Unique Games Conjecture (UGC), proposed by [Khot02], lies in the center of many inapproximability results. At the heart of UGC lies approximability of MAX-CUT, which is a special instance of Unique Game.[KhotKMO04, MosselOO05] showed that assuming Unique Games Conjecture, it is NP-hard to distinguish between MAX-CUT instance that has a value $1-\epsilon$ vs. $1-\Omega(\sqrt{\epsilon})$. [CharikarMM06] then showed a matching polynomial time algorithm using Semi-Definite Programming. Towards resolving UGC, it has been long conjectured that inapproximability of MAX-CUT and UGC are equivalent. Assuming the equivalence, it suffices to exhibit lower bounds on MAX-CUT towards resolving UGC.
Towards showing the equivalence of the hardness of MAX-CUT and UGC, we initiate the study of symmetric parallel repetition, which is parallel repetition without coordinate. In particular, we show that symmetric parallel repetition beats strong parallel repetition in certain regimes, that is the value decays $(1-\epsilon^c)^{\Omega(r)}$ with $c < 2$, exhibiting the first separation between symmetric parallel repetition and usual parallel repetition. This is in sharp contrast to the usual parallel repetition as shown by [Holenstein07, Rao08a] where the best upper bound known for the value of the game $\mathcal{G}^{\otimes r}$ is $(1-\epsilon^2 / 2)^{\Omega(r)}$ for projection games. The counterexample shown by [Raz08] gives a lower bound of $\val(\mathcal{G}^{\otimes r}) \geq (1-\epsilon^2)^{O(r)}$ for $r = \Omega(n^2)$ where $n$ is the size of the graph. This also implies that the odd cycle game is not a counterexample for symmetric parallel repetition.
The main technical tool is the analysis of the Birthday Repetition in high intersection regime first introduced in [AIM14] subsequently improved in [MR16]. From a technical perspective we show that (a) if the set size is slightly larger than $\sqrt{n}$, then the value decays strong exponentially (i.e. $(1-\epsilon)^{\tilde{\Omega}(r)}$) in the expected intersection size; (b) if the set becomes large as in the whole vertex set, then the value decays strong exponentially in the number of edges that are checked by the verifier. Then we use prove a translation lemma to translate these technical results to corollaries in symmetric parallel repetition.
This exhibits a dichotomy between the usual parallel repetition and symmetric parallel repetition. In particular, it shows that the avenue of attack for showing the equivalence between the hardness of MAX-CUT and the Unique Games Conjecture using some model of repetition is still open.
