# Maximum Weighted Matching with Few Edge Crossings for 2-Layered Bipartite Graph

## TL;DR

This paper presents polynomial-time algorithms for finding maximum weighted matchings in 2-layered bipartite graphs with limited edge crossings, optimizing for specific crossing constraints.

## Contribution

It introduces new polynomial-time algorithms for maximum weighted matchings with at most one or two edge crossings in 2-layered bipartite graphs.

## Key findings

- Algorithms run in O((k+m)log n+n) for c=1
- Algorithms run in O(k^3+k^2n+m(m+log n)) for c=2
- Effective solutions for crossing-constrained matchings in bipartite graphs

## Abstract

Let c denote a non-negative constant. Suppose that we are given an edge-weighted bipartite graph G=(V,E) with its 2-layered drawing and a family X of intersecting edge pairs. We consider the problem of finding a maximum weighted matching M* such that each edge in M* intersects with at most c other edges in M*, and that all edge crossings in M* are contained in X. In the present paper, we propose polynomial-time algorithms for the cases of c=1 and 2. The time complexities of the algorithms are O((k+m)log n+n) for c=1 and O(k^3+k^2n+m(m+log n)) for c=2, respectively, where n=|V|, m=|E| and k=|X|.