Finding Hall blockers by matrix scaling

TL;DR

This paper extends the use of the Sinkhorn algorithm to identify Hall blockers in bipartite graphs, providing polynomial-time methods to detect perfect matchings and their obstructions.

Contribution

It demonstrates that Sinkhorn iterations can identify Hall blockers and parametric Hall blockers in bipartite graphs, extending previous results on perfect matching detection.

Findings

Sinkhorn iterations identify Hall blockers in polynomial time.

The method detects all parametric Hall blockers maximizing a specific function.

Connections established between Sinkhorn limit, network flow, and graph decompositions.

Abstract

For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $D_1AD_2$ for some positive diagonal matrices $D_1,D_2$.The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ and column-normalization $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$ alternatively. By this algorithm, $A$ converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with $A$ has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph $G$, which is identified with the $0,1$-matrix $A_G$.Linial, Samorodnitsky, and Wigderson showed that $O(n^2 \log n)$ iterations for $A_G$ decide whether $G$ has a perfect matching. Here $n$ is the number of vertices in one of the color classes of $G$. In this paper, we show an extension of this result:If $G$ has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker -- a vertex subset $X$ having neighbors $\Gamma(X)$ with $|X| > |\Gamma(X)|$. Specifically, we show that $O(n^2 \log n)$ iterations can identify one Hall blocker, and that further polynomial iterations can also identify all parametric Hall blockers $X$ of maximizing $(1-\lambda) |X| - \lambda |\Gamma(X)|$ for $\lambda \in [0,1]$.The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for KL-divergence (Csisz\'{a}r and Tusn\'{a}dy 1984, Gietl and Reffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.