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Pandora's Problem with Combinatorial Cost

Ben Berger, Tomer Ezra, Michal Feldman, Federico Fusco

TL;DR

Pandora's problem is extended to settings with combinatorial costs, exploring how classic Weitzman-style insights translate when costs are not additive. The authors show submodular costs retain a fixed non-adaptive exploration order, preserving structural simplicity, while more general cost classes (subadditive or XOS) may require adaptive strategies. They also establish strong computational hardness: even for submodular costs, approximating the optimal utility demands super-polynomial cost queries, ruling out polynomial-time approximation under standard oracle access. The paper introduces impulsive strategies and a reduction framework to Bernoulli instances, establishing both structural results and hardness by connecting general distributions to Bernoulli reductions. Together, these results delineate the boundary between tractable and intractable Pandora-like search problems under combinatorial costs and point to future directions in extensions and special-cost classes.

Abstract

Pandora's problem is a fundamental model in economics that studies optimal search strategies under costly inspection. In this paper we initiate the study of Pandora's problem with combinatorial costs, capturing many real-life scenarios where search cost is non-additive. Weitzman's celebrated algorithm [1979] establishes the remarkable result that, for additive costs, the optimal search strategy is non-adaptive and computationally feasible. We inquire to which extent this structural and computational simplicity extends beyond additive cost functions. Our main result is that the class of submodular cost functions admits an optimal strategy that follows a fixed, non-adaptive order, thus preserving the structural simplicity of additive cost functions. In contrast, for the more general class of subadditive (or even XOS) cost functions, the optimal strategy may already need to determine the search order adaptively. On the computational side, obtaining any approximation to the optimal utility requires super polynomially many queries to the cost function, even for a strict subclass of submodular cost functions.

Pandora's Problem with Combinatorial Cost

TL;DR

Pandora's problem is extended to settings with combinatorial costs, exploring how classic Weitzman-style insights translate when costs are not additive. The authors show submodular costs retain a fixed non-adaptive exploration order, preserving structural simplicity, while more general cost classes (subadditive or XOS) may require adaptive strategies. They also establish strong computational hardness: even for submodular costs, approximating the optimal utility demands super-polynomial cost queries, ruling out polynomial-time approximation under standard oracle access. The paper introduces impulsive strategies and a reduction framework to Bernoulli instances, establishing both structural results and hardness by connecting general distributions to Bernoulli reductions. Together, these results delineate the boundary between tractable and intractable Pandora-like search problems under combinatorial costs and point to future directions in extensions and special-cost classes.

Abstract

Pandora's problem is a fundamental model in economics that studies optimal search strategies under costly inspection. In this paper we initiate the study of Pandora's problem with combinatorial costs, capturing many real-life scenarios where search cost is non-additive. Weitzman's celebrated algorithm [1979] establishes the remarkable result that, for additive costs, the optimal search strategy is non-adaptive and computationally feasible. We inquire to which extent this structural and computational simplicity extends beyond additive cost functions. Our main result is that the class of submodular cost functions admits an optimal strategy that follows a fixed, non-adaptive order, thus preserving the structural simplicity of additive cost functions. In contrast, for the more general class of subadditive (or even XOS) cost functions, the optimal strategy may already need to determine the search order adaptively. On the computational side, obtaining any approximation to the optimal utility requires super polynomially many queries to the cost function, even for a strict subclass of submodular cost functions.
Paper Structure (32 sections, 17 theorems, 86 equations)

This paper contains 32 sections, 17 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Our Techniques
  4. Related Work
  5. Preliminaries
  6. Combinatorial functions.
  7. Computational setting.
  8. Impulsive Optimal Strategies for Bernoulli Instances
  9. Setup for Lemma \ref{['lem:one_fork_wlog']}
  10. Impulsive Strategies with Dummies.
  11. Proof of Lemma \ref{['lem:one_fork_wlog']}
  12. Reduction to Bernoulli Instances
  13. Transformation 1:
  14. Transformation 2:
  15. Computational Results
  16. ...and 17 more sections

Key Result

Theorem 3.1

For every Bernoulli instance with a submodular cost function there exists an optimal strategy that is impulsive.

Theorems & Definitions (63)

  • Example 1
  • Example 2
  • Theorem 3.1
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['thm:MTT_Submodular']}
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 53 more