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Engineering Algorithms for Dynamic Greedy Set Cover

Amitai Uzrad

Abstract

In the dynamic set cover problem, the input is a dynamic universe of elements and a fixed collection of sets. As elements are inserted or deleted, the goal is to efficiently maintain an approximate minimum set cover. While the past decade has seen significant theoretical breakthroughs for this problem, a notable gap remains between theoretical design and practical performance, as no comprehensive experimental study currently exists to validate these results. In this paper, we bridge this gap by implementing and evaluating four greedy-based dynamic algorithms across a diverse range of real-world instances. We derive our implementations from state-of-the-art frameworks (such as GKKP, STOC 2017; SU, STOC 2023; SUZ, FOCS 2024), which we simplify by identifying and modifying intricate subroutines that optimize asymptotic bounds but hinder practical performance. We evaluate these algorithms based on solution quality (set cover size) and efficiency, which comprises update time (the time required to update the solution following each insertion or deletion) and recourse (the number of changes made to the solution per update). Each algorithm uses a parameter $β$ to balance quality against efficiency; we investigate the influence of this tradeoff parameter on each algorithm and then perform a comparative analysis to evaluate the algorithms against each other. Our results provide the first practical insights into which algorithmic strategies provide the most value in realistic scenarios.

Engineering Algorithms for Dynamic Greedy Set Cover

Abstract

In the dynamic set cover problem, the input is a dynamic universe of elements and a fixed collection of sets. As elements are inserted or deleted, the goal is to efficiently maintain an approximate minimum set cover. While the past decade has seen significant theoretical breakthroughs for this problem, a notable gap remains between theoretical design and practical performance, as no comprehensive experimental study currently exists to validate these results. In this paper, we bridge this gap by implementing and evaluating four greedy-based dynamic algorithms across a diverse range of real-world instances. We derive our implementations from state-of-the-art frameworks (such as GKKP, STOC 2017; SU, STOC 2023; SUZ, FOCS 2024), which we simplify by identifying and modifying intricate subroutines that optimize asymptotic bounds but hinder practical performance. We evaluate these algorithms based on solution quality (set cover size) and efficiency, which comprises update time (the time required to update the solution following each insertion or deletion) and recourse (the number of changes made to the solution per update). Each algorithm uses a parameter to balance quality against efficiency; we investigate the influence of this tradeoff parameter on each algorithm and then perform a comparative analysis to evaluate the algorithms against each other. Our results provide the first practical insights into which algorithmic strategies provide the most value in realistic scenarios.

Paper Structure

This paper contains 21 sections, 4 theorems, 6 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Static Set Cover.
  3. Dynamic Set Cover.
  4. Previous Results in Theory.
  5. Previous Results in Practice.
  6. Our Contribution.
  7. Paper Organization.
  8. The Static Greedy Algorithm
  9. Basic Dynamic Algorithms
  10. The "Robust" Algorithm
  11. Level-Based Dynamic Algorithms
  12. The "Local" Algorithm
  13. The "Partial" Algorithm
  14. The "Global" Algorithm
  15. Experimental Evaluation
  16. ...and 6 more sections

Key Result

Lemma 6

staticalg produces a solution $\mathcal{C}$ with approximation factor $\beta \cdot (\ln (n) + 1)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (6)

  • Figure 1: Normalized trade-off curves representing solution quality vs. efficiency. The left plot displays mean normalized set cover size vs. mean normalized update time, and the right plot displays mean normalized set cover size vs. mean normalized recourse. Each curve represents the geometric mean across 120 instances for the full range of $\beta$ values; curves closer to the bottom-left corner indicate superior overall performance.
  • Figure 2: Performance profiles for amortized set cover size (left), amortized update time (center) and amortized recourse (right) of all four algorithms, each with the chosen $\beta$.
  • Figure 3: Performance profiles for each algorithm evaluated across all three quality metrics. Each row represents a specific algorithm (top to bottom: robust, local, partial, and global), divided into six variants with varying values of $\beta$. The columns represent amortized set cover size (left), update time (center), and recourse (right). Detailed legends for $\beta$ values are provided beneath each row.
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 6
  • Theorem 8
  • Claim 9
  • Claim 10
  • Definition 11
  • Lemma 13
  • Claim 14
  • ...and 3 more