Three-dimensional matching is NP-Hard

TL;DR

This paper presents a linear-time reduction for 3-dimensional matching, improving the lower bound on its computational complexity under the exponential time hypothesis.

Contribution

It introduces a more efficient reduction from 3SAT to 3DM, enhancing the understanding of 3DM's computational hardness.

Findings

Provides a linear-time reduction from 3SAT to 3DM

Improves the exponential time lower bound from $2^{o( oot4 m{m})}$ to $2^{o(m)}$

Strengthens the NP-hardness evidence for 3DM

Abstract

The standard proof of NP-Hardness of 3DM provides a power-$4$ reduction of 3SAT to 3DM. In this note, we provide a linear-time reduction. Under the exponential time hypothesis, this reduction improves the runtime lower bound from $2^{o(\sqrt[4]{m})}$ (under the standard reduction) to $2^{o(m)}$.