树状超图上的 Scarf 算法
Scarf's Algorithm on Arborescence Hypergraphs
AI总结：
本文研究 Scarf 算法在树状超图上的运行，证明其可在多项式时间内找到整数稳定匹配，并发现更一般网络超图的基与转轴结构性质。
摘要：
Scarf 算法是一种转轴过程，用于在下单调多面体中寻找支配极点，可用于证明超图中分数稳定匹配的存在性。然而，在超图中寻找分数稳定匹配的问题是 PPAD 完全的。在本工作中，我们研究 Scarf 算法在树状超图上的行为；树状超图是指超边对应于某棵树状图中路径的一类超图。对于树状超图，我们证明 Scarf 算法可以被实现为在多项式时间内找到一个整数稳定匹配。在得到该结果的过程中，我们揭示了更一般的网络超图族在基和转轴方面的新结构性质。我们的工作首次证明了 Scarf 算法在超图稳定匹配问题上的多项式时间收敛性，为 Scarf 算法在其他多面体族上实现多项式时间收敛的可能性带来了希望。
英文摘要：
Scarf's algorithm--a pivoting procedure that finds a dominating extreme point in a down-monotone polytope--can be used to show the existence of a fractional stable matching in hypergraphs. The problem of finding a fractional stable matching in a hypergraph, however, is PPAD-complete. In this work, we study the behavior of Scarf's algorithm on arborescence hypergraphs, the family of hypergraphs in which hyperedges correspond to the paths of an arborescence. For arborescence hypergraphs, we prove that Scarf's algorithm can be implemented to find an integral stable matching in polynomial time. En route to our result, we uncover novel structural properties of bases and pivots for the more general family of network hypergraphs. Our work provides the first proof of polynomial-time convergence of Scarf's algorithm on hypergraphic stable matching problems, giving hope to the possibility of polynomial-time convergence of Scarf's algorithm for other families of polytope.
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