旅行商问题的信息论公式化
Information-theoretic formulation of the Traveling Salesman Problem
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中文总结 AI 辅助
该研究将旅行商问题（TSP）纳入信息论框架，通过定义最大熵分布、推导平均场近似及实现可微循环惩罚，在多项式时间内求解TSP，多数实例达到最优解，且提供处理硬约束的通用方法。
中文摘要 AI 辅助
旅行商问题（TSP）要求找到一条最短路线，恰好访问一组城市各一次。它将简单的局部规则——每个城市必须被访问一次——与严格的全局约束——所有城市必须在单个循环内遍历——相结合。我们将该问题置于概率信息论框架中。局部与全局约束的共存正是该问题在此框架中难以解决的原因：局部规则可通过顶点级约束强制执行，而全局约束无法由独立边概率捕获。我们表明，通过在图上定义最大熵概率分布可克服这一障碍，其中边成本和度约束生成类分配集合，描述严格约束的全局项将该集合偏向哈密顿循环。为使构造易于处理，我们推导了基于边占用的平均场近似，并实现了可抑制子循环的可微循环惩罚。这产生了一个自洽数值过程，其输出不仅是候选路线，还是编码竞争边和退化解的概率矩阵。我们在合成集合和TSPLIB实例上测试了该方法。该算法在多项式时间内收敛到连通路线，在大多数实例中匹配已知最佳解，否则保持较小的相对差距。除了具有竞争力的性能外，所提出的框架为处理严格约束提供了通用方法，同时将困难的组合优化问题简化为更简单的问题。
英文摘要
The Traveling Salesman Problem (TSP) asks for the shortest route to visit a set of cities exactly once. It combines a simple local rule - each city must be visited once - with a hard, global constraint- all cities must be traversed within a single cycle. We cast the problem within a probabilistic, information-theoretic framework. The coexistence of local and global constraints is precisely what makes the problem difficult to address in this framework: the local rule can be enforced through vertex-level constraints, whereas the global constraint cannot be captured by independent edge probabilities. We show that this obstacle can be overcome by defining a maximum-entropy probability distribution over graphs, in which edge costs and degree constraints generate an assignment-like ensemble, and a global term, describing the hard constraint, tilts this ensemble toward Hamiltonian cycles. To make the construction tractable, we derive a mean-field approximation in terms of edge occupancies and implement a differentiable cycle penalty that suppresses sub-tours. This leads to a self-consistent numerical procedure whose output is not only a candidate tour but also a probability matrix encoding competing edges and degenerate solutions. We test the method on synthetic ensembles and on TSPLIB instances. The algorithm converges to connected tours in polynomial time, matching the best-known solution in the majority of instances and remaining within a small relative gap otherwise. Beyond its competitive performance, the proposed framework offers a general approach for handling hard constraints while reducing hard combinatorial optimization problems to simpler ones.
