We show that NP-hardness of approximating Boolean unique games on small set expanders can be amplified to the full Unique Games Conjecture on small set expanders. The latter conjecture is known to imply hardness results for problems like Balanced-Separator, Minimum-Linear-Rearrangement and Small-Set-Expansion that are not known under the Unique Games Conjecture. Our result follows from: (1) A transformation that fortifies any unique game on a small set expander. (2) An improved parallel repetition theorem for fortified games on small set expanders. Our approach avoids the known limitation on strong parallel repetition of unique games by enlarging the alphabet considerably before repetition. The limitation does not hold for unique games with a sufficiently large alphabet. Prior to this work, parallel repetition from fortification was only known for projection games, and seemed hopeless for unique games. Perhaps surprisingly, the idea for fortification of unique games comes from a reduction of Raghavendra and Steurer from Small-Set-Expansion to Unique Games.

翻译：我们显示,在小型组合扩展器上接近布林独有游戏的硬度可以放大到小组合扩展器上独一游戏的完整独一猜想。 后一种猜想已知意味着平衡- 分隔器、 最小利那重新组合和小型探索等在独特游戏猜想下不为人知的问题的硬度结果。 我们的结果来自:(1) 一种变异,在小型组合扩展器上强化任何独一游戏。 (2) 在小型组合扩展器上改进平行重复强化游戏的理论。 我们的方法避免了已知的对独一游戏的强烈平行重复的限制,在重复之前大大扩展了字母。 限制并不适用于具有足够大字母的独特游戏。 在这项工作之前,与强化平行的重复只为投影游戏所知道,而对独特游戏来说似乎毫无希望。 令人惊讶的是, 使独特游戏的强化理念来自从小型探索到独特游戏的拉加文德朗和斯蒂尔的减少。