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

Authors Kei Kimura , Kazuhisa Makino

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

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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)


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.

Subject Classification

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


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