Why Linear Programming cannot solve large instances of NP-complete problems in polynomial time

TL;DR

This paper examines the limitations of linear programming approaches in solving large NP-complete problems, demonstrating that despite recent methods, polynomial-time solutions for infinite instances remain infeasible.

Contribution

It provides a critical analysis showing that LP-based methods cannot solve arbitrarily large NP-complete problems in polynomial time, clarifying their limitations.

Findings

LP methods solve large but finite instances efficiently

They cannot solve infinitely large instances in polynomial time

Recent LP approaches do not extend to polynomial-time solutions for all NP-complete problems

Abstract

This article discusses ability of Linear Programming models to be used as solvers of NP-complete problems. Integer Linear Programming is known as NP-complete problem, but non-integer Linear Programming problems can be solved in polynomial time, what places them in P class. During past three years there appeared some articles using LP to solve NP-complete problems. This methods use large number of variables (O(n^9)) solving correctly almost all instances that can be solved in reasonable time. Can they solve infinitively large instances? This article gives answer to this question.