[论文解读] How to Play Unique Games against a Semi-Random Adversary

Alexandra Kolla, Konstantin Makarychev+1 more|arXiv (Cornell University)|Apr 19, 2011

Complexity and Algorithms in Graphs参考文献 23被引用 4

一句话总结

本文提出了一种新颖的半随机唯一博弈模型，其中在完全可满足的实例构建完成后，对手会部分地破坏约束。该文提出了一种非标准的SDP松弛方法，并结合混合SDP-LP舍入方案，使得在平均度足够高且ε低于绝对常数时，能够以多项式时间满足(1−δ)比例的约束，优于标准SDP方法在半随机实例上的表现。

ABSTRACT

In this paper, we study the average case complexity of the Unique Games problem. We propose a natural semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an epsilon-fraction of all edges, and finally replaces ("corrupts") the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1-epsilon) satisfiable instance, so then the problem is as hard as in the worst case. In our semi-random model, one of the steps is random, and all other steps are adversarial. We show that known algorithms for unique games (in particular, all algorithms that use the standard SDP relaxation) fail to solve semi-random instances of Unique Games. We present an algorithm that with high probability finds a solution satisfying a (1-delta) fraction of all constraints in semi-random instances (we require that the average degree of the graph is Omega(log k). To this end, we consider a new non-standard SDP program for Unique Games, which is not a relaxation for the problem, and show how to analyze it. We present a new rounding scheme that simultaneously uses SDP and LP solutions, which we believe is of independent interest. Our result holds only for epsilon less than some absolute constant. We prove that if epsilon > 1/2, then the problem is hard in one of the models, the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1-epsilon) satisfiable. We present an algorithm that with high probability, distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1-delta) satisfiable instance if epsilon > c delta.

研究动机与目标

• 研究在结合对抗性与随机性元素的半随机模型下，唯一博弈的平均情况复杂度。
• 开发一种高效求解标准SDP方法失效的半随机唯一博弈实例的算法。
• 区分几乎可满足的半随机实例与仅中等可满足的任意实例。
• 分析在半随机模型下近似解的极限，特别是基于2-to-2猜想。

提出的方法

• 提出一种半随机模型，其中在完全可满足的唯一博弈实例构建后，对手会以随机约束破坏ε比例的边。
• 提出一种非标准SDP松弛，其并非原问题的松弛，但可在半随机模型下进行分析。
• 设计一种混合舍入方案，同时利用SDP和LP解以提高约束满足率。
• 应用定理4.4的修改版本，以界定解中'短'边的数量，证明松弛SDP解中测度的集中性。
• 通过概率分析表明，在高概率下，半随机实例中至少可实现ε/32的SDP目标值。
• 采用一种松弛的SDP解，其中重新定义部分顶点的向量以保持范数边界，从而增强对破坏的鲁棒性。

实验结果

研究问题

• RQ1 标准SDP-based算法能否解决唯一博弈的半随机实例，即使有常数比例的约束被对抗性破坏？
• RQ2 当平均度足够高时，是否存在一种多项式时间算法，可满足半随机唯一博弈实例中(1−δ)比例的约束？
• RQ3 当δ ≪ ε时，能否高效区分ε比例约束被破坏的半随机实例与(1−δ)-可满足的任意实例？
• RQ4 在假设2-to-2猜想成立的前提下，半随机唯一博弈近似难度的阈值ε在何值以上时，问题将变得难以近似？

主要发现

• 标准SDP-based算法即使在实例仅被ε比例破坏且ε低于绝对常数时，也无法解决半随机唯一博弈实例。
• 所提出的算法在半随机实例中可满足(1−O(ε+η/log k))比例的约束，其中η是与SDP解相关的参数，且平均度为Ω̃(log k)。
• 该算法即使在被破坏的约束上也能实现高满足率，满足总约束中(1−O(ε+η/log k))的比例。
• 当ε < 1/2时，问题在半随机模型下仍保持可解；但若ε ≥ 1/2，则在2-to-2猜想下问题变得困难，不存在多项式时间算法可区分(1−ε)-可满足的半随机实例与δ-可满足实例。
• 在ε|E|条边被随机破坏的半随机实例中，SDP值以高概率至少为ε/32，从而可在δ < cε（c为某绝对常数）时有效区分于(1−δ)-可满足实例。