A Randomized Rounding Approach for DAG Edge Deletion

2025-07-10

AI Summary

This paper studies the edge-weighted $k$-path deletion problem on directed acyclic graphs (DAGs): given an edge-weighted DAG and an integer $k$, delete a minimum-weight set of edges to eliminate all directed paths of length exactly $k$. Being NP-hard, we propose a novel approximation framework based on randomized rounding of vertex labels in $[0,1]$. Our key innovation is the design and analysis of an improved independent label distribution, achieving for the first time a $0.549(k+1)$-approximation ratio on general DAGs—approaching the theoretical lower bound of this framework. For bipartite DAGs and certain structured instances, the ratio improves to $0.5(k+1)$. Unlike prior approaches, our method uniformly handles the rounding of linear programming relaxation solutions, yielding strictly better guarantees than all previously known results.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter $k$, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length $k$. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a $k$-approximation and showed that it is UGC-Hard to approximate better than $lfloor 0.5k floor$ for any constant $k ge 4$ using a work of Svensson from 2012. The approximation ratio was improved to $frac{2}{3}(k+1)$ by Klein and Wexler in 2016. In this work, we introduce a randomized rounding framework based on distributions over vertex labels in $[0,1]$. The most natural distribution is to sample labels independently from the uniform distribution over $[0,1]$. We show this leads to a $(2-sqrt{2})(k+1) approx 0.585(k+1)$-approximation. By using a modified (but still independent) label distribution, we obtain a $0.549(k+1)$-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below $0.542(k+1)$. Finally, we show a $0.5(k+1)$-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.

Problem

• Minimize edge deletions to remove paths of length k in DAGs
• Improve approximation ratios for DAG edge deletion problem
• Develop randomized rounding framework for vertex label distributions

Innovation

• Randomized rounding with uniform vertex labels
• Modified independent label distribution approach
• Specialized approximation for bipartite graphs