On Feasibility, Boundedness and Redundancy of Systems of Linear Constraints over R ²-Plane

Abstract

A system of linear constraints is represented as a set of half-planes S = {(a ^j X + b ^j Y + c ^j ≤ 0) j = 1...N}. Therefore, in the context of linear constraints two terms “set” and “system” can be used interchangeably whenever one of two is suitable. A set of linear constraints represents a convex polygon that is the intersection of all half-planes in the set. The convex polygon represented by S is called the feasible polygon of S. Such sets of linear constraints can be used as a new way of represent spatial data. These sets need to be manipulated efficiently and stored using minimal storage. It is natural to store only sets of linear constraints which are feasible and in irredundant format. Therefore, it is very important to find out if a given system is feasible and/or bounded and to find the minimal (irredundant) set of linear constraint which have the same feasible area with the given one. LASSEZ and MAHER (1988) have investigated algorithms to check if a system of linear constraints over multidimensional R ^d is feasible. LASSEZ et al (1989) have investigated algorithms to eliminate redundant constraints from a system of linear constraints over R ^d. Their algorithms are based on the Fourier variable elimination (similar with Gaussian elimination in solving the linear system of equations) and therefore have the running time O(N ²) where N is the number of constraints, and as such it is not efficient. DYER (1984) and MEGIDDO (1983) have independent proposed linear time algorithms to solve the linear programming problem in 2- and 3-dimension cases.