[Paper Review] An algebraic Monte-Carlo algorithm for the Bipartite Partition Adjacency Matrix realization problem

Éva Czabarka, Łászló A. Székely+2 more|arXiv (Cornell University)|Aug 28, 2017

Advanced Combinatorial Mathematics17 references3 citations

TL;DR

This paper presents an algebraic Monte-Carlo algorithm that solves the bipartite Partition Adjacency Matrix realization and construction problems in polynomial time for a fixed number of partition classes. It also extends the approach to a related perfect matching problem with prescribed inter-class edge counts, offering a randomized, efficient solution for combinatorial graph realization under structural constraints.

ABSTRACT

Consider a set $W,$ natural numbers $d(w)$ associated with each $w\in W$, and a $W_i:i\in I$ partition of $W$ with natural numbers $c(W_i,W_j)$ associated to every unordered pair of partition classes. The Partition Adjacency Matrix realization problem asks whether there is a simple graph on the vertex set $W$ with degree sequence $d(w)$ for $w\in W$, with exactly $c(W_i,W_j)$ edges with endpoints in $W_i$ and $W_j$, respectively; the Partition Adjacency Matrix construction problem asks for such a graph, if it exists. Here we provide an algebraic Monte-Carlo algorithm for the bipartite Partition Adjacency Matrix realization and construction problems, which runs in polynomial time for fixed number of partition classes. Furthermore, we provide a similar algorithm for the following problem: Given a bipartite graph, with partitions on the partite vertex classes, and prescribed numbers for all the pairs of partition classes, is there a perfect matching, such that the number of matching edges between the partition classes are exactly the prescribed numbers?

Motivation & Objective

• To address the computational challenge of realizing a simple graph on a vertex set W with specified degree sequences and inter-partition edge counts between predefined partition classes.
• To provide a constructive algorithm that not only determines the existence of such a graph but also builds it explicitly when possible.
• To extend the framework to a related problem: finding a perfect matching in a bipartite graph with exact prescribed numbers of edges between given partition classes of the two partite sets.
• To achieve polynomial-time complexity for fixed numbers of partition classes, leveraging algebraic and randomized techniques.

Proposed method

• Formulate the Partition Adjacency Matrix realization problem as a system of constraints on edge distributions between partition classes and vertex degrees.
• Use algebraic techniques involving the permanent of a matrix and polynomial identity testing to encode the existence of valid edge configurations.
• Apply a Monte-Carlo randomized algorithm based on polynomial identity testing over a finite field to check whether a solution exists with high probability.
• Reduce the bipartite realization and matching problems to systems of multivariate polynomial equations that can be evaluated efficiently via random sampling.
• Utilize the Schwartz-Zippel lemma to bound the error probability of the randomized algorithm, ensuring correctness with high confidence.
• Construct the actual graph or matching by iteratively sampling and verifying solutions using the same algebraic framework.

Experimental results

Research questions

• RQ1 Can the bipartite Partition Adjacency Matrix realization problem be solved in polynomial time using algebraic and randomized methods?
• RQ2 Is there an efficient algorithm to construct a simple graph with given vertex degrees and prescribed inter-partition edge counts between partition classes?
• RQ3 Can the approach be extended to the problem of finding a perfect matching in a bipartite graph with exact inter-class edge counts?
• RQ4 What is the computational complexity of these problems when the number of partition classes is fixed?
• RQ5 How can algebraic techniques such as polynomial identity testing be effectively applied to combinatorial graph realization problems?

Key findings

• The proposed algorithm solves the bipartite Partition Adjacency Matrix realization and construction problems in polynomial time when the number of partition classes is fixed.
• The algorithm uses a randomized approach based on polynomial identity testing, ensuring correctness with high probability and bounded error via the Schwartz-Zippel lemma.
• The method extends to the perfect matching problem with prescribed inter-class edge counts, providing a similar polynomial-time solution for fixed partition counts.
• The core technique relies on encoding graph realization constraints as multivariate polynomials and testing their non-identical-zero status using algebraic methods.
• The approach achieves efficiency by reducing complex combinatorial constraints to algebraic evaluations over finite fields.
• The algorithm is both theoretically efficient and practically implementable, offering a novel bridge between algebraic computation and graph realization problems.