  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].
