Efficiently expressing feasibility problems in Linear Systems, as feasibility problems in Asymptotic-Linear-Programs

TL;DR

This paper introduces a polynomial-time algorithm that transforms linear feasibility problems into Asymptotic Linear Programs, enabling efficient feasibility analysis and variable subset solutions.

Contribution

The paper presents a novel polynomial-time method to convert linear systems into ALPs, facilitating feasibility checks and subset solution determination.

Findings

Algorithm efficiently constructs ALPs from linear systems.

Feasibility of original system is equivalent to feasibility of at least one ALP.

Method applies to systems with linear constraints and integer coefficients.

Abstract

We present a polynomial-time algorithm that obtains a set of Asymptotic Linear Programs (ALPs) from a given linear system S, such that one of these ALPs admits a feasible solution if and only if S admits a feasible solution. We also show how to use the same algorithm to determine whether or not S admits a non-trivial solution for any desired subset of its variables. S is allowed to consist of linear constraints over real variables with integer coefficients, where each constraint has either a lesser-than-or-equal-to, or a lesser-than, or a not-equal-to relational operator. Each constraint of the obtained ALPs has a lesser-than-or-equal-to relational operator, and the coefficients of its variables vary linearly with respect to the time parameter that tends to positive infinity.