A faster FPT algorithm for Bipartite Contraction

TL;DR

This paper introduces a faster, simpler randomized fixed-parameter tractable algorithm for the Bipartite Contraction problem, significantly improving the previous double-exponential dependence on the parameter k.

Contribution

The authors develop a new randomized FPT algorithm for Bipartite Contraction with improved exponential running time, and provide a derandomization method.

Findings

Achieves a $2^{O(k^2)} n m$ running time, avoiding double-exponential dependence.

Simplifies the conceptual approach compared to previous algorithms.

Can be derandomized using standard techniques.

Abstract

The \textsc{Bipartite Contraction} problem is to decide, given a graph $G$ and a parameter $k$, whether we can can obtain a bipartite graph from $G$ by at most $k$ edge contractions. The fixed-parameter tractability of the problem was shown by [Heggernes et al. 2011], with an algorithm whose running time has double-exponential dependence on $k$. We present a new randomized FPT algorithm for the problem, which is both conceptually simpler and achieves an improved $2^{O(k^2)} n m$ running time, i.e., avoiding the double-exponential dependence on $k$. The algorithm can be derandomized using standard techniques.