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The Cost of Consistency: Submodular Maximization with Constant Recourse

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

TL;DR

This work seeks bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make, at most, a constant number of updates per step and shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

Abstract

In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of $\tfrac{2}{3}$ for general monotone submodular functions, and an improved (also tight) bound of $\tfrac{3}{4}$ for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a $0.51$-approximation. Combined with an information-theoretic hardness of $\tfrac{1}{2}$ for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

The Cost of Consistency: Submodular Maximization with Constant Recourse

TL;DR

This work seeks bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make, at most, a constant number of updates per step and shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

Abstract

In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of for general monotone submodular functions, and an improved (also tight) bound of for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a -approximation. Combined with an information-theoretic hardness of for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

Paper Structure

This paper contains 33 sections, 23 theorems, 88 equations, 3 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Tight Information-Theoretic Bound.
  4. Submodular vs. Coverage Functions.
  5. Separation for Efficient Algorithms.
  6. Further Related Work
  7. Preliminaries
  8. Submodular Functions.
  9. Coverage Functions.
  10. Approximation Guarantee.
  11. Consistency.
  12. Techniques and Ideas
  13. Addition-Robust Submodular Maximization.
  14. Tight-Information Theoretic Bounds
  15. Coverage Functions.
  16. ...and 18 more sections

Key Result

Theorem 3.2

Suppose $\mathcal{A}$ is an $\alpha$-addition-robust algorithm. Then Check-Point with precision parameter $\varepsilon \in (0,1)$ is $O(1/\varepsilon^2)$-consistent and provides an $(\alpha-O(\varepsilon))$-approximation.

Figures (3)

  • Figure 1: Visualization of the perfect alignment phenomenon described in \ref{['ex:alignment']}.
  • Figure 2: Visualization of MinMaxSampling on the perfect alignment instance.
  • Figure 3: Visualization of the Greedy-with-Certificate algorithm.

Theorems & Definitions (45)

  • Definition 3.1
  • Theorem 3.2
  • Example 3.3: Perfect Alignment
  • Theorem 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.0
  • Lemma 5.1
  • ...and 35 more