Weighted Directed Tree Augmentation

• WDTAP is a directed set covering problem that selects minimum-cost links to cover every fundamental dicut in an oriented tree.
• The approach leverages linear programming relaxations with a base 2-approximation and improves to (1.75+ε)-approximation using willow and visibly k-wide instance techniques.
• Techniques from WDTAP extend to cross-free set covering and weighted multi 2-TAP, with implications for survivable network design in directed settings.

The weighted Directed Tree Augmentation Problem (WDTAP) is a directed analogue of the classical Weighted Tree Augmentation Problem (WTAP). Given an oriented tree $T = (V, A)$ and a set of directed links $L \subseteq V \times V$ each with associated positive costs, the objective is to select a subset of links of minimum total cost such that every fundamental dicut of the tree is entered by at least one link. WDTAP not only captures the essence of directed set covering for cross-free set families, but also provides a template for solving weighted multi $2$-TAP (where every undirected edge of a tree must be doubly covered). Approximation algorithms for WDTAP achieve a factor of $2$ by standard approaches; recent advances establish an improved $(1.75+\varepsilon)$-approximation for instances with bounded link costs through an overview of LP integrality on “willow” instances, dynamic programming for “visibly $k$-wide” cases, and a cascade of decomposition and rounding techniques (Neuwohner et al., 8 Nov 2025).

1. Formal Problem Definition

WDTAP is specified as follows:

• Input:
  • An oriented tree $T = (V, A)$ with each arc $a \in A$ directed.
  • A set of directed links $L \subseteq V \times V$, with each $\ell = (u, v) \in L$ assigned a cost $c_\ell > 0$.
• Objective: Select a subset $F \subseteq L$ of minimum total cost such that every fundamental dicut is entered by at least one link in $F$.

A fundamental dicut for an arc $a = (x, y) \in A$ is constructed by removing $a$ from $T$ to create two weakly connected components; let $U$ contain the tail $x$, then the pair $(U,\ V \setminus U)$ constitutes the fundamental dicut. A link $\ell = (u, v)$ enters this dicut if $v \in U$ and $u \notin U$; equivalently, the unique tree-path from $v$ to $u$ traverses $a$ in its forward direction. Thus, feasibility requires each tree-arc $a$ to be covered in the backward sense by at least one link from the chosen subset.

2. Linear Programming Relaxation

The natural set-covering LP for WDTAP is formulated as:

$$\min\ \sum_{\ell \in L} c_\ell x_\ell \ \text{s.t.}\ \sum_{\ell \in L} \chi_{a\ell}\,x_\ell \geq 1 \quad \forall\, a \in A \ x_\ell \geq 0 \quad \forall\, \ell \in L,$$

where $\chi_{a\ell} = 1$ if $\ell$ “backs over” $a$ (i.e., covers $a$ in the backward sense), and zero otherwise.

This LP has $O(|A| \times |L|)$ variables and constraints. Without exploiting further structure, the integrality gap of this relaxation is exactly $2$, and a simple 2-approximation is obtained by splitting the problem on up-arcs and down-arcs and rounding independently.

Approach	Integrality Gap	Polynomial Solvable
General WDTAP, standard LP	$2$	Yes (LP relaxation)
Willow or visibly $k$-wide instance	$1$	Yes

3. Tractable Instance Classes: Willows and Visibly $k$-Wide

3.1 Willow Instances

A rooted instance $(T, L, c, r)$ is a willow if there exists a set $W \subseteq V$ of up- or down-independent vertices such that every link in $L$ is:

• an up-link with head in $W$, or
• a down-link with tail in $W$, or
• a $W$-cross-link whose apex is in $W$.

Up-independence (resp. down-independence) at vertex $v$ means no link simultaneously covers both an up-arc (resp. down-arc) in the subtree rooted at $v$ and such an arc outside the subtree.

Crucially, for willow instances, the constraint matrix $(\chi_{a\ell})$ is totally unimodular. By Ghouila–Houri’s criterion and standard network-matrix arguments, this property ensures that the LP relaxation has integral optimal solutions. Consequently, WDTAP is solvable in polynomial time on willow instances using an LP solver.

3.2 Visibly $k$-Wide Instances

An instance is visibly $k$-wide if, for all vertices $v$, the visible width (the maximum size of an ancestor-free set of visible up- or down-arcs under $v$) is at most $k$, where visibility is defined with respect to a subset $L'$ of links illuminating a given subtree.

For visibly $k$-wide instances, bottom-up dynamic programming (DP) applies efficiently:

• For each node $v$ and each boundary condition (at most $2k$-sized interface $Y \subseteq L$), subproblems are defined recursively.
• Overall, the DP solves $n \cdot |L|^{O(k)}$ subproblems, each updatable in $|L|^{O(k)}$ time, yielding total runtime $O(n \cdot |L|^{O(k)})$.

4. Improved Approximation: $(1.75+\varepsilon)$-Algorithm for Bounded Costs

For instances with bounded cost ratio $\Delta = \max c_\ell\,/\min c_\ell$, a $(1.75 + \varepsilon)$-approximation is achieved via a sequence of partial decompositions, instance-splitting, and rounding based on willow and visibly $k$-wide substructures.

The algorithm consists of:

1. Solve the LP relaxation to obtain a fractional solution $x$.
2. Phase I (Light-Arc Splitting):
   • Heavily covered arcs ($\geq \zeta_1$) are identified and contracted (solved integrally).
   • Split links at vertices as dictated by relative $x$-mass crossing arcs, to ensure most vertices are up- or down-independent or have small visible width ($\leq k$), except possibly at parent arcs with high mass in the “wrong” direction.
3. Phase II (Core Splitting):
   • Partition tree into up- and down-arborescences; within each, determine base arcs with high mass in the wrong direction.
   • Each core is recursively split at core-roots and trunk arcs to ensure resultant links do not simultaneously cover a core-arc in both directions.
   • Each core-root has links only outward or inward, making these vertices up- or down-independent.
4. Best-of-Three Routines:
   • Partition links into four classes: those covering core-arcs in the right direction ($\overrightarrow L$), wrong direction ($\overleftarrow L$), non-core cross-links ($L_{\text{cross}}$), and the rest ($L_{\text{rest}}$).
   • Three solutions are constructed:
     • (A) Removing all of $$\overrightarrow L,\, \overleftarrow L,\, L_{\text{cross}}$$ yields a visibly $k$-wide instance (solved by DP).
     • (B) Removing $\overleftarrow L,\, L_{\text{rest}}$ yields a willow (solved by LP).
     • (C) Removing $\overrightarrow L,\, L_{\text{rest}}$ and performing apex-splitting yields a willow (solved by LP).
   • Output is the minimum-cost result of these three, plus that for contracted arcs from Phase I.

This best-of-three approach, with parameter choices such that $k = O(1/\varepsilon^2)$, drives the approximation factor to $1.375$ post-Phase II. Incorporating the minor cost from contracted arcs in Phase I, the final guarantee is $(1.75+\varepsilon)$. All operations in both Phases and solution composition are polynomial time ($$\text{poly}(n, |L|, \Delta^{O(1)}, \varepsilon^{-O(1)})$$).

Step	Key Operation	Solution Type
Light-arc splitting	Contract heavily covered arcs; split	Creates up-/down-independence
Core splitting	Partition, identify cores, split	Forces single-direction covering
Best-of-three	DP on k-wide, LP on willow instances	Min cost among three cases

5. Algorithm Pseudocode and Runtime

The algorithm for the $(1.75+\varepsilon)$-approximation, as described by Neuwohner–Silina–Zlatin, can be expressed as:

Input: (T,L,c,r), cost-ratio Δ, ε>0 1. Solve LP to get x. 2. Contract arcs covered ≥ζ1 by x. (ζ1≈ε) 3. Apply Light-Splitting (Phase I) → support L* and x*. 4. Apply Core-Splitting (Phase II) → support L** and x**. 5. Partition L** into L_cross, →L, ←L, L_rest. 6. Construct 3 instances: (A) split off {→L∪←L∪L_cross}, call DP on k-wide. (B) split off {←L∪L_rest}, solve as willow by LP. (C) split off {→L∪L_rest}, solve as willow by LP. 7. Output best of (A),(B),(C) plus the contracted-arc solution.

• The LP in step 1 is polynomial in $(n, |L|)$.
• Both splitting phases require only single-pass updates through the tree, also polynomial.
• DP on $k$-wide instances runs in $n \cdot |L|^{O(k)}$.
• All willow subproblems are solved by LP in polynomial time.
• The overall runtime is $$\text{poly}(n,|L|, \Delta^{O(1)}, \varepsilon^{-O(1)})$$.

6. Connections to Set Covering and Multi Cover Problems

WDTAP generalizes several fundamental combinatorial covering settings:

• Cross-free set covering: By the theorem of Edmonds–Giles, every cross-free set family admits a tree representation, and WDTAP on the resulting oriented tree captures the directed covering version for such families.
• Weighted multi $2$-TAP: The multi $2$-TAP, where each undirected tree edge must be covered twice, reduces to WDTAP by duplicating each edge and link with both orientations.

This suggests that techniques developed for WDTAP, especially decompositions based on visible width and willow structures, may inform more general survivable network design questions, particularly in directed graphs.

7. Future Directions and Open Problems

Critical open directions identified include:

• Determining whether the $1.75$ approximation barrier can be improved further for bounded-cost WDTAP.
• Extending visibility-based dynamic programming or willow-decomposition techniques to more general requirements, such as ensuring strong $k$-connectivity in the underlying digraph, particularly when the structure of the required cuts is laminar or cross-free.
• Generalizing these tools to broader families of Survivable Network Design problems for directed graphs, potentially by further structural analysis of visible width and interlinking dependencies.

These inquiries reflect both algorithmic and combinatorial frontiers in the study of directed augmentation problems.