NP-hard sets are not sparse unless P=NP: An exposition of a simple proof of Mahaney's Theorem, with applications

TL;DR

This paper presents a simple, accessible proof of Mahaney's Theorem, demonstrating that NP-hard sets cannot be sparse unless P equals NP, and explores its applications in complexity theory and representation theory.

Contribution

It provides an easy-to-teach proof of Mahaney's Theorem and illustrates its applications in fundamental questions of complexity and representation theory.

Findings

NP-hard sets are not sparse unless P=NP

The proof is simple enough for undergraduate teaching

Applications to representation theory and geometric complexity theory

Abstract

Mahaney's Theorem states that, assuming $\mathsf{P} \neq \mathsf{NP}$, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind ("Geometric sets of low information content," Theoret. Comp. Sci., 1996). This proof is so simple that it can easily be taught to undergraduates or a general graduate CS audience - not just theorists! - in about 10 minutes, which the author has done successfully several times. We also include applications of Mahaney's Theorem to fundamental questions that bright undergraduates would ask which could be used to fill the remaining hour of a lecture, as well as an application (due to Ikenmeyer, Mulmuley, and Walter, arXiv:1507.02955) to the representation theory of the symmetric group and the Geometric Complexity Theory Program. To this author, the fact that sparsity results on NP-complete sets have an application to classical questions in representation theory says that they are not only a gem of classical theoretical computer science, but indeed a gem of mathematics.