Heuristic to reduce the complexity of complete bipartite graphs to accelerate the search for maximum weighted matchings with small error

TL;DR

This paper introduces a heuristic to reduce the edge count in complete bipartite graphs, significantly lowering the complexity of maximum weighted matching algorithms while maintaining small error margins.

Contribution

The paper proposes a simple heuristic method to sparsify complete bipartite graphs from n^2 edges to n log n edges, reducing computational complexity.

Findings

Edge reduction from n^2 to n log n in bipartite graphs.

Complexity lowered from O(n^3) to O(n^2 log n).

Applicable when edge weights are uniformly distributed in [0,1].

Abstract

A maximum weighted matching for bipartite graphs $G=(A \cup B,E)$ can be found by using the algorithm of Edmonds and Karp with a Fibonacci Heap and a modified Dijkstra in $O(nm + n^2 \log{n})$ time where n is the number of nodes and m the number of edges. For the case that $|A|=|B|$ the number of edges is $n^2$ and therefore the complexity is $O(n^3)$. In this paper we want to present a simple heuristic method to reduce the number of edges of complete bipartite graphs $G=(A \cup B,E)$ with $|A|=|B|$ such that $m = n\log{n}$ and therefore the complexity of such that $m = n\log{n}$ and therefore the complexity of $O(n^2 \log{n})$. The weights of all edges in G must be uniformly distributed in [0,1].