A Flow-Network-Based Polynomial-Time Approximation Algorithm for the Minimum Constrained Input Structural Controllability Problem

Abstract

This paper deals with minimum cost constrained input selection (minCCIS) for state-space structured systems. Our goal is to optimally select an input set from the given inputs such that the system is structurally controllable when the set of states that each input can influence is prespecified and each input has a cost associated with it. This problem is known to be NP-hard. First, we give a flow-network-based novel necessary and sufficient graph theoretic condition for checking structural controllability. Using this condition, we propose a polynomial time reduction of the problem to a known NP-hard problem: the minimum-cost fixed-flow problem (MCFF). Subsequently, we show that approximation schemes of MCFF directly apply to the minCCIS problem. Using the special structure of the flow-network constructed for the structured system, we formulate a polynomial time approximation algorithm based on minimum weight bipartite matching problem and a greedy scheme for solving the MCFF problem on the system flow-network. The proposed algorithm gives a so-called $\Delta$ -approximate solution to MCFF, where $\Delta$ denotes the maximum in-degree of input vertices in the flow-network of the structured system.