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].

翻译：我们考虑最大2-CSP问题的近似问题，其中每个变量最多出现在$d$个约束中（但字母表大小可能任意大）。该问题存在一个简单的$(\frac{d+1}{2})$近似算法。对于任意足够大的$d$，我们证明以下结论：在唯一博弈猜想（UGC）下，该问题在随机归约下是NP难的，无法在因子$\left(\frac{d}{2} - o(d)\right)$内近似；在不假设唯一博弈猜想的情况下，该问题在因子$\left(\frac{d}{3} - o(d)\right)$内近似也是NP难的（基于随机归约）。借助已知关联性[Dvorak等, Algorithmica 2023]，我们建立了在$k$-无爪图上近似最大独立集问题的以下难度结果：在唯一博弈猜想（UGC）下，该问题在随机归约下是NP难的，无法在因子$\left(\frac{k}{4} - o(k)\right)$内近似；在不假设唯一博弈猜想的情况下，该问题在因子$\left(\frac{k}{3 + 2\sqrt{2}} - o(k)\right) \geq \left(\frac{k}{5.829} - o(k)\right)$内近似也是NP难的（基于随机归约）。作为对比，已知多项式时间近似算法可达到$\left(\frac{k}{2} - o(k)\right)$的近似比[Neuwohner, STACS 2021; Thiery和Ward, SODA 2023]，拟多项式时间算法可达到$(\frac{k}{3} + o(k))$的近似比[Cygan等, SODA 2013]。