A Reply to Hofman On: "Why LP cannot solve large instances of NP-complete problems in polynomial time"

TL;DR

This paper critiques Hofman's claim that NP-complete problems cannot be solved by polynomial-sized linear programs, demonstrating his counter-example to TSP and QAP formulations is flawed and invalid.

Contribution

The paper refutes Hofman's proposed counter-example, reinforcing the limitations of linear programming approaches for NP-complete problems.

Findings

Hofman's construct is flawed

His counter-example to TSP and QAP is invalid

Linear programming cannot solve large NP-complete problems in polynomial time

Abstract

Using an approach that seems to be patterned after that of Yannakakis, Hofman argues that an NP-complete problem cannot be formulated as a polynomial bounded-sized linear programming problem. He then goes on to propose a "construct" that he claims to be a counter-example to recently published linear programming formulations of the Traveling Salesman Problem (TSP) and the Quadratic Assignment Problems (QAP), respectively. In this paper, we show that Hofman's construct is flawed, and provide further proof that his "counter-example" is invalid.