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Optimal 4-Approximation for the Correlated Pandora's Problem

Nikhil Bansal, Zhiyi Huang, Zixuan Zhu

TL;DR

This work tackles the Correlated Pandora's Problem, where box outcomes are drawn from a general, potentially correlated distribution, by establishing an optimal $4$-approximation. The authors depart from prior combinatorial analyses and develop LP-based relaxations (Unit-Cost CP and General CP) together with a Poisson Rounding scheme and a Balanced Stopping rule to achieve the bound. They show that their $4$-approximation matches the MSSC lower bound, and they provide a cohesive framework that extends from unit-cost/unit-time settings to general costs and volumes. The results yield a conceptually simpler and tighter approximation that closes the gap to the MSSC benchmark and advances the understanding of stopping decisions under correlation.

Abstract

The Correlated Pandora's Problem posed by Chawla et al. (2020) generalizes the classical Pandora's Problem by allowing the numbers inside the Pandora's boxes to be correlated. It also generalizes the Min Sum Set Cover problem, and is related to the Uniform Decision Tree problem. This paper gives an optimal 4-approximation for the Correlated Pandora's Problem, matching the lower bound of 4 from Min Sum Set Cover.

Optimal 4-Approximation for the Correlated Pandora's Problem

TL;DR

This work tackles the Correlated Pandora's Problem, where box outcomes are drawn from a general, potentially correlated distribution, by establishing an optimal -approximation. The authors depart from prior combinatorial analyses and develop LP-based relaxations (Unit-Cost CP and General CP) together with a Poisson Rounding scheme and a Balanced Stopping rule to achieve the bound. They show that their -approximation matches the MSSC lower bound, and they provide a cohesive framework that extends from unit-cost/unit-time settings to general costs and volumes. The results yield a conceptually simpler and tighter approximation that closes the gap to the MSSC benchmark and advances the understanding of stopping decisions under correlation.

Abstract

The Correlated Pandora's Problem posed by Chawla et al. (2020) generalizes the classical Pandora's Problem by allowing the numbers inside the Pandora's boxes to be correlated. It also generalizes the Min Sum Set Cover problem, and is related to the Uniform Decision Tree problem. This paper gives an optimal 4-approximation for the Correlated Pandora's Problem, matching the lower bound of 4 from Min Sum Set Cover.

Paper Structure

This paper contains 29 sections, 25 theorems, 152 equations, 7 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Previous Approaches Using Combinatorial Arguments
  3. Our Results and Techniques
  4. Preliminaries
  5. Correlated Pandora's Problem
  6. Partially Adaptive Algorithms
  7. Unit-Cost Case with Clairvoyant Stopping
  8. Linear Program Relaxation
  9. (Discrete-Time) Poisson Rounding
  10. Clairvoyant Stopping
  11. Brief Discussion on Stopping Algorithms
  12. General Case
  13. Linear and Convex Program Relaxations
  14. Poisson Rounding
  15. Balanced Stopping
  16. ...and 14 more sections

Key Result

lemma 1

For any unit-cost instance of Correlated Pandora's Problem, we have $\mathrm{CP}_{\textsc{Unit}} \le \mathrm{OPT}$.

Figures (7)

  • Figure 1: Illustration of combinatorial amortization analysis
  • Figure 2: Timeline of box $i$
  • Figure 3: Illustration of the threshold $t(v)$ and the optimal $z_i(t \,|\, v)$ for scenario $v$ given $x_i(t)$
  • Figure 4: Illustration of the threshold $t(v)$ and the optimal $Z_i(t\,|\,v)$ for scenario $v$ given $X_i(t)$
  • Figure 5: Illustration of events in the Poisson and real time horizons
  • ...and 2 more figures

Theorems & Definitions (39)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • proof
  • theorem 6
  • proof
  • remark 1
  • theorem 7
  • ...and 29 more