NP-Completeness of deciding the feasibility of Linear Equations over binary-variables with coefficients and constants that are 0, 1, or -1

TL;DR

This paper demonstrates that deciding the feasibility of linear systems with binary-variable coefficients of 0, 1, or -1 is strongly NP-Complete by polynomial-time reductions from NP-Complete problems like SUBSET-SUM and 3-SAT.

Contribution

It establishes the NP-Completeness of a specific class of linear feasibility problems with restricted coefficients through polynomial reductions from classic NP-Complete problems.

Findings

Feasibility decision problem is strongly NP-Complete.

Polynomial-time reductions from SUBSET-SUM and 3-SAT.

Method to reduce coefficient magnitudes in Integer Linear Programs.

Abstract

We convert, within polynomial-time and sequential processing, NP-Complete Problems into a problem of deciding feasibility of a given system S of linear equations with constants and coefficients of binary-variables that are 0, 1, or -1. S is feasible, if and only if, the NP-Complete problem has a feasible solution. We show separate polynomial-time conversions to S, from the SUBSET-SUM and 3-SAT problems, both of which are NP-Complete. The number of equations and variables in S is bounded by a polynomial function of the size of the NP-Complete problem, showing that deciding the feasibility of S is strongly-NP-Complete. We also show how to apply the approach used for the SUBSET-SUM problem to decide the feasibility of Integer Linear Programs, as it involves reducing the coefficient-magnitudes of variables to the logarithm of their initial values, though the number of variables and equations are increased.