[Paper Review] Perfect Matchings in Õ(n^{1.5}) Time in Regular Bipartite Graphs

Ashish Goel, Michael Kapralov+1 more|ArXiv.org|Feb 10, 2009

Complexity and Algorithms in Graphs15 references4 citations

TL;DR

This paper presents a randomized algorithm that finds a perfect matching in d-regular bipartite graphs in Õ(n^{1.5}) expected time, significantly improving upon prior bounds. The approach uses a two-stage sampling scheme—first uniform sampling to reduce edge count, then non-uniform sampling to preserve matchings—followed by Hopcroft-Karp on the sparse subgraph, achieving optimal performance for dense regular graphs.

ABSTRACT

We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m \sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$ expected time, an expression that is bounded by $ ilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(\min\{m, \frac{n^2\ln^3 n}{d}\})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((n\ln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(n\ln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $ ilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $ ilde{O}(\min\{m, \frac{n^2}{d} \})$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Hall's theorem ``witnesses'' for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n imes n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $ ilde{O}(m + n^{1.5})$.

Motivation & Objective

• To design a faster algorithm for finding perfect matchings in d-regular bipartite graphs, especially when d is large.
• To improve upon the O(m) and O(n^{1.75})-time bounds previously known for this problem.
• To reduce the problem size while preserving the existence of a perfect matching with high probability.
• To provide a tighter analysis of the Hopcroft-Karp algorithm in the context of sampled regular graphs.
• To establish a new correspondence between cuts and Hall’s theorem witnesses, enabling structured sampling.

Proposed method

• Propose a two-stage sampling scheme: first uniform sampling to reduce graph size, then non-uniform sampling to preserve matching structure.
• Use Benczúr-Karger style sampling on the reduced graph to ensure at least one edge from every Hall witness set is retained.
• Leverage a novel correspondence theorem linking cuts and Hall’s theorem witnesses to validate the sampling correctness.
• Apply the Hopcroft-Karp algorithm on the final sparse subgraph with O(n ln n) edges to recover a perfect matching.
• Analyze the Hopcroft-Karp algorithm on the sampled graph with a tighter bound, showing O(n^{1.5} ln²n / d) time whp.
• Use the Birkhoff-von Neumann decomposition framework to extend results to doubly stochastic matrices.

Experimental results

Research questions

• RQ1 Can the perfect matching problem in d-regular bipartite graphs be solved in time better than O(m) for large d?
• RQ2 Can a two-stage sampling process preserve perfect matchings with high probability while reducing edge count to O(n ln n)?
• RQ3 Is there a structural correspondence between cuts and Hall’s theorem witnesses that enables efficient sampling?
• RQ4 Can the Hopcroft-Karp algorithm be analyzed more tightly on sampled regular graphs to achieve Õ(n^{1.5}) time?
• RQ5 Can the sampling framework be extended to find matchings in the support of doubly stochastic matrices?

Key findings

• The paper presents an O(min{m, n² ln³n / d}) expected time algorithm for perfect matching in d-regular bipartite graphs.
• For d = Ω(√n ln n), the running time is O((n ln n)^{1.5}), achieving Õ(n^{1.5}) time as a function of n alone.
• The two-stage sampling scheme reduces the graph to O(n ln n) edges while preserving a perfect matching with high probability.
• A new correspondence theorem between cuts and Hall’s witnesses is proven, which is central to the correctness of the sampling process.
• The Hopcroft-Karp algorithm runs in O(n^{1.5} ln²n / d) time on the sampled graph, improving the standard analysis.
• The method extends to finding a perfect matching in the support of an n×n doubly stochastic matrix in Õ(m + n^{1.5}) expected time.