Total Unimodularity and Strongly Polynomial Solvability of Constrained Minimum Input Selections for Structural Controllability: An LP-Based Method

This paper investigates several cost-sparsity induced optimal input selection problems for structured systems. Given an autonomous system and a prescribed set of links, where each link has non-negative cost, the include selecting minimum cost links with smallest possible while bounding their cardinality to achieve structural controllability. Current studies show that in dedicated case (i.e., can actuate only one state variable), former problem is polynomially solvable by some graph-theoretic algorithms, general nontrivial constrained largely unexplored. We these be formulated as equivalent integer linear programming (ILP) problems. Subject 'source strongly-connected component grouped constraint', which contains special case, we demonstrate constraint matrices ILPs are totally unimodular . property allows us solve efficiently via (LP) relaxations, leading unifying algebraic method polynomial time complexity. further solved strongly time, independent size costs bounds. Finally, provide example illustrate power proposed method.