TY - JOUR

T1 - Trichotomy for the reconfiguration problem of integer linear systems

AU - Kimura, Kei

AU - Suzuki, Akira

N1 - Funding Information:
The first author is partially supported by JSPS KAKENHI Grant Number JP17K12636 , Japan. The second author is partially supported by JST CREST Grant Number JPMJCR1402 , and JSPS KAKENHI Grant Numbers JP17K12636 and JP18H04091 , Japan.

PY - 2021/2/8

Y1 - 2021/2/8

N2 - 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 then 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.

AB - 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 then 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.

KW - Complexity trichotomy

KW - Integer linear system

KW - Reconfiguration

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U2 - 10.1016/j.tcs.2020.12.025

DO - 10.1016/j.tcs.2020.12.025

M3 - Article

AN - SCOPUS:85098178527

VL - 856

SP - 88

EP - 109

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -