Blocking optimal $k$-arborescences

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.

💡 Research Summary

The paper addresses a combinatorial optimization problem that lies at the intersection of arborescence packing, matroid theory, and transversal (hitting‑set) problems. Given a directed graph D = (V, A), a positive integer k, and a non‑negative cost function c on the arcs, a k‑arborescence is defined as the disjoint union of k spanning arborescences (each arborescence is a directed spanning tree with at most one incoming arc per vertex and exactly one root). The classic result of Edmonds and Fulkerson shows that a minimum‑cost k‑arborescence can be found in polynomial time via a matroid‑intersection algorithm.

The authors consider the blocking version of this problem: find a smallest set T ⊆ A that intersects (i.e., contains at least one arc from) every minimum‑cost k‑arborescence. In other words, T is a minimum transversal of the family of optimal k‑arborescences. For k = 1 this problem was solved earlier (Bernáth and Pap, IPCO 2013). The present work extends the study to arbitrary k, providing an algorithm whose running time is polynomial when k is treated as a fixed constant. The general case where k is part of the input remains open.

Equivalence of two formulations
Two natural formulations are introduced: (1) blocking optimal k‑arborescences (no distinguished root) and (2) blocking optimal s‑rooted k‑arborescences (the root vertex s is fixed). The paper proves that the two problems are polynomial‑time equivalent. The reduction from (2) to (1) is trivial (delete all arcs entering s). The opposite reduction augments the original digraph with a new source s and α = |A| + k parallel arcs from s to every vertex, assigning them a very large cost β. In the resulting (α, β)‑extension, any minimum‑cost s‑rooted k‑arborescence must use exactly k arcs leaving s, which forces the remaining arcs to be a minimum‑cost k‑arborescence of the original graph. Hence solving the rooted version solves the unrooted version and vice‑versa.

Laminar structure of optimal arborescences
A central technical contribution is a structural description of optimal k‑arborescences using laminar families. By considering the linear program that minimizes c·x subject to the standard arborescence constraints (0 ≤ x_a ≤ 1, and for every non‑empty Z ⊆ V \ {s}, the in‑degree sum x(δ^{in}(Z)) ≥ k), the authors invoke total dual integrality (TDI) and complementary slackness. They show that there exists an optimal dual solution whose variables corresponding to the cut constraints have a laminar support L. Translating back to the primal side yields three sets: * A₁ — arcs that must appear in every optimal solution (dual variable z_a > 0); * A₀ — arcs that can never appear (dual reduced cost c_a − ∑_{Z∋a} y_Z < 0); * L — a laminar family of vertex subsets such that any optimal arborescence must be L‑tight, i.e., for each W ∈ L the number of arcs entering W is exactly k.

Thus an arborescence F is optimal iff A₁ ⊆ F ⊆ A \ A₀ and F is L‑tight. This description reduces the blocking problem to “find a minimum hitting set for the family of L‑tight k‑arborescences that avoids A₀ and respects the mandatory arcs A₁”.

Matroid‑restricted k‑arborescences
To handle the L‑tightness condition, the authors introduce matroid‑restricted k‑arborescences. For each vertex v, a matroid M_v is defined on the set of arcs entering v. A set F of arcs is M‑restricted if F ∩ δ^{in}(v) is independent in M_v for every v. The problem of finding an M‑restricted k‑arborescence can be expressed as a matroid intersection between: * M₁ — the k‑fold circuit matroid of the underlying undirected graph (ensuring the edge‑disjoint spanning‑tree condition); * M₂ — the direct sum of the k‑shortened matroids M_v (which cap the rank at k).

Edmonds’ matroid‑intersection theorem yields a necessary and sufficient condition: for every subpartition 𝔛 of V, \