Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Better-Than-2 Approximation for the Directed Tree Augmentation Problem

Meike Neuwohner, Olha Silina, Michael Zlatin

TL;DR

This work introduces the Weighted Directed Tree Augmentation Problem (WDTAP) and proves a near-optimal approximation for the bounded-cost regime by developing new structural notions. The key ideas are partial decomposition through splitting, the concepts of visible width and willow instances, and a partial-separation framework that yields a $(1.75+\varepsilon)$-approximation. A central technical contribution is identifying conditions under which the LP becomes integral (willows) and enabling exact dynamic programming for constant-width instances. The approach blends polyhedral insights (total unimodularity for willows) with combinatorial decompositions and a sophisticated LP-based oracle to tame heavy coverage in the wrong direction. The results advance directed network augmentation by surpassing the longstanding 2-approximation barrier in the bounded-cost setting and connect to broader cross-free directed set-cover problems.

Abstract

We introduce and study a directed analogue of the weighted Tree Augmentation Problem (WTAP). In the weighted Directed Tree Augmentation Problem (WDTAP), we are given an oriented tree $T = (V,A)$ and a set of directed links $L \subseteq V \times V$ with positive costs. The goal is to select a minimum cost set of links which enters each fundamental dicut of $T$ (cuts with one leaving and no entering tree arc). WDTAP captures the problem of covering a cross-free set family with directed links. It can also be used to solve weighted multi $2$-TAP, in which we must cover the edges of an undirected tree at least twice. WDTAP can be approximated to within a factor of $2$ using standard techniques. We provide an improved $(1.75+ \varepsilon)$-approximation algorithm for WDTAP in the case where the links have bounded costs, a setting that has received significant attention for WTAP. To obtain this result, we discover a class of instances, called "willows'', for which the natural set covering LP is an integral formulation. We further introduce the notion of "visibly $k$-wide'' instances which can be solved exactly using dynamic programming. Finally, we show how to leverage these tractable cases to obtain an improved approximation ratio via an elaborate structural analysis of the tree.

A Better-Than-2 Approximation for the Directed Tree Augmentation Problem

TL;DR

This work introduces the Weighted Directed Tree Augmentation Problem (WDTAP) and proves a near-optimal approximation for the bounded-cost regime by developing new structural notions. The key ideas are partial decomposition through splitting, the concepts of visible width and willow instances, and a partial-separation framework that yields a -approximation. A central technical contribution is identifying conditions under which the LP becomes integral (willows) and enabling exact dynamic programming for constant-width instances. The approach blends polyhedral insights (total unimodularity for willows) with combinatorial decompositions and a sophisticated LP-based oracle to tame heavy coverage in the wrong direction. The results advance directed network augmentation by surpassing the longstanding 2-approximation barrier in the bounded-cost setting and connect to broader cross-free directed set-cover problems.

Abstract

We introduce and study a directed analogue of the weighted Tree Augmentation Problem (WTAP). In the weighted Directed Tree Augmentation Problem (WDTAP), we are given an oriented tree and a set of directed links with positive costs. The goal is to select a minimum cost set of links which enters each fundamental dicut of (cuts with one leaving and no entering tree arc). WDTAP captures the problem of covering a cross-free set family with directed links. It can also be used to solve weighted multi -TAP, in which we must cover the edges of an undirected tree at least twice. WDTAP can be approximated to within a factor of using standard techniques. We provide an improved -approximation algorithm for WDTAP in the case where the links have bounded costs, a setting that has received significant attention for WTAP. To obtain this result, we discover a class of instances, called "willows'', for which the natural set covering LP is an integral formulation. We further introduce the notion of "visibly -wide'' instances which can be solved exactly using dynamic programming. Finally, we show how to leverage these tractable cases to obtain an improved approximation ratio via an elaborate structural analysis of the tree.

Paper Structure

This paper contains 36 sections, 55 theorems, 46 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Directed Tree Augmentation Problem
  3. Further related work
  4. Organization of the paper
  5. Preliminaries
  6. Comparison with previous work
  7. A decomposition-based approach for WTAP with bounded cost ratio $\dots$
  8. $\dots$ and why it doesn't work for DTAP
  9. Our contribution
  10. New notions of partial decomposition: visible width and willows
  11. Visible width
  12. Willows
  13. Our approach
  14. Blue-sky version
  15. Coming down to earth
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $\Delta\ge 1$ and let $\varepsilon > 0$. There exists a polynomial-time $(1.75+\varepsilon)$-approximation algorithm for WDTAP, restricted to instances with cost ratio at most $\Delta$.

Figures (8)

  • Figure 1.1: A DTAP instance is shown on the left, with links drawn as dashed lines. A feasible solution is shown on the right. Colors indicate the tree arcs that are covered by each link.
  • Figure 3.1: An instance of WDTAP, with arcs indicated by solid arrows and links shown as dashed arrows. A solution to \ref{['eq:WDTAP_LP']} is indicated next to the links. All links have cost $1$. The subtree hanging off the vertex $v$ contains a large number $p$ of leaves that are connected to $v$ via up-arcs. In the given LP solution, these up-arcs are (partially) covered by the orange links, each of which covers the down-arc entering $v$ in the wrong direction. Splitting all of the orange links at $v$ is too expensive.
  • Figure 4.1: The visible width at $v$ is at least 4, as certified by the four orange-shaded arcs in its subtree. This is an ancestor-free set of down-arcs which are all visible to $v$. However, the black down-arc incident to $x$, as well as the down-arc above, are not visible to $v$ since they are not covered by a link $\ell$ for which $v$ is an inner vertex of $\overline{P}_\ell$. The down-arc incident to $y$ is visible to $v$, but does not form an ancestor-free set with the shaded down-arc above.
  • Figure 4.2: A willow (choosing $W = \{r, u, v\}$). Notice that $u$ is down-independent, $v$ is up-independent, and the root $r$ is both. All links are either up-links, down-links, or have their apex in $W$.
  • Figure 4.3: Illustration of \ref{['theorem:weak_dream']}. (Top) A DTAP instance and a solution $x$ to \ref{['eq:WDTAP_LP']} whose support is shown as dashed links. (Bottom) The resulting solution $x^*$ with $W=\{a,c,d,r\}$, where arcs $a_e$ and $a_d$ are $\zeta_2$-heavy (light blue). The links contained in the support of $x^*$, but not in the support of $x$, are shown in purple. All vertices except $a,c, d, e$ have visible width at most $k=3$. After splitting $W$-cross-links, $a$ and $c$ will have visible width $0$, $d$ will have visible up-width $0$ and $e$ will have visible down-width $0$.
  • ...and 3 more figures

Theorems & Definitions (143)

  • Theorem 1.1
  • Theorem 4.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Definition 5.1
  • Definition 5.2
  • Theorem 5.3: unimodularity theorem
  • Proof 1: Proof of \ref{['theorem:unimodularity']}
  • Claim 5.4
  • ...and 133 more