Blocking optimal arborescences

TL;DR

This paper presents a polynomial-time algorithm for a special case of a complex matroid covering problem, focusing on finding minimal arc sets intersecting all minimum-cost arborescences in a directed graph.

Contribution

The paper introduces a polynomial-time algorithm for a specific case of covering minimum cost arborescences, which was previously NP-complete in general.

Findings

Algorithm runs in $O(n^3T(n,m))$ time for graphs with $n$ nodes and $m$ arcs.

Successfully solves a weighted version minimizing $w(H)$.

Provides a new polynomial-time solution for a special case of a matroid covering problem.

Abstract

The problem of covering minimum cost common bases of two matroids is NP-complete, even if the two matroids coincide, and the costs are all equal to 1. In this paper we show that the following special case is solvable in polynomial time: given a digraph $D=(V,A)$ with a designated root node $r\in V$ and arc-costs $c:A\to \mathbb{R}$, find a minimum cardinality subset $H$ of the arc set $A$ such that $H$ intersects every minimum $c$-cost $r$-arborescence. By an $r$-arborescence we mean a spanning arborescence of root $r$. The algorithm we give solves a weighted version as well, in which a nonnegative weight function $w:A\to \mathbb{R}_+$ (unrelated to $c$) is also given, and we want to find a subset $H$ of the arc set such that $H$ intersects every minimum $c$-cost $r$-arborescence, and $w(H)=\sum_{a\in H}w(a)$ is minimum. The running time of the algorithm is $O(n^3T(n,m))$, where $n$ and $m$ denote the number of nodes and arcs of the input digraph, and $T(n,m)$ is the time needed for a minimum $s-t$ cut computation in this digraph. A polyhedral description is not given, and seems rather challenging.