整数线性系统重构问题的三分法
Trichotomy for the reconfiguration problem of integer linear systems
摘要：
本文研究整数线性系统的重构问题。在这个问题中，我们给出了一个整数线性系统$I$和两个可行解$\boldsymbol{s}$和$I$的$\boldsymbol{t}$，然后要求转换$\boldsymbol{s}$到$\boldsymbol{t}$通过改变一个变量的值在同一时间，而保持一个可行的解决方案$I$始终。$I$的$Z(I)$是Kimura和Makino（Discrete Applied Mathematics 200：67- 78，2016）引入的复杂度指数，由输入矩阵的符号模式定义。基于给定$I$的复杂性指标$Z(I)$，分析了整数线性系统重构问题的复杂性.然后，我们表明，该问题是（i）可解的常数时间，如果$Z(I)$是小于一，（ii）弱coNP-完全和伪多项式可解的，如果$Z(I)$正好是一个，和（iii）PSPACE-完全，如果$Z(I)$是大于一。由于Horn和两变量par-inequality整数线性系统的复杂性指数最多为1，我们的结果意味着这些系统的重构是coNP和伪多项式可解的.此外，据我们所知，这是第一个结果，揭示了一个重新配置问题的coNP完全性。
英文摘要：
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 $\boldsymbol{s}$ and $\boldsymbol{t}$ of $I$, and then asked to transform $\boldsymbol{s}$ to $\boldsymbol{t}$ 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.
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