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

2025-11-08

AI Summary

This paper studies the Weighted Directed Tree Augmentation Problem (WDTAP): given a directed tree and a set of directed links, select minimum-cost links to cover all fundamental directed cuts—i.e., cuts with out-degree one and in-degree zero. WDTAP is equivalent to the directed covering problem for cross-free set families and closely related to the Weighted Multi-2-TAP. To overcome the long-standing 2-approximation barrier, we introduce a structural instance classification framework: we define the “willow” class—instances for which the natural set cover LP is inherently integral—and the “visibly k-wide” class—instances amenable to exact dynamic programming. By integrating tree decomposition, LP rounding, and refined structural analysis, we achieve a $(1.75+varepsilon)$-approximation under bounded link costs, the first algorithm to break the theoretical 2-approximation barrier for WDTAP.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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 $L subseteq V imes V$ 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.

Problem

• Approximating minimum cost directed tree augmentation
• Covering fundamental dicuts with bounded cost links
• Improving approximation ratio via structural analysis

Innovation

• Introduces approximation algorithm for directed tree augmentation
• Uses dynamic programming for visibly k-wide instances
• Leverages structural analysis to improve approximation ratio