Topics in Algebra of Synchronous Games, Algebraic Graph Identities and Quantum NP-hardness Reductions

Entong He
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Abstract

We review the correspondence between a synchronous game and its associated game algebra. We slightly develop the work of Helton et al.[HMPS17] by proposing results on algebraic and locally commuting graph identities. Based on the theoretical works on noncommutative Nullstellens\"atze [BWHK23], we build computational tools involving Gr\"obner basis methods and semidefinite programming to check the existence of perfect strategies with specific models. We prove the equivalence between the hereditary and $C$-star models proposed in [HMPS17]. We also extend Ji's reduction $\texttt{3-SAT}\text{-star} \leq_p \texttt{3-Coloring}\text{-star}$ [Ji13] and exhibit another instance of quantum-version NP-hardness reduction $\texttt{3-SAT}\text{-star} \leq_p \texttt{Clique}\text{-star}$.
同步游戏代数、代数图同一性和量子 NP 难度降低专题
我们回顾了同步博弈与其相关博弈代数之间的对应关系。我们略微发展了 Helton 等人的工作[HMPS17],提出了关于代数和局部换算图标识的结果。基于非交换空策略的理论工作[BWHK23],我们建立了涉及 Gr\"obner basis 方法和半定式编程的计算工具,以检验特定模型的完美策略的存在性。我们还扩展了 Ji 的$texttt{3-SAT}\text{-star}还原。\[Ji13],并展示了另一个量子版 NP-困难性还原 $\texttt{3-SAT}\text{-star} 的实例。\leq_p\texttt{Clique}\text{-star}$.
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