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Commitment Gap via Correlation Gap

Shuchi Chawla, Dimitris Christou, Trung Dang

TL;DR

This work studies Costly Information Combinatorial Selection (CICS), where a decision maker selects a feasible subset of stochastic items while learning values through costly probes. It develops a universal reduction to Bayesian Combinatorial Selection (BCS) with free information, tying the CICS commitment gap to ex ante prophet-inequality benchmarks via the correlation gap and frugality concepts. The authors show that for many standard feasibility constraints, committing policies yield near-optimal performance and can be computed efficiently by first solving the ex ante relaxation, constructing surrogate values, and then applying one-sided semi-online algorithms. They also connect CICS to query-commit models for matching, obtaining near-best-known competitive ratios and providing reductions that hold under general MDPs, not just acyclic ones. These results yield efficient approximation schemes for CICS across matroid, knapsack, k-system, and matching constraints, with practical implications for resource-aware stochastic optimization.

Abstract

Selection problems with costly information, dating back to Weitzman's Pandora's Box problem, have received much attention recently. We study the general model of Costly Information Combinatorial Selection (CICS) that was recently introduced by Chawla et al. [2024] and Bowers et al. [2025]. In this problem, a decision maker needs to select a feasible subset of stochastic variables, and can only learn information about their values through a series of costly steps, modeled by a Markov decision process. The algorithmic objective is to maximize the total value of the selection minus the cost of information acquisition. However, determining the optimal algorithm is known to be a computationally challenging problem. To address this challenge, previous approaches have turned to approximation algorithms by considering a restricted class of committing policies that simplify the decision-making aspects of the problem and allow for efficient optimization. This motivates the question of bounding the commitment gap, measuring the worst case ratio in the performance of the optimal committing policy and the overall optimal. In this work, we obtain improved bounds on the commitment gap of CICS through a reduction to a simpler problem of Bayesian Combinatorial Selection where information is free. By establishing a close relationship between these problems, we are able to relate the commitment gap of CICS to ex ante free-order prophet inequalities. As a consequence, we obtain improved efficient approximations for arbitrary instances of the CICS under various feasibility constraints.

Commitment Gap via Correlation Gap

TL;DR

This work studies Costly Information Combinatorial Selection (CICS), where a decision maker selects a feasible subset of stochastic items while learning values through costly probes. It develops a universal reduction to Bayesian Combinatorial Selection (BCS) with free information, tying the CICS commitment gap to ex ante prophet-inequality benchmarks via the correlation gap and frugality concepts. The authors show that for many standard feasibility constraints, committing policies yield near-optimal performance and can be computed efficiently by first solving the ex ante relaxation, constructing surrogate values, and then applying one-sided semi-online algorithms. They also connect CICS to query-commit models for matching, obtaining near-best-known competitive ratios and providing reductions that hold under general MDPs, not just acyclic ones. These results yield efficient approximation schemes for CICS across matroid, knapsack, k-system, and matching constraints, with practical implications for resource-aware stochastic optimization.

Abstract

Selection problems with costly information, dating back to Weitzman's Pandora's Box problem, have received much attention recently. We study the general model of Costly Information Combinatorial Selection (CICS) that was recently introduced by Chawla et al. [2024] and Bowers et al. [2025]. In this problem, a decision maker needs to select a feasible subset of stochastic variables, and can only learn information about their values through a series of costly steps, modeled by a Markov decision process. The algorithmic objective is to maximize the total value of the selection minus the cost of information acquisition. However, determining the optimal algorithm is known to be a computationally challenging problem. To address this challenge, previous approaches have turned to approximation algorithms by considering a restricted class of committing policies that simplify the decision-making aspects of the problem and allow for efficient optimization. This motivates the question of bounding the commitment gap, measuring the worst case ratio in the performance of the optimal committing policy and the overall optimal. In this work, we obtain improved bounds on the commitment gap of CICS through a reduction to a simpler problem of Bayesian Combinatorial Selection where information is free. By establishing a close relationship between these problems, we are able to relate the commitment gap of CICS to ex ante free-order prophet inequalities. As a consequence, we obtain improved efficient approximations for arbitrary instances of the CICS under various feasibility constraints.

Paper Structure

This paper contains 26 sections, 20 theorems, 45 equations, 2 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. References for \ref{['table:res']}
  3. Related Work
  4. Organization
  5. Notation
  6. Combinatorial Selection with Costly Information
  7. Bayesian Combinatorial Selection
  8. Online Selection Algorithms.
  9. Online Selection Algorithms and Prophet Inequalities.
  10. One-Sided Semi-Online and Frugal Algorithms.
  11. Relating CICS and BCS
  12. Computing an Ex Ante Optimal Policy for CICS
  13. Comparison to Previous Work.
  14. Computing Optimal Solutions for Local Games
  15. One-Sided Semi-Online Matching Is Query-Commit Matching
  16. ...and 11 more sections

Key Result

Lemma 3.1

Let $\mathbbm{I} = (\mathcal{F},\mathbb{M})$ be any CICS instance and assume that $\mathcal{F}$ admits an efficient separation oracle. Then, we can efficiently compute a commitment $\mathcal{P}\in\mathcal{C}(\mathbbm{I})$ such that

Figures (2)

  • Figure 1: Connections between different benchmarks for Bayesian Combinatorial Selection.
  • Figure 2: A min-CICS instance for which the commitment gap approaches zero as $N\rightarrow\infty$. Black arrows denote costly actions (although all actions have cost $0$ in this example) and dotted lines denote random transitions.

Theorems & Definitions (39)

  • Definition 1: Commitment Gap
  • Lemma 3.1
  • Definition 2: Surrogate Values
  • Definition 3: Promise of Payment
  • Lemma 3.2
  • Definition 4: BCS and Benchmarks
  • Definition 5: Correlation Gap
  • Lemma 4.1
  • Definition 6: Free-Order Online Algorithms
  • Definition 7: One-Sided Semi-Online Algorithms
  • ...and 29 more