A faster fixed parameter algorithm for two-layer crossing minimization

TL;DR

This paper presents a faster fixed parameter algorithm for determining if a bipartite graph's crossing number is at most k, improving the runtime from exponential in k log k to exponential in k.

Contribution

The authors introduce a more efficient algorithm with a runtime of 2^{O(k)} + n^{O(1)} for the bipartite crossing number decision problem, based on a new combinatorial bound.

Findings

Algorithm runs in 2^{O(k)} + n^{O(1)} time, faster than previous methods.

Provides a combinatorial upper bound on two-layer drawings with bounded crossing number.

Improves fixed parameter tractability for bipartite crossing minimization.

Abstract

We give an algorithm that decides whether the bipartite crossing number of a given graph is at most $k$. The running time of the algorithm is upper bounded by $2^{O(k)} + n^{O(1)}$, where $n$ is the number of vertices of the input graph, which improves the previously known algorithm due to Kobayashi et al. (TCS 2014) that runs in $2^{O(k \log k)} + n^{O(1)}$ time. This result is based on a combinatorial upper bound on the number of two-layer drawings of a connected bipartite graph with a bounded crossing number.