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Practical and Parallelizable Algorithms for Non-Monotone Submodular Maximization with Size Constraint

Yixin Chen, Alan Kuhnle

TL;DR

This work proposes a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in O(log(n) adaptive rounds with high probability, and provides two approximation algorithms.

Abstract

We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to $0.193 - \varepsilon$. The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in $O( \log(n) )$ adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio $1/6 - \varepsilon$, adaptivity $O( \log (n) )$, and query complexity $O( n \log (k) )$, while the second has approximation ratio $0.193 - \varepsilon$, adaptivity $O( \log^2 (n) )$, and query complexity $O(n \log (k))$. Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

Practical and Parallelizable Algorithms for Non-Monotone Submodular Maximization with Size Constraint

TL;DR

This work proposes a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in O(log(n) adaptive rounds with high probability, and provides two approximation algorithms.

Abstract

We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to . The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio , adaptivity , and query complexity , while the second has approximation ratio , adaptivity , and query complexity . Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.

Paper Structure

This paper contains 23 sections, 21 theorems, 66 equations, 6 figures, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. The ThreshSeq Algorithm
  5. Algorithm Overview
  6. Theoretical Guarantees
  7. The AdaptiveSimpleThreshold Algorithm
  8. The AdaptiveThresholdGreedy Algorithm
  9. Empirical Evaluation
  10. Algorithm Setup for AST and ATG
  11. Comparison Algorithms and Other Settings
  12. Applications and Datasets
  13. Main Results
  14. Comparison of Different Threshold Sampling Procedures.
  15. Discussion and Future Directions
  16. ...and 8 more sections

Key Result

Theorem 1

For each $\varepsilon > 0$, there is an algorithm that achieves a $(1/2 - \varepsilon )$-approximation for unconstrained submodular maximization using $\mathcal{O}\left( \log (1/ \varepsilon ) / \varepsilon \right)$ adaptive rounds and $\mathcal{O}\left( n \log^3 (1/ \varepsilon ) / \varepsi

Figures (6)

  • Figure 1: Value of $\tau_0$ and $\tau_\ell$. It is satisfied that $\tau_0 \ge \textsc{OPT}\xspace / k$ and $\tau_\ell \le \textsc{OPT}\xspace / (ck)$.
  • Figure 2: Comparison of objective value (normalized by the IteratedGreedy objective value), total queries, and adaptive rounds on web-Google for the maxcut application for both small and large $k$ values. The large $k$ values are given as a fraction of the number of nodes in the network. The algorithm of ? (? ) is run with oracle access to the multilinear extension and its gradient; total queries reported for this algorithm are queries to these oracles, rather than the original set function. The legend in Fig. \ref{['fig:legend']} applies to all other subfigures.
  • Figure 3: Results for revenue maximization on ca-Astro, for both small and large $k$ values. Large $k$ values are indicated by a fraction of the total number $n$ of nodes. The legends in Fig. \ref{['fig:main']} and \ref{['fig:apx-exp-maxcut']} apply.
  • Figure 4: Additional results for maximum cut on BA and ca-GrQc with ParCardinal algorithms.
  • Figure 5: Results of AST and ATG with four threshold sampling procedures on two datasets. The algorithms are run strictly following pseudocode. The legends in Fig. \ref{['fig:val-BA-2-ast']} and \ref{['fig:val-BA-2-atg']} apply to all other subfigures.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Theorem 1: ? (? )
  • Definition 2: Threshold
  • Theorem 3
  • Lemma 3
  • Lemma 3
  • Lemma 3
  • proof : Proof of Success Probability (Property 1)
  • proof : Proof of Adaptivity and Query Complexity (Property 2)
  • proof : Proof of Marginal Gains (Property 3 and 4)
  • Theorem 4
  • ...and 26 more