Algebraic Algorithms for Matching

Given a set of nodes and edges between them, what’s the maximum of number of disjoint edges? This problem is known as the graph matching problem, and its study has had an enormous impact on the develpoment of algorithms, combinatorics, optimization theory, and even complexity theory. Mathematicians have been interested in the matching problem since the 19th century, leading to celebrated theorems in graph theory due to Tutte, Menger, König, and Egerváry in the mid-twentieth century[18] However, the algorithmic aspects of the matching problem were not examined until the 60’s, and the first efficient algorithm for the (unweighted) matching problem was described in Jack Edmond’s seminal 1965 paper “Paths, Flowers and Trees” [4]1. Since Edmond’s paper, there has been a long and extensive literature on the algorithms for computing matchings. There are many different types of matching problems: bipartite versus non-bipartite, perfect matching versus maximum matching, weighted versus unweighted, and so on. Many of the algorithms developed to solve graph matching have been combinatorial, meaning the operation and analysis involve graphs and combinatorial structures. Examples include Edmond’s algorithm [4], Micali and Vazirani’s [10] algorithm, and the Ford-Fulkerson maximum flow algorithm applied to bipartite matching. However, many modern algorithms for matching are algebraic, involving techniques that superficially have little to do with graphs and matchings: computing determinants, updating matrix inverses, creating random matrices. The use of algebra has led to elegant and fast matching algorithms and in certain settings, the fastest known. The development of algebraic matching algorithms has in turn spurred advances in linear algebraic techniques that have wide applicability to algorithms design for other combinatorial and graph-theoretic problems. In this survey, we explore an old combinatorial problem, but through a recent algebraic lens. Nearly all the algebraic methods for solving matching have taken advantage of an important connection between determinants and matching discovered by Tutte in 1947 [18]: an undirected simple graph G = (V,E) has a perfect matching if and only if its associated Tutte matrix is nonsingular. This characterization of the existence of perfect matchings immediately leads to a simple randomized algorithm for testing whether a graph has a perfect matching, and serves as the foundation for algorithms to find such a matching (if it exists), to find a maximum matching, and more. The Tutte matrix will also serve as a starting point for our survey. The second key piece of technology used for the results covered here is the equivalence between computing the determinant, multiplying two matrices, inverting a matrix, and performing Gaussian Elimination. It is an extremely interesting and nontrivial fact that these problems all take O(n) time, where ω is called the matrix multiplication constant, and is somewhere between 2 and 2.373 [20]2. The paper will proceed as follows: (1) Using the Tutte matrix to detect and construct a perfect matching. (2) Computing maximum matchings via Gaussian Elimination inO(n) time. (3) Computing perfect matchings via “lazy updates” of the inverse Tutte matrix in O(n) time. (4) Solving the dynamic matching problem via a dynamic matrix inverse updating algorithm. (5) Counting perfect matchings in a graph via the determinant. (6) Open problems.