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Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint

Max Klimm, Martin Knaack

TL;DR

A greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint is relied on and tight approximation guarantees are obtained in the setting of a submodular objective function with curvature.

Abstract

This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop a policy with a robustness factor that is decreasing in the curvature $c$ of the submodular function. For the extreme cases $c=0$ corresponding to an additive objective function, it matches a previously known and best possible robustness factor of $1/2$. For the other extreme case of $c=1$ it yields a robustness factor of $\approx 0.35$ improving over the best previously known robustness factor of $\approx 0.06$. The analysis of our policy relies on a greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint. We obtain tight approximation guarantees for both of these algorithms in the setting of a submodular objective function with curvature $c$.

Maximizing a Submodular Function with Bounded Curvature under an Unknown Knapsack Constraint

TL;DR

A greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint is relied on and tight approximation guarantees are obtained in the setting of a submodular objective function with curvature.

Abstract

This paper studies the problem of maximizing a monotone submodular function under an unknown knapsack constraint. A solution to this problem is a policy that decides which item to pack next based on the past packing history. The robustness factor of a policy is the worst case ratio of the solution obtained by following the policy and an optimal solution that knows the knapsack capacity. We develop a policy with a robustness factor that is decreasing in the curvature of the submodular function. For the extreme cases corresponding to an additive objective function, it matches a previously known and best possible robustness factor of . For the other extreme case of it yields a robustness factor of improving over the best previously known robustness factor of . The analysis of our policy relies on a greedy algorithm that is a slight modification of Wolsey's greedy algorithm for the submodular knapsack problem with a known knapsack constraint. We obtain tight approximation guarantees for both of these algorithms in the setting of a submodular objective function with curvature .
Paper Structure (11 sections, 11 theorems, 71 equations, 4 figures, 5 algorithms)

This paper contains 11 sections, 11 theorems, 71 equations, 4 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Further Related Work
  4. Preliminaries
  5. Submodular Functions
  6. Curvature
  7. Submodular Maximization under a Knapsack Constraint
  8. Submodular Knapsack Problem with Known Capacity
  9. Submodular Knapsack Problem with Unknown Capacity
  10. Indispensable Items
  11. A Robust Policy

Key Result

Lemma 2.1

For a normalized, monotonic, and submodular function $f \colon 2^N \to \mathbb{R}_{\geq 0}$ with curvature $c \in [0,1]$, the following inequalities are satisfied:

Figures (4)

  • Figure 1: Robustness factors $\alpha$ of deterministic policies as a function of the curvature $c$ achieved by this and previous work.
  • Figure 2: Modified Greedy Algorithm MGreedy
  • Figure 3: Visualization of the functions $g$ and $h$ with $f(S^*)=B=1$ and for different curvatures $c \in \{0,0.5,1\}$. We fixed values of $y$ and graphs with the same color belong to the same value of $y$. The increasing graphs are function $g$ and the decreasing graphs are function $h$; for $c=1$ the three graphs of $h$ are identical. The horizontal axis $z$ represents the total size of $G_k$ and the vertical axis the approximation guarantee. We can see that minimum of the maximum of $g$ and $h$ (the dots at the intersection) get lower if we decrease the value of $y$.
  • Figure 4: Coverage function $f$ with four items $i_1,\dotsc,i_4$.

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 11 more