A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

TL;DR

This paper introduces a new multilayered PCP construction to establish the NP-hardness of approximating the hypergraph vertex cover problem within a factor close to the optimal, extending previous hardness results.

Contribution

It presents a novel multilayered PCP framework that extends the Raz verifier, proving tight hardness of approximation for hypergraph vertex cover.

Findings

Proves NP-hardness of approximating E$k$-Vertex-Cover within factor $(k-1-)$ for all $k \u2265 3$.

Uses biased Long-Code and combinatorial properties of intersecting families.

Shows the hardness result is essentially tight with existing approximation algorithms.

Abstract

Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is essentially tight as this problem can be easily approximated within factor $k$. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of $s$-wise $t$-intersecting families of subsets.