The UNIQUE GAMES problem is a central problem in algorithms and complexity theory. Given an instance of UNIQUE GAMES, the STRONG UNIQUE GAMES problem asks to find the largest subset of vertices, such that the UNIQUE GAMES instance induced on them is completely satisfiable. In this paper, we give new algorithmic and hardness results for STRONG UNIQUE GAMES. Given an instance with label set size $k$ where a set of $(1 - ε)$-fraction of the vertices induce an instance that is completely satisfiable, our first algorithm produces a set of $1 - \widetilde{O}({k^2}) ε\sqrt{\log n}$ fraction of the vertices such that the UNIQUE GAMES induced on them is completely satisfiable. In the same setting, our second algorithm produces a set of $1 - \widetilde{O}({k^2}) \sqrt{ε\log d}$ (here $d$ is the largest vertex degree of the graph) fraction of the vertices such that the UNIQUE GAMES induced on them is completely satisfiable. The technical core of our results is a new connection between STRONG UNIQUE GAMES and Small-Set-Vertex-Expansion in graphs. Complementing this, assuming the Unique Games Conjecture, we prove that it is NP-hard to compute a set of size larger than $1 - Ω( \sqrt{ε\log k \log d})$ for which all the constraints induced on this set are satisfied.
Given an undirected graph $G(V,E)$ the ODD CYCLE TRANSVERSAL problem asks to delete the least fraction of vertices to make the induced graph on the remaining vertices bipartite. As a corollary to our main algorithmic results, we obtain an algorithm that outputs a set $S$ such the graph induced on $V \setminus S$ is bipartite, and $|S|/n \leq O(\sqrt{ε\log d})$ (here $d$ is the largest vertex degree and $ε$ is the optimal fraction of vertices that need to be deleted). Assuming the Unique Games Conjecture, we prove a matching (up to constant factors) hardness.
How to Play Unique Games against a Semi-Random Adversary
在本文中，我们研究了唯一对策问题的平均情况复杂性。我们提出了一个自然的半随机模型，其中一个独特的游戏实例是在几个步骤中生成的。首先，对手选择一个完全可满足的唯一游戏实例，然后她选择所有边的ε分数，最后用新的约束替换（“破坏”）与这些边对应的约束。如果所有步骤都是对抗的，对手可以获得任何（1-n）可满足的实例，那么问题就和最坏情况一样难。在我们的半随机模型中，其中一个步骤是随机的，所有其他步骤都是对抗性的。我们发现，已知的算法独特的游戏（特别是，所有的算法，使用标准的SDP放松）无法解决半随机的独特的游戏。我们提出了一个算法，以高概率找到一个解决方案，满足（1-delta）部分的所有约束在半随机的情况下（我们要求图的平均度是欧米茄（log k）。为此，我们考虑了一个新的非标准的SDP程序的独特的游戏，这是不是一个放松的问题，并显示如何分析它。我们提出了一个新的舍入方案，同时使用SDP和LP的解决方案，我们相信是独立的利益。我们的结果只适用于小于某个绝对常数的情形。我们证明了当λ> 1/2时，在其中一个模型中，问题是困难的，其结果假设2-to-2猜想。最后，我们研究了最多（1-n）可满足的唯一博弈的半随机实例。我们提出了一个算法，以高概率，区分的情况下，当实例是一个半随机的实例和情况下，当实例是（任意）（1-delta）可满足的实例，如果δ> c δ。
本文改进了Arora，Khot，Kolla，Steurer，Tulsiani和Vishnoi关于扩张器上唯一博弈问题的一个结果。给定具有约束图$G$的唯一博弈的$(1-\varepsilon)$可满足实例，如果$\varepsilon < c λ_G$，则我们的算法找到满足所有约束的至少一个$1- C \varepsilon/h_G$分数的分配，其中$h_G$是$G$的边扩展，$λ_G$是$G$的拉普拉斯算子的第二小特征值，并且$C$和$c$是一些绝对常数。
Can We Break Fine-Grained and NP-Hardness Barriers if We've Seen the Graph Before? The Isomorphic-Priors Model
Max Planck Institute for Informatics（马克斯·普朗克 informatics 研究所）；Saarland Informatics Campus（萨尔兰 informatics 校园）；TU Dresden（德累斯顿工业大学）；University of Liverpool（利物浦大学）
