Constant Approximating k-Clique is W[1]-hard

TL;DR

This paper proves that approximating the maximum clique size within any constant factor is W[1]-hard, showing no fixed-parameter tractable algorithm can efficiently distinguish between large and small cliques unless FPT=W[1].

Contribution

It introduces a simple parameterized reduction that demonstrates the W[1]-hardness of constant-factor approximation for the k-Clique problem.

Findings

Constant factor approximation is W[1]-hard.

No fixed-parameter tractable algorithm can distinguish large from small cliques unless FPT=W[1].

Provides a reduction with specific bounds on the size of the constructed graph.

Abstract

For every graph $G$, let $\omega(G)$ be the largest size of complete subgraph in $G$. This paper presents a simple algorithm which, on input a graph $G$, a positive integer $k$ and a small constant $\epsilon>0$, outputs a graph $G'$ and an integer $k'$ in $2^{\Theta(k^5)}\cdot |G|^{O(1)}$-time such that (1) $k'\le 2^{\Theta(k^5)}$, (2) if $\omega(G)\ge k$, then $\omega(G')\ge k'$, (3) if $\omega(G)<k$, then $\omega(G')< (1-\epsilon)k'$. This implies that no $f(k)\cdot |G|^{O(1)}$-time algorithm can distinguish between the cases $\omega(G)\ge k$ and $\omega(G)<k/c$ for any constant $c\ge 1$ and computable function $f$, unless $FPT= W[1]$.