### 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 language | English |
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Title of host publication | WALCOM |

Subtitle of host publication | Algorithms and Computation - 14th International Conference, WALCOM 2020, Proceedings |

Editors | M. Sohel Rahman, Kunihiko Sadakane, Wing-Kin Sung |

Publisher | Springer |

Pages | 336-341 |

Number of pages | 6 |

ISBN (Print) | 9783030398804 |

DOIs | |

Publication status | Published - 2020 Jan 1 |

Event | 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020 - Singapore, Singapore Duration: 2020 Mar 31 → 2020 Apr 2 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12049 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020 |
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Country | Singapore |

City | Singapore |

Period | 20/3/31 → 20/4/2 |

### Keywords

- Combinatorial reconfiguration
- Complexity index
- Integer linear systems

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*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