Blocking optimal $k$-arborescences

TL;DR

This paper introduces a polynomial-time algorithm for finding a minimum cardinality subset of arcs that intersects all minimum cost $k$-arborescences in a directed graph, extending previous work from the case $k=1$ to fixed $k$.

Contribution

It generalizes the blocking problem for optimal arborescences to arbitrary fixed $k$, providing a polynomial-time solution.

Findings

Algorithm works in polynomial time for fixed $k$

Extends previous $k=1$ case to general fixed $k$

Provides a new approach to blocking optimal $k$-arborescences

Abstract

Given a digraph $D=(V,A)$ and a positive integer $k$, an arc set $F\subseteq A$ is called a \textbf{$k$-arborescence} if it is the disjoint union of $k$ spanning arborescences. The problem of finding a minimum cost $k$-arborescence is known to be polynomial-time solvable using matroid intersection. In this paper we study the following problem: find a minimum cardinality subset of arcs that contains at least one arc from every minimum cost $k$-arborescence. For $k=1$, the problem was solved in [A. Bern\'ath, G. Pap , Blocking optimal arborescences, IPCO 2013]. In this paper we give an algorithm for general $k$ that has polynomial running time if $k$ is fixed.