization problems. DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 129.

Digital Library

[8]

Gutin, G., and Yeo, A. 2001. TSP tour domination and hamilton cycle decompositions of regular digraphs. Oper. Res. Lett. 28, 3, 107--111.

Digital Library

[9]

Gutin, G., and Yeo, A. 2002. Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number. Discr. Appl. Math. 119, 1--2, 107--116.

Digital Library

[10]

Gutin, G., Yeo, A., and Zverovich, A. 2002a. Exponential neighborhoods and domination analysis for the TSP. In The Traveling Salesman Problem and its Variations, G. Gutin and A. Punnen, Eds. Kluwer Academic Publishers, Boston, Chapter 6, 223--256.

[11]

Gutin, G., Yeo, A., and Zverovich, A. 2002b. Traveling salesman should not be greedy: Domination analysis of greedy-type heuristics for the TSP. Disc. Appl. Math. 117, 1--3, 81--86.

[12]

Hassin, R., and Khuller, S. 2001. z-approximations. J. Algor. 41, 429--442.

Digital Library

[13]

Ibarra, O., and Kim, C. 1975. Fast approximation algorithms for knapsack and sum of subset problems. J. ACM 22, 463--468.

Digital Library

[14]

Levin, L. 1986. Average case complete problems. SIAM J. Computing 15, 285--286.

Digital Library

[15]

Phan, V., Skiena, S., and Sumazin, P. 2003. A model for analyzing black-box optimization. In Lecture Notes in Computer Science, vol. 2748. Springer-Verlag, New York, pp. 424--438.

[16]

Punnen, A., Margot, F., and Kabadi, S. 2001. TSP heuristics: Domination analysis and complexity. Res. rep. 2001-06, Dept. of Mathematics, Univ. Kentucky, Mar.

[17]

Rublineckii, V. 1973. Estimates of the accuracy of procedures in the traveling salesman problem. Num. Math. Comput. Tech. (in Russian) 4, 18--23.

[18]

Sarvanov, V., and Doroshko, N. 1981a. The approximate solution of the traveling salesman problem by a local algorithm that searches neighborhoods of exponential cardinality in quadratic time. Softw.: Algor. Prog. (in Russian) 31, 8--11.

[19]

Sarvanov, V., and Doroshko, N. 1981b. The approximate solution of the traveling salesman problem by a local algorithm that searches neighborhoods of factorial cardinality in cubic time. Soft.: Algor. Prog. (in Russian) 31, 11--13.

[20]

Sperner, E. 1928. Ein satzber untermegen einer endlichen menge. Math. Zeitschr. 27, 285--286.

[21]

Valient, L. 1979. The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189--201.

[22]

Zemel, E. 1981. Measuring the quality of approximate solutions to zero-one programming problems. Math. Oper. Res. 6, 319--332.

Digital Library

Cited By

Gutin G(2024)Domination Analysis in Combinatorial OptimizationEncyclopedia of Optimization10.1007/978-3-030-54621-2_136-1(1-12)Online publication date: 8-May-2024
https://doi.org/10.1007/978-3-030-54621-2_136-1
Berend DSkiena STwitto Y(2018)Dominance Certificates for Combinatorial Optimization ProblemsOptimization Problems in Graph Theory10.1007/978-3-319-94830-0_6(107-122)Online publication date: 28-Sep-2018
https://doi.org/10.1007/978-3-319-94830-0_6
Punnen ASripratak PKarapetyan D(2015)Average value of solutions for the bipartite boolean quadratic programs and rounding algorithmsTheoretical Computer Science10.1016/j.tcs.2014.11.008565:C(77-89)Online publication date: 2-Feb-2015
https://dl.acm.org/doi/10.1016/j.tcs.2014.11.008
Show More Cited By

Index Terms

Combinatorial dominance guarantees for problems with infeasible solutions
1. Theory of computation
  1. Design and analysis of algorithms

Recommendations

A discrete gravitational search algorithm for solving combinatorial optimization problems

Metaheuristics are general search strategies that, at the exploitation stage, intensively exploit areas of the solution space with high quality solutions and, at the exploration stage, move to unexplored areas of the solution space when necessary. The ...
Coloring graphs by iterated local search traversing feasible and infeasible solutions

Graph coloring is one of the hardest combinatorial optimization problems for which a wide variety of algorithms has been proposed over the last 30 years. The problem is as follows: given a graph one has to assign a label to each vertex such that no ...
Evolving heuristically difficult instances of combinatorial problems
GECCO '09: Proceedings of the 11th Annual conference on Genetic and evolutionary computation

When evaluating a heuristic for a combinatorial problem, randomly generated instances of the problem may not provide a thorough exploration of the heuristic's performance, and it may not be obvious what kinds of instances challenge or confound the ...

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 5, Issue 1

November 2008

281 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/1435375

Issue’s Table of Contents

Copyright © 2008 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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 12 December 2008

Accepted: 01 September 2007

Revised: 01 August 2007

Received: 01 March 2006

Published in TALG Volume 5, Issue 1

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
315
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)1

Reflects downloads up to 25 Dec 2024

Other Metrics

View Author Metrics

Citations

Cited By

Gutin G(2024)Domination Analysis in Combinatorial OptimizationEncyclopedia of Optimization10.1007/978-3-030-54621-2_136-1(1-12)Online publication date: 8-May-2024
https://doi.org/10.1007/978-3-030-54621-2_136-1
Berend DSkiena STwitto Y(2018)Dominance Certificates for Combinatorial Optimization ProblemsOptimization Problems in Graph Theory10.1007/978-3-319-94830-0_6(107-122)Online publication date: 28-Sep-2018
https://doi.org/10.1007/978-3-319-94830-0_6
Punnen ASripratak PKarapetyan D(2015)Average value of solutions for the bipartite boolean quadratic programs and rounding algorithmsTheoretical Computer Science10.1016/j.tcs.2014.11.008565:C(77-89)Online publication date: 2-Feb-2015
https://dl.acm.org/doi/10.1016/j.tcs.2014.11.008
Wang Y(2015)An approximate method to compute a sparse graph for traveling salesman problemExpert Systems with Applications: An International Journal10.1016/j.eswa.2015.02.03742:12(5150-5162)Online publication date: 15-Jul-2015
https://dl.acm.org/doi/10.1016/j.eswa.2015.02.037
Kühn DOsthus DPatel V(2015)A domination algorithm for {0,1}-instances of the travelling salesman problemRandom Structures & Algorithms10.1002/rsa.2060048:3(427-453)Online publication date: 8-Oct-2015
https://doi.org/10.1002/rsa.20600
Wang Y(2013)A Representation Model for TSP2013 IEEE 10th International Conference on High Performance Computing and Communications & 2013 IEEE International Conference on Embedded and Ubiquitous Computing10.1109/HPCC.and.EUC.2013.38(204-209)Online publication date: Nov-2013
https://doi.org/10.1109/HPCC.and.EUC.2013.38
Punnen ASripratak PKarapetyan D(2013)Domination analysis of algorithms for bipartite boolean quadratic programsProceedings of the 19th international conference on Fundamentals of Computation Theory10.1007/978-3-642-40164-0_26(271-282)Online publication date: 19-Aug-2013
https://dl.acm.org/doi/10.1007/978-3-642-40164-0_26
Twitto Y(2008)Dominance guarantees for above-average solutionsDiscrete Optimization10.1016/j.disopt.2007.11.0095:3(563-568)Online publication date: 1-Aug-2008
https://dl.acm.org/doi/10.1016/j.disopt.2007.11.009
Gutin GGoldengorin BHuang J(2008)Worst case analysis of Max-Regret, Greedy and other heuristics for Multidimensional Assignment and Traveling Salesman ProblemsJournal of Heuristics10.1007/s10732-007-9033-314:2(169-181)Online publication date: 1-Apr-2008
https://dl.acm.org/doi/10.1007/s10732-007-9033-3

View Options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents