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A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems

Authors Kei Kimura , Kazuhisa Makino

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LIPIcs.ISAAC.2023.47.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Mathematical optimization
Keywords
  • linear programming problem
  • certifying algorithm
  • Horn system

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Abstract

Linear programming (LP) problems with gainfree Leontief substitution systems have been intensively studied in economics and operations research, and include the feasibility problem of a class of Horn systems, which arises in, e.g., polyhedral combinatorics and logic. This subclass of LP problems admits a strongly polynomial time algorithm, where devising such an algorithm for general LP problems is one of the major theoretical open questions in mathematical optimization and computer science. Recently, much attention has been paid to devising certifying algorithms in software engineering, since those algorithms enable one to confirm the correctness of outputs of programs with simple computations. Devising a combinatorial certifying algorithm for the feasibility of the fundamental class of Horn systems remains open for almost a decade. In this paper, we provide the first combinatorial (and strongly polynomial time) certifying algorithm for LP problems with gainfree Leontief substitution systems. As a by-product, we resolve the open question on the feasibility of the class of Horn systems.

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Kei Kimura and Kazuhisa Makino. A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 47:1-47:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.47

Author Details

Kei Kimura
  • Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan
Kazuhisa Makino
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Funding

Partially supported by JSPS KAKENHI Grant Number JP19K22841.
  • Kimura, Kei: Partially supported by JST, ACT-X Grant Number JPMJAX200C, Japan, and JSPS KAKENHI Grant Number JP21K17700.
  • Makino, Kazuhisa: Partially supported by JSPS KAKENHI Grant Number JP20H05967.

References

  1. Ilan Adler and Steven Cosares. A strongly polynomial algorithm for a special class of linear programs. Operations Research, 39:955-960, 1991. Google Scholar
  2. Richard Bellman. On a routing problem. Quarterly of applied mathematics, 16(1):87-90, 1958. Google Scholar
  3. Bart Bogaerts, Stephan Gocht, Ciaran McCreesh, and Jakob Nordström. Certified symmetry and dominance breaking for combinatorial optimisation. In Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI'22), 2022. Google Scholar
  4. Riccardo Cambini, Giorgio Gallo, and Maria Grazia Scutellà. Flows on hypergraphs. Mathematical Programming, 78(2):195-217, 1997. Google Scholar
  5. R. Chandrasekaran and K. Subramani. A combinatorial algorithm for horn programs. Discrete Optimization, 10:85-101, 2013. Google Scholar
  6. Maria Chudnovsky, Jan Goedgebeur, Oliver Schaudt, and Mingxian Zhong. Obstructions for three-coloring graphs with one forbidden induced subgraph. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 1774-1783. SIAM, 2016. Google Scholar
  7. Derek G Corneil, Barnaby Dalton, and Michel Habib. Ldfs-based certifying algorithm for the minimum path cover problem on cocomparability graphs. SIAM Journal on Computing, 42(3):792-807, 2013. Google Scholar
  8. Richard W. Cottle and Arthur F. Veinott, Jr. Polyhedral sets having a least element. Mathematical Programming, 3:238-249, 1972. Google Scholar
  9. George B. Dantzig. Optimal solution of a dynamic leontief model with substitution. Econometrica, 23(3):295-302, 1955. Google Scholar
  10. Marcel Dhiflaoui, Stefan Funke, Carsten Kwappik, Kurt Mehlhorn, Michael Seel, Elmar Schömer, Ralph Schulte, and Dennis Weber. Certifying and repairing solutions to large lps how good are lp-solvers? In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, pages 255-256, 2003. Google Scholar
  11. Lester R Ford Jr. Network flow theory. Technical report, Rand Corp Santa Monica Ca, 1956. Google Scholar
  12. Loukas Georgiadis and Robert E Tarjan. Dominator tree certification and divergent spanning trees. ACM Transactions on Algorithms (TALG), 12(1):1-42, 2015. Google Scholar
  13. Fred Glover. A bound escalation method for the solution of integer linear programs. Cahiers du Centre d'Etudes de Recherche Operationelle, 6(3):131-168, 1964. Google Scholar
  14. Andrew V Goldberg. Scaling algorithms for the shortest paths problem. SIAM Journal on Computing, 24(3):494-504, 1995. Google Scholar
  15. Pratik Bijaiprakash Gupta. A certifying algorithm for Horn constraint systems. Master’s thesis, The University of Texas at Dallas, 2014. Google Scholar
  16. Robert G Jeroslow, Kipp Martin, Ronald L Rardin, and Jinchang Wang. Gainfree leontief substitution flow problems. Mathematical Programming, 57(1):375-414, 1992. Google Scholar
  17. Haim Kaplan and Yahav Nussbaum. Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs. Discrete Applied Mathematics, 157(15):3216-3230, 2009. Google Scholar
  18. Kei Kimura and Kazuhisa Makino. Trichotomy for integer linear systems based on their sign patterns. Discrete Applied Mathematics, 200:67-78, 2016. URL: https://doi.org/10.1016/j.dam.2015.07.004.
  19. Kei Kimura and Kazuhisa Makino. A combinatorial certifying algorithm for linear programming problems with gainfree leontief substitution systems. arXiv, 2023. URL: https://arxiv.org/abs/2306.03368.
  20. Dieter Kratsch, Ross M McConnell, Kurt Mehlhorn, and Jeremy P Spinrad. Certifying algorithms for recognizing interval graphs and permutation graphs. SIAM Journal on Computing, 36(2):326-353, 2006. Google Scholar
  21. Wassily W Leontief. The structure of American economy, 1919-1939. Oxford University Press, second edition, 1951. Google Scholar
  22. Ross M McConnell, Kurt Mehlhorn, Stefan Näher, and Pascal Schweitzer. Certifying algorithms. Computer Science Review, 5(2):119-161, 2011. Google Scholar
  23. Kurt Mehlhorn, Stefan Naher, and Stefan Näher. LEDA: A platform for combinatorial and geometric computing. Cambridge university press, 1999. Google Scholar
  24. Kurt Mehlhorn, Adrian Neumann, and Jens M Schmidt. Certifying 3-edge-connectivity. Algorithmica, 77(2):309-335, 2017. Google Scholar
  25. Antoine Miné. The octagon abstract domain. Higher-order and symbolic computation, 19(1):31-100, 2006. Google Scholar
  26. Edward F Moore. The shortest path through a maze. In Proc. Int. Symp. Switching Theory, 1959, pages 285-292, 1959. Google Scholar
  27. Jens M Schmidt. Contractions, removals, and certifying 3-connectivity in linear time. SIAM Journal on Computing, 42(2):494-535, 2013. Google Scholar
  28. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1998. Google Scholar
  29. K Subramani and Piotr Wojciechowski. A combinatorial certifying algorithm for linear feasibility in utvpi constraints. Algorithmica, 78(1):166-208, 2017. Google Scholar
  30. K Subramani and James Worthington. A new algorithm for linear and integer feasibility in horn constraints. In International Conference on AI and OR Techniques in Constriant Programming for Combinatorial Optimization Problems, pages 215-229. Springer, 2011. Google Scholar
  31. J.D. Ullman and A. Van Gelder. Efficient test for top-down termination of logical rules. Journal of the Association for Computing Machinery, 35:345-373, 1988. Google Scholar
  32. Hans Van Maaren and Chuangyin Dang. Simplicial pivoting algorithms for a tractable class of integer programs. Journal of Combinatorial Optimization, 6(2):133-142, 2002. Google Scholar
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