Hardness of Approximating Bounded-Degree Max 2-CSP and Independent Set on k-Claw-Free Graphs

TL;DR

This paper establishes new hardness of approximation results for Max 2-CSP with bounded degree and for Maximum Independent Set on k-claw-free graphs, showing these problems are harder to approximate than previously known under certain conjectures.

Contribution

It proves stronger NP-hardness of approximation bounds for Max 2-CSP and Independent Set on k-claw-free graphs, assuming the Unique Games Conjecture, surpassing previous results.

Findings

NP-hard to approximate Max 2-CSP within (d/2 - o(d)) assuming UGC

NP-hard to approximate Max 2-CSP within (d/3 - o(d))

NP-hard to approximate Independent Set on k-claw-free graphs within (k/4 - o(k)) assuming UGC

Abstract

We consider the question of approximating Max 2-CSP where each variable appears in at most $d$ constraints (but with possibly arbitrarily large alphabet). There is a simple $(\frac{d+1}{2})$-approximation algorithm for the problem. We prove the following results for any sufficiently large $d$: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{d}{2} - o(d)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{d}{3} - o(d)\right)$. Thanks to a known connection [Dvorak et al., Algorithmica 2023], we establish the following hardness results for approximating Maximum Independent Set on $k$-claw-free graphs: - Assuming the Unique Games Conjecture (UGC), it is NP-hard (under randomized reduction) to approximate this problem to within a factor of $\left(\frac{k}{4} - o(k)\right)$. - It is NP-hard (under randomized reduction) to approximate the problem to within a factor of $\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$. In comparison, known approximation algorithms achieve $\left(\frac{k}{2} - o(k)\right)$-approximation in polynomial time [Neuwohner, STACS 2021; Thiery and Ward, SODA 2023] and $(\frac{k}{3} + o(k))$-approximation in quasi-polynomial time [Cygan et al., SODA 2013].