What Data Enables Optimal Decisions? An Exact Characterization for Linear Optimization

TL;DR

This paper provides a geometric characterization of when a dataset is sufficient to determine optimal decisions in linear programming, enabling efficient data selection for decision-making tasks.

Contribution

It introduces a sharp geometric criterion for dataset sufficiency in linear optimization and develops an algorithm to construct minimal, task-specific datasets.

Findings

Small, well-chosen datasets can fully determine optimal decisions

The geometric characterization identifies key directions of the cost vector

Practical algorithm for constructing minimal sufficient datasets

Abstract

We study the fundamental question of how informative a dataset is for solving a given decision-making task. In our setting, the dataset provides partial information about unknown parameters that influence task outcomes. Focusing on linear programs, we characterize when a dataset is sufficient to recover an optimal decision, given an uncertainty set on the cost vector. Our main contribution is a sharp geometric characterization that identifies the directions of the cost vector that matter for optimality, relative to the task constraints and uncertainty set. We further develop a practical algorithm that, for a given task, constructs a minimal or least-costly sufficient dataset. Our results reveal that small, well-chosen datasets can often fully determine optimal decisions -- offering a principled foundation for task-aware data selection.