Redundancy elimination with a lexicographic solved form

Abstract

Detection of redundant linear constraints is important for a variety of applications including linear programming, variable elimination, canonical forms, and convex hull algorithms to name a few. Systematic semantic methods (i.e. optimization) exist to identify all redundant constraints but they are generally computationally expensive. It is thus desirable to identify syntactic criteria to reduce the need for semantic methods. This paper presents new syntactic criteria to detect redundant and non-redundant constraints. The results exploit a lexicographic solved form for linear programming (originally proposed to detect implicit equalities) which makes it possible to define syntactic criteria much stronger than those proposed in the past. In particular, one of our results shows that a system in a lexicographic solved form with n constraints and m non-basic variables has at most min(2n-1,n+m) redundant variables. In addition, the paper presents new syntactic criteria to reduce the size of the constraint system before further applications of syntactic and semantic methods. These theoretical results have been integrated in a complete system to detect redundant linear constraints. Experimental evaluation shows that our syntactic criteria decide the redundancy of about 90 constraints and may produce orders of magnitude improvement in efficiency over previous work whose criteria were deciding between 24 and 56 increase with the size and with the sparsity of the constraint system and are especially dramatic when the number of variables is large compared to the number of constraints.