Scarf's Algorithm on Arborescence Hypergraphs

TL;DR

The paper investigates the convergence of Scarf's algorithm for hypergraphic stable matching, focusing on arborescence hypergraphs where hyperedges are directed paths. By introducing a depth-first ordering and a First Forward Leaving pivot rule, the authors prove that Scarf's algorithm terminates in at most |V| iterations and runs in O(|V||E|) time, yielding an integral stable matching due to total unimodularity in the network-hypergraph setting. This result constitutes the first polynomial-time convergence proof for Scarf's algorithm on hypergraphic stable matching problems and hinges on structural properties of network bases and pivots. The work also shows that the fractional stable matching polytope can be non-integral even in interval/hypergraph contexts, underscoring the importance of the polynomial-time algorithm for arborescence hypergraphs and suggesting directions for extending the approach to broader hypergraph families. Overall, the paper contributes a nontrivial intersection of combinatorial optimization, polyhedral theory, and hypergraph stability with implications for algorithmic design in related domains.

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.

Figures (10)

• Figure 1: An example of a hypergraph. The circles around the nodes are singletons. Line segments represent edges. Other hyperedges (only $\{1,3,4\}$ in this example) are indicated by splinegons.
• Figure 2: An example of a network hypergraph. On the left we present the principal tree using grey arcs. On the right we have a hypergraph with $V=[8]$ and 12 hyperedges (8 of them are singletons). We illustrate the hyperedges $e_4=\{4\}$, $e_6={6}$ and $e'=\{1,2,3\}$ on the right by the paths $f_4$, $f_6$ and $f'$ using black arcs on the left, respectively. In the principal tree we present, the unique source is $v_5$. The sinks are $v_1,v_6,v_7,v_9$. There are two branching vertices, namely $v_3$ and $v_4$.
• Figure 3: An example of an arborescence and an arborescence hypergraph. On the left we present an arborescence $\mathcal{T}$ with the root $v_5$. On the right we have a hypergraph $H$ with principal tree $\mathcal{T}$ with $V=[6]$ and 9 hyperedges (6 of them are singletons). We present the arcs in the arborescence which are corresponding to the singletons in the hypergraph.
• Figure 4: An example of an interval hypergraph. On the left we present the principal tree. On the right we have an interval hypergraph with $V=[4]$. The hyperedge $\{1,2,3\}$ can be seen as an interval $[1,3]$, and correspond to the black arc/grey path from $v_4$ to $v_1$ on the principal tree.
• Figure 5: A depth-first arborescence $\mathcal{T}=(U,\mathcal{A}_0)$. $U=\{v_1,\dots,v_{15}\}$ and $r=v_{15}$ is the root. $\mathcal{A}_0=\{f_1,\dots,f_{14}\}$. Notice that for every $i\in[14]$, the head of $f_i$ is $v_i$, yet the tail of $f_i$ may not be $v_{i+1}$ (for example, $f_2=(v_{14},v_2)$).

Theorems & Definitions (64)

• Theorem 1
• Definition 2: Standard form, cardinal basis, extreme point
• Definition 3: Ordinal matrix, ordinal basis, utility vector
• Definition 4: Dominating basis
• Theorem 5: Scarf's Lemma
• Definition 6
• Theorem 7: aharoni2003lemma
• Example 8
• Definition 9: Cardinal pivot
• Lemma 10: scarf1967core