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Consistent Submodular Maximization

Paul Dütting, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Morteza Zadimoghaddam

TL;DR

This paper provides algorithms in this setting with different trade-offs between consistency and approximation quality, and complements the theoretical results with an experimental analysis showing the effectiveness of these algorithms in real-world instances.

Abstract

Maximizing monotone submodular functions under cardinality constraints is a classic optimization task with several applications in data mining and machine learning. In this paper we study this problem in a dynamic environment with consistency constraints: elements arrive in a streaming fashion and the goal is maintaining a constant approximation to the optimal solution while having a stable solution (i.e., the number of changes between two consecutive solutions is bounded). We provide algorithms in this setting with different trade-offs between consistency and approximation quality. We also complement our theoretical results with an experimental analysis showing the effectiveness of our algorithms in real-world instances.

Consistent Submodular Maximization

TL;DR

This paper provides algorithms in this setting with different trade-offs between consistency and approximation quality, and complements the theoretical results with an experimental analysis showing the effectiveness of these algorithms in real-world instances.

Abstract

Maximizing monotone submodular functions under cardinality constraints is a classic optimization task with several applications in data mining and machine learning. In this paper we study this problem in a dynamic environment with consistency constraints: elements arrive in a streaming fashion and the goal is maintaining a constant approximation to the optimal solution while having a stable solution (i.e., the number of changes between two consecutive solutions is bounded). We provide algorithms in this setting with different trade-offs between consistency and approximation quality. We also complement our theoretical results with an experimental analysis showing the effectiveness of our algorithms in real-world instances.
Paper Structure (18 sections, 5 theorems, 37 equations, 3 figures, 4 algorithms)

This paper contains 18 sections, 5 theorems, 37 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Our Techniques.
  4. Preliminaries
  5. Impossibility Result
  6. Encompassing-Set
  7. Chasing-Local-Opt
  8. $S_\tau$ is a local optimum.
  9. $S_{\tau}$ is not a local optimum.
  10. Experiments
  11. Influence Maximization.
  12. Summarizing Geolocation Data.
  13. Experimental Results.
  14. Conclusion
  15. Instability of known algorithms
  16. ...and 3 more sections

Key Result

Theorem 3.1

Fix any constant $C$ and precision parameter $\varepsilon \in (0,1)$. No $C$-consistent (deterministic) algorithm provides a $(2-\varepsilon)$-approximation.

Figures (3)

  • Figure 1: Experimental Results. The first row reports the objective values, the second one the cumulative consistency.
  • Figure 2: Experimental results on the weighted covering instance of \ref{['ex:swapping']}, for $\delta = 0.01$, $i = 7$, and $\varepsilon = 0.1$.
  • Figure 3: Further Experimental Results. The first row reports the objective values, the second one the cumulative consistency.

Theorems & Definitions (22)

  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Definition 5.1
  • Theorem 5.2
  • ...and 12 more