对抗SAT问题中的SAT-UNSAT转换
The SAT-UNSAT transition in the adversarial SAT problem
摘要：
对抗性SAT（AdSAT）是可满足性（SAT）问题的一种推广，其中两个玩家试图使布尔公式为真（分别为false）通过控制它们各自的变量集。AdSAT在多项式层次中属于比SAT更高的复杂性类，因此临界区域和过渡的性质不容易与SAT的性质相混淆，值得独立研究。AdSAT还为量子可满足性问题（QSAT）的转换阈值提供了一个上界。我们给出了一个完整的AdSAT算法，证明了2-AdSAT是在$\mathbf{P}$，然后研究了两种随机算法（模拟退火及其改进的变体），并详细比较了它们的性能3-AdSAT。改变条款$α$的密度，我们发现一个尖锐的SAT-UNSAT过渡在一个临界值，其上限是$α_c \lesssim 1.5$，从而提供了一个更严格的上限QSAT过渡比以前发现的。
英文摘要：
Adversarial SAT (AdSAT) is a generalization of the satisfiability (SAT) problem in which two players try to make a boolean formula true (resp. false) by controlling their respective sets of variables. AdSAT belongs to a higher complexity class in the polynomial hierarchy than SAT and therefore the nature of the critical region and the transition are not easily paralleled to those of SAT and worth of independent study. AdSAT also provides an upper bound for the transition threshold of the quantum satisfiability problem (QSAT). We present a complete algorithm for AdSAT, show that 2-AdSAT is in $\mathbf{P}$, and then study two stochastic algorithms (simulated annealing and its improved variant) and compare their performances in detail for 3-AdSAT. Varying the density of clauses $α$ we find a sharp SAT-UNSAT transition at a critical value whose upper bound is $α_c \lesssim 1.5$, thus providing a much stricter upper bound for the QSAT transition than those previously found.
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