利用布尔加权约束满足问题中的结构：一种基于约束复合图的方法
Exploiting Structure in the Boolean Weighted Constraint Satisfaction Problem: A Constraint Composite Graph-Based Approach
AI总结：
本研究提出基于约束复合图（CCG）的方法，利用布尔加权约束满足问题（WCSP）的宏观与微观结构，通过核化、改进消息传递等方式提升求解效率，为量子优势研究提供更好基线。
AI中文摘要：
什么是“结构”，以及如何在组合优化中利用它？系统设计、状态估计和预测等推理任务可被转化为组合优化问题（COPs），传统上由专用算法解决，但这些算法在其狭窄子类之外失效，且研究等价问题的群体往往会重复发明彼此的技术。加权约束满足问题（WCSP）是一个通用框架，涵盖了来自多个领域的COPs，可表示从自旋玻璃到社交网络的复杂物理和非物理系统。这种通用性是一种诅咒吗？我们的答案是利用“结构”：通用WCSP算法应自动模仿其输入所属子类的专用算法。WCSP具有宏观（图形）结构（变量如何交互）和微观（数值）结构（变量交互的方式），不同的研究流派分别利用其中一种结构，从未同时利用两者。2008年提出的约束复合图（CCG）将两者统一：它是在WCSP实例变量及辅助变量上的无向图，通过最小加权顶点覆盖（MWVC）可求解原实例，其可高效构建但尚未被充分利用。本论文对三个问题给出肯定回答：CCG除了识别可处理类外具有理论优势（本文证明了新性质）；它具有实际实用性（高效实现与实验）；它有望扩展到非布尔变量（新编码）。我们利用CCG完成：（a）对WCSP实例进行核化，在搜索开始前通过最大流确定部分变量的最优值；（b）改进最小和消息传递；（c）利用整数线性规划求解器；（d）在量子退火器上求解COPs。更快地经典求解通用COPs也为备受争议的量子优势提供了更好的基线。
英文摘要：
What is "structure," and how can we exploit it in combinatorial optimization? Reasoning tasks such as system design, state estimation, and prediction can be cast as combinatorial optimization problems (COPs), traditionally attacked by dedicated algorithms that fail outside their narrow subclass, while communities working on equivalent problems reinvent each other's techniques. The weighted constraint satisfaction problem (WCSP) is a general framework that subsumes COPs from many communities and represents complex physical and non-physical systems, from spin glasses to social networks. Is such generality a curse? Our answer is to exploit "structure": a general-purpose WCSP algorithm should automatically imitate the specialized algorithm for whatever subclass its input belongs to. A WCSP has macro (graphical) structure, which variables interact, and micro (numerical) structure, how they interact. Separate schools of thought exploit one or the other, never both. The constraint composite graph (CCG), introduced in 2008, unifies them: it is an undirected graph over a WCSP instance's variables plus auxiliary ones, on which minimum weighted vertex cover (MWVC) solves the original instance. It is efficiently constructible, but largely unexploited. This dissertation answers three questions affirmatively. The CCG has theoretical advantages beyond identifying tractable classes (new properties proved here); it is practically useful (efficient implementation and experiments); and it extends promisingly to non-Boolean variables (new encodings). We use the CCG to (a) kernelize a WCSP instance, fixing optimal values of some variables by maxflow before search begins, (b) improve min-sum message passing, (c) exploit integer linear programming solvers, and (d) solve COPs on quantum annealers. Solving general COPs faster classically also yields better baselines for the debated quantum advantage.
