Scarf's Algorithm on Arborescence Hypergraphs

TL;DR

This paper demonstrates that Scarf's algorithm can efficiently find integral stable matchings in arborescence hypergraphs, a significant step in understanding its convergence on hypergraph-based problems.

Contribution

It proves polynomial-time convergence of Scarf's algorithm on arborescence hypergraphs, revealing new structural properties and advancing the understanding of hypergraphic stable matching problems.

Findings

Scarf's algorithm finds integral stable matchings in polynomial time for arborescence hypergraphs.

Structural properties of bases and pivots in network hypergraphs are uncovered.

First proof of polynomial-time convergence of Scarf's algorithm on hypergraphic stable matching problems.

Abstract

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.