## Blocking optimal arborescences

### 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 $c:A\to \Rset$, 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 $w:A\to \Rset_+$ (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.

This is a revised and extended version of report no. 2011-13.

Bibtex entry:

@techreport{egres-15-06,
AUTHOR = {Bern{\'a}th, Attila and Pap, Gyula},
TITLE = {Blocking optimal arborescences},
NOTE= {{\tt www.cs.elte.hu/egres}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2015},
NUMBER = {TR-2015-06}
}

Last modification: 29.12.2019. Please email your comments to Tamás Király!