Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments

TL;DR

This article gives a new algorithm for Feedback Vertex Set in Bipartite Tournaments and obtains the fastest currently known exact exponential-time algorithm for the problem, with running time O(1.3820^n).

Abstract

A {\em bipartite tournament} is a directed graph $T:=(A \cup B, E)$ such that every pair of vertices $(a,b), a\in A,b\in B$ are connected by an arc, and no arc connects two vertices of $A$ or two vertices of $B$. A {\em feedback vertex set} is a set $S$ of vertices in $T$ such that $T - S$ is acyclic. In this article we consider the {\sc Feedback Vertex Set} problem in bipartite tournaments. Here the input is a bipartite tournament $T$ on $n$ vertices together with an integer $k$, and the task is to determine whether $T$ has a feedback vertex set of size at most $k$. We give a new algorithm for {\sc Feedback Vertex Set in Bipartite Tournaments}. The running time of our algorithm is upper-bounded by $O(1.6181^k + n^{O(1)})$, improving over the previously best known algorithm with running time $2^kk^{O(1)} + n^{O(1)}$ [Hsiao, ISAAC 2011]. As a by-product, we also obtain the fastest currently known exact exponential-time algorithm for the problem, with running time $O(1.3820^n)$.

Theorems & Definitions (67)

• lemma 1
• lemma 2: folklore
• lemma 3
• lemma 4
• definition 1: $t$-wise independent
• theorem 1
• definition 2: $M$-equivalent
• definition 3: $(M,X)$-equivalent
• definition 4: $(M,X)$-conflicting
• definition 5: $M$-consistent