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A Randomized Rounding Approach for DAG Edge Deletion

Authors Sina Kalantarzadeh , Nathan Klein , Victor Reis

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Mathematics of computing → Approximation algorithms
Keywords
  • Approximation Algorithms
  • Randomized Algorithms
  • Linear Programming
  • Graph Algorithms
  • Scheduling

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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 ⌊0.5k⌋ for any constant k ≥ 4 using a work of Svensson from 2012. The approximation ratio was improved to 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-√2)(k+1) ≈ 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.

Cite As Get BibTex

Sina Kalantarzadeh, Nathan Klein, and Victor Reis. A Randomized Rounding Approach for DAG Edge Deletion. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.18

Author Details

Sina Kalantarzadeh
  • University of Waterloo, Canada
Nathan Klein
  • Boston University, MA, USA
Victor Reis
  • Microsoft Research, Seattle, WA, USA

References

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