Blocking optimal structures

Abstract

We consider weighted blocking problems (a.k.a. weighted transversal problems) of the following form. Given a finite set $S$, weights $w:S\to {\mathbb{R}}_{+}$, and a family $\mathcal{F}\subseteq 2^S$, find $min\{w\left(H\right):H\subseteq S,H\text{ intersects every member of }\mathcal{F}\}$. In our problems $S$ is the set of edges of a (directed or undirected) graph and $\mathcal{F}$ is the family of optimal solutions of a combinatorial optimization problem with respect to a cost function $c:S\to {\mathbb{R}}_{+}$. Note that the cost function $c$ that defines the family and the weight function $w$ in the weighted transversal problem are not related.

In particular, we study the following five kinds of families: minimum cost $k$-spanning trees (unions of $k$ edge-disjoint spanning trees), minimum cost $s$-rooted $k$-arborescences (unions of $k$ arc-disjoint arborescences rooted at node $s$), minimum cost (directed or undirected) $k$-braids between nodes $s$ and $t$ (unions of $k$ edge-disjoint $s$-$t$ paths), and minimum cost (directed or undirected) $k$-edge-connected subgraphs. We focus on the special cases when either $c$ or $w$ is uniform. For the case $c\equiv 0$ (i.e. we want to block all combinatorial objects, not just the optimal ones), we show that most of the problems are NP-complete. In the other case, when $w\equiv 1$ (a minimum cardinality transversal problem for $\mathcal{F}$), most of our problems turn out to be polynomial-time solvable.