Trichotomy for the reconfiguration problem of integer linear systems

Kei Kimura, Akira Suzuki

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we consider the reconfiguration problem of integer linear systems. In this problem, we are given an integer linear system I and two feasible solutions s and t of I, and then asked to transform s to t by changing a value of only one variable at a time, while maintaining a feasible solution of I throughout. Z(I) for I is the complexity index introduced by Kimura and Makino (Discrete Applied Mathematics 200:67–78, 2016), which is defined by the sign pattern of the input matrix. We analyze the complexity of the reconfiguration problem of integer linear systems based on the complexity index Z(I) of given I. We show that the problem is (i) solvable in constant time if Z(I) is less than one, (ii) weakly coNP-complete and pseudo-polynomially solvable if Z(I) is exactly one, and (iii) PSPACE-complete if Z(I) is greater than one. Since the complexity indices of Horn and two-variable-par-inequality integer linear systems are at most one, our results imply that the reconfiguration of these systems are in coNP and pseudo-polynomially solvable. Moreover, this is the first result that reveals coNP-completeness for a reconfiguration problem, to the best of our knowledge.

Original languageEnglish
Title of host publicationWALCOM
Subtitle of host publicationAlgorithms and Computation - 14th International Conference, WALCOM 2020, Proceedings
EditorsM. Sohel Rahman, Kunihiko Sadakane, Wing-Kin Sung
PublisherSpringer
Pages336-341
Number of pages6
ISBN (Print)9783030398804
DOIs
Publication statusPublished - 2020 Jan 1
Event14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020 - Singapore, Singapore
Duration: 2020 Mar 312020 Apr 2

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12049 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020
CountrySingapore
CitySingapore
Period20/3/3120/4/2

Keywords

  • Combinatorial reconfiguration
  • Complexity index
  • Integer linear systems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Kimura, K., & Suzuki, A. (2020). Trichotomy for the reconfiguration problem of integer linear systems. In M. S. Rahman, K. Sadakane, & W-K. Sung (Eds.), WALCOM: Algorithms and Computation - 14th International Conference, WALCOM 2020, Proceedings (pp. 336-341). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 12049 LNCS). Springer. https://doi.org/10.1007/978-3-030-39881-1_29