满足子集合与子集和的重构：容易找到，难以连接
Reconfiguration of Satisfying Assignments and Subset Sums: Easy to Find, Hard to Connect
摘要：
我们考虑了重构问题的计算复杂性，其中一个是给定的两个组合配置满足某些约束，并要求转换成另一个使用初等变换，同时满足约束在任何时候。这样的问题在许多环境中自然出现，例如模型检验、运动规划、枚举和采样以及娱乐数学。我们在这个家庭，其中的约束和操作是特别简单的问题提供硬度的结果。更确切地说，我们证明了PSPACE的完整性以下决策问题：$\bullet$给定两个满意的分配到一个平面单调的情况下，不全等于3-SAT，可以一个分配转化为其他单变量“翻转”（分配变化），保持可满足性在每一步？$\bullet$给定一个整数集合S的两个子集具有相同的和，一个子集是否可以通过一次添加或删除S的最多三个元素而转换为另一个子集，使得中间子集也具有相同的和？给定$\bullet$中的两个点包含在由常数个线性不等式指定的多面体P中，在n-超立方体中是否有一条路连接这两个点并包含在P中？这些问题可以被解释为NP标准问题的重构类似物。有趣的是，在我们的约简中，作为重新配置问题的输入出现的NP问题的实例可以被证明位于P中。特别是，S的元素和定义P的不等式的系数可以被限制为具有对数位长。
英文摘要：
We consider the computational complexity of reconfiguration problems, in which one is given two combinatorial configurations satisfying some constraints, and is asked to transform one into the other using elementary transformations, while satisfying the constraints at all times. Such problems appear naturally in many contexts, such as model checking, motion planning, enumeration and sampling, and recreational mathematics. We provide hardness results for problems in this family, in which the constraints and operations are particularly simple. More precisely, we prove the PSPACE-completeness of the following decision problems: $\bullet$ Given two satisfying assignments to a planar monotone instance of Not-All-Equal 3-SAT, can one assignment be transformed into the other by single variable `flips' (assignment changes), preserving satisfiability at every step? $\bullet$ Given two subsets of a set S of integers with the same sum, can one subset be transformed into the other by adding or removing at most three elements of S at a time, such that the intermediate subsets also have the same sum? $\bullet$ Given two points in $\{0,1\}^n$ contained in a polytope P specified by a constant number of linear inequalities, is there a path in the n-hypercube connecting the two points and contained in P? These problems can be interpreted as reconfiguration analogues of standard problems in NP. Interestingly, the instances of the NP problems that appear as input to the reconfiguration problems in our reductions can be shown to lie in P. In particular, the elements of S and the coefficients of the inequalities defining P can be restricted to have logarithmic bit-length.
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