Using Matching to Detect Infeasibility of Some Integer Programs

A novel matching based heuristic algorithm designed to detect specially formulated infeasible {0, 1} IPs is presented. The algorithm's input is a set of nested doubly stochastic subsystems and a set E of instance defining variables set at zero level. The algorithm deduces additional variables at zero level until either a constraint is violated (the IP is infeasible), or no more variables can be deduced zero (the IP is undecided). All feasible IPs, and all infeasible IPs not detected infeasible are undecided. We successfully apply the algorithm to a small set of specially formulated infeasible {0, 1} IP instances of the Hamilton cycle decision problem. We show how to model both the graph and subgraph isomorphism decision problems for input to the algorithm. Increased levels of nested doubly stochastic subsystems can be implemented dynamically. The algorithm is designed for parallel processing, and for inclusion of techniques in addition to matching. 1. Introduction. We present a novel matching based heuristic algorithm deigned to detect specially formulated infeasible {0, 1} IPs. It either detects an infeasible IP or exits undecided. It does not solve an IP. We call it the triple overlay matching based closure algorithm (the algorithm). Input to the algorithm is an IP whose constraints are a set of nested doubly stochastic boolean subsystems [12] together with a set E of instance defining variables set at zero level. The IP's solution set is a subset of the set of n! nxn permutation matrices P , written as n 2 xn 2 block permutation matrices Q each with block structure P. The algorithm is a polynomial time search that deduces additional variables at zero level via matching until either a constraint is violated in which case the IP is infeasible, or we can go no further in which case the IP is undecided. If the IP is decided infeasible, a set of variables deduced to be at zero level can be used to test and display a set of violated constraints. If the IP is undecided, additional variables deduced zero can be added to E, and nothing more can be concluded. While some infeasible IPs may fail to be detected infeasible (not yet found), feasible IPs can only fall in the undecided category. In section 2 we present the generic IP required as input to the algorithm, and we view the set of all solutions of the IP as an n …