结构能控性约束最小输入选择的全幺模性与强多项式可解性：一种基于线性规划的方法
Total Unimodularity and Strongly Polynomial Solvability of Constrained Minimum Input Selections for Structural Controllability: an LP-based Method
AI总结：
本文将结构能控性下带代价与基数约束的最优输入选择问题建模为整数线性规划，并证明在特定输入配置条件下约束矩阵全幺模，从而通过LP松弛实现强多项式时间求解。
摘要：
本文研究了结构化系统中若干由代价稀疏性诱导的最优输入选择问题。给定一个自治系统和一组预先指定的输入链路，其中每条输入链路具有非负代价。这些问题包括：选择最小代价的输入链路，以及在其基数有上界约束的情况下选择代价尽可能小的输入链路，所有这些都旨在保证系统的结构能控性。当前研究表明，在专用输入情形下（即每个输入只能驱动一个状态变量），前一个问题可通过一些图论算法在多项式时间内求解，而一般的非平凡约束情形在很大程度上尚未被探索。我们证明这些问题可以表述为等价的整数线性规划（ILP）问题。在预先指定的输入配置满足一定条件（该条件将专用输入情形作为特例包含在内）的前提下，我们证明这些ILP的约束矩阵是全幺模的。这一性质使我们能够仅通过其线性规划（LP）松弛来高效求解这些ILP，从而为这些问题提供一种具有多项式时间复杂度的统一代数方法。进一步表明，这些问题可以在强多项式时间内求解，其复杂度与代价的大小和基数上界无关。最后，给出一个例子来说明所提方法的有效性。
英文摘要：
This paper investigates several cost-sparsity induced optimal input selection problems for structured systems. Given are an autonomous system and a prescribed set of input links, where each input link has a non-negative cost. The problems include selecting the minimum cost of input links, and selecting the input links with the smallest possible cost with a bound on their cardinality, all to ensure system structural controllability. Current studies show that in the dedicated input case (i.e., each input can actuate only a state variable), the former problem is polynomially solvable by some graph-theoretic algorithms, while the general nontrivial constrained case is largely unexploited. We show these problems can be formulated as equivalent integer linear programming (ILP) problems. Subject to a certain condition on the prescribed input configurations that contains the dedicated input one as a special case, we demonstrate that the constraint matrices of these ILPs are totally unimodular. This property allows us to solve those ILPs efficiently simply via their linear programming (LP) relaxations, leading to a unifying algebraic method for these problems with polynomial time complexity. It is further shown those problems could be solved in strongly polynomial time, independent of the size of the costs and cardinality bounds. Finally, an example is provided to illustrate the power of the proposed method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
