[Paper Review] A polynomial-time approximation algorithm for the number of k-matchings in bipartite graphs

Shmuel Friedland, Daniel Levy|arXiv (Cornell University)|Jul 28, 2006

Markov Chains and Monte Carlo Methods7 references7 citations

TL;DR

This paper presents a polynomial-time approximation algorithm for counting k-matchings in bipartite graphs by reducing the problem to computing the permanent of a larger bipartite graph, leveraging a novel transformation that relates k-matchings to perfect matchings. The key result is that the number of k-matchings in a bipartite graph can be approximated in fully polynomial time using the known fpras for permanents of nonnegative matrices.

ABSTRACT

We show that the number of $k$-matching in a given undirected graph $G$ is equal to the number of perfect matching of the corresponding graph $G_k$ on an even number of vertices divided by a suitable factor. If $G$ is bipartite then one can construct a bipartite $G_k$. For bipartite graphs this result implies that the number of $k$-matching has a polynomial-time approximation algorithm. The above results are extended to permanents and hafnians of corresponding matrices.

Motivation & Objective

• To establish the existence of a fully polynomial-time randomized approximation scheme (fpras) for counting k-matchings in bipartite graphs.
• To extend the fpras for permanents of nonnegative matrices to the computation of k-matchings and subpermanents.
• To demonstrate that the complexity of counting k-matchings in general graphs is equivalent to that of computing perfect matchings via hafnians.
• To provide a theoretical foundation for approximating the matching polynomial and pressure functions in statistical mechanics models.
• To clarify the structural relationship between permanents, hafnians, and k-matchings using matrix transformations.

Proposed method

• Construct a transformed bipartite graph $G_k$ with $2n - 2k$ vertices such that the number of perfect matchings in $G_k$ is proportional to the number of $k$-matchings in the original graph $G$.
• Use the identity $\mathop{\mathrm{perm}}\nolimits_k B = \frac{\mathop{\mathrm{perm}}\nolimits B_k}{(m-k)!(n-k)!}$ to relate $k$-matchings to the permanent of a larger matrix $B_k$.
• Apply the fpras for permanents of nonnegative matrices (from Jerrum, Sinclair, and Vigoda) to approximate $\mathop{\mathrm{perm}}\nolimits_k B$ efficiently.
• For general graphs, define a matrix $A_k$ such that $\mathop{\mathrm{haf}}\nolimits_k A = \frac{\mathop{\mathrm{haf}}\nolimits A_k}{(m-2k)!}$, linking $k$-matchings to hafnians of larger matrices.
• Use the structure of symmetric positive semidefinite matrices and block matrix constructions to preserve weight and matching correspondence.
• Leverage known results on hyperbolic polynomials and matrix permanents to justify the theoretical validity of the reduction.

Experimental results

Research questions

• RQ1 Can the number of $k$-matchings in a bipartite graph be approximated in polynomial time?
• RQ2 What is the relationship between $k$-matchings and perfect matchings in an extended graph construction?
• RQ3 Does the existence of an fpras for permanents imply an fpras for $k$-matchings in bipartite graphs?
• RQ4 How do hafnians and permanents relate to the computation of $k$-matchings in general graphs?
• RQ5 Is the complexity of counting $k$-matchings in arbitrary graphs equivalent to that of computing perfect matchings?

Key findings

• The number of $k$-matchings in a bipartite graph $G$ is equal to $\frac{\mathop{\mathrm{perm}}\nolimits B_k}{(m-k)!(n-k)!}$, where $B_k$ is a matrix constructed from the incidence matrix of $G$.
• This transformation allows the use of the fpras for permanents of nonnegative matrices to approximate the number of $k$-matchings in polynomial time.
• For general graphs, $\mathop{\mathrm{haf}}\nolimits_k A = \frac{\mathop{\mathrm{haf}}\nolimits A_k}{(m-2k)!}$, showing that computing $k$-matchings is as hard as computing perfect matchings via hafnians.
• The existence of an fpras for $k$-matchings in bipartite graphs is therefore established, given the fpras for permanents.
• The method extends to weighted graphs and provides a computable approximation for the generating matching polynomial of bipartite graphs.
• The result implies that the pressure function in statistical mechanics models on infinite lattices can be approximated via this framework.