Repeating Patterns in Linear Programs that express NP-Complete Problems

TL;DR

This paper explores how repeating patterns in linear programs can represent NP-Complete problems efficiently, potentially enabling new methods for feasibility decision despite exponential inequality bounds.

Contribution

It introduces a class of linear programs with repeating coefficient patterns that encode NP-Complete problems, offering a novel approach to analyze their feasibility.

Findings

Linear programs can encode NP-Complete problems with repeating coefficient patterns.

Efficient description of these linear programs is possible despite exponential inequality bounds.

Two conjectures are proposed to aid in deciding the feasibility of such linear programs.

Abstract

One of my recent papers transforms an NP-Complete problem into the question of whether or not a feasible real solution exists to some Linear Program. The unique feature of this Linear Program is that though there is no explicit bound on the minimum required number of linear inequalities, which is most probably exponential to the size of the NP-Complete problem, the Linear Program can still be described efficiently. The reason for this efficient description is that coefficients keep repeating in some pattern, even as the number of inequalities is conveniently assumed to tend to Infinity. I discuss why this convenient assumption does not change the feasibility result of the Linear Program. I conclude with two Conjectures, which might help to make an efficient decision on the feasibility of this Linear Program.