Approximating the minimum directed tree cover

Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a rooted directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be $NP$-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combinatorial approximation algorithm for the DTCP and proving that the approximation ratio of the algorithm is $\max{2, \ln(D^+)}$ with $D^+$ is the maximum outgoing degree of the nodes in $G$.

💡 Research Summary

The paper studies the Minimum Directed Tree Cover Problem (DTCP), where a directed graph (G=(V,A)) with non‑negative arc costs is given and the goal is to find a rooted directed tree (T) such that for every arc ((u,v)\in A) at least one of its endpoints belongs to (T). The cost of (T) is the sum of the costs of the arcs that constitute the tree, and the objective is to minimize this sum.

First, the authors establish a tight connection between DTCP and the classic Weighted Set Cover Problem (SCP). By constructing a gadget that maps each element of the universe to a vertex and each set to a directed edge together with a small auxiliary tree, they show that any instance of SCP can be transformed into an instance of DTCP while preserving the optimal cost. Consequently, the known NP‑hardness of SCP carries over to DTCP, and the logarithmic lower bound on approximability ((\Omega(\log n))) for set cover also applies to DTCP. This reduction also clarifies why one should not expect an approximation ratio better than (\ln D^{+}), where (D^{+}) denotes the maximum out‑degree of the input graph.

Having identified the theoretical limits, the paper proceeds to design a purely combinatorial approximation algorithm for DTCP. The algorithm is greedy in nature and can be described as follows:

1. Initialise the cover tree (T) with a designated root (r).
2. While there exist uncovered arcs, compute for each candidate arc ((x,y)) the cover efficiency – the number of currently uncovered arcs that would become covered if ((x,y)) (and the necessary connecting vertices) were added to (T), divided by the cost of ((x,y)).
3. Select the arc with the highest efficiency, insert it together with any required vertices into (T), and mark all newly covered arcs as satisfied.

The algorithm does not rely on linear programming relaxations or primal‑dual constructions; it only needs to maintain a list of uncovered arcs and recompute efficiencies after each insertion.

The approximation analysis splits into two regimes based on the maximum out‑degree (D^{+}).

• Low‑degree regime ((\ln D^{+}<2)). In each iteration the chosen arc costs at most half of the remaining optimal cost, because the efficiency measure guarantees that the ratio of added cost to newly covered arcs is bounded by 2. Summing over all iterations yields a total cost at most (2\cdot\text{OPT}).

• High‑degree regime ((\ln D^{+}\ge 2)). The greedy choice mimics the classic set‑cover greedy algorithm. By the standard analysis of set‑cover, the total cost incurred is at most (\ln D^{+}) times the optimal cost.

Thus the overall approximation factor of the algorithm is (\displaystyle \max{2,;\ln D^{+}}).

Regarding computational complexity, each iteration scans the remaining uncovered arcs to recompute efficiencies, which takes (O(|A|)) time. In the worst case the algorithm performs at most (|V|) iterations, leading to a total running time of (O(|A|\cdot|V|)). This is polynomial and, importantly, much simpler to implement than LP‑based methods.

The paper’s contributions can be summarised as follows:

1. Hardness transfer: By explicitly reducing weighted set cover to DTCP, the authors confirm that DTCP inherits the same logarithmic inapproximability barrier.
2. Combinatorial algorithm: They present a greedy algorithm that achieves the best possible approximation ratio up to constant factors, matching the lower bound for the high‑degree case and providing a constant‑factor (2) guarantee when the graph is sparse.
3. Practical relevance: The algorithm’s simplicity and its dependence only on the maximum out‑degree make it attractive for applications such as network design, broadcast tree construction, and power‑grid monitoring, where the underlying digraph often has bounded out‑degree.

The paper also discusses limitations. When (D^{+}) is large, the (\ln D^{+}) factor can be substantial, and the algorithm does not exploit any additional structural properties of the input graph (e.g., planarity, tree‑width) that could lead to better ratios. Moreover, the analysis assumes exact computation of cover efficiencies; in massive graphs, approximate efficiency evaluation or sampling techniques might be needed to keep the runtime practical.

In conclusion, the work bridges the gap between the theoretical hardness of DTCP and a concrete, implementable approximation method. It clarifies the role of the maximum out‑degree in determining the achievable approximation quality and opens several avenues for future research, including refined analyses for special graph classes, hybrid algorithms that combine greedy selection with LP‑based rounding, and empirical studies on real‑world network instances.