combinatorial-optimization-using-comparison-oracles/theorem-boolean-linear-optimization.md

# Research Challenge: Combinatorial linear optimization with comparison queries

> **Source paper:** Combinatorial Optimization using Comparison Oracles  
> **Reference:** [arXiv:2511.15142](https://arxiv.org/abs/2511.15142) — Theorem boolean.linear.optimization  
> **Task:** Prove the theorem stated below. This document is self-contained: all definitions and notation needed to understand the statement are included. The proof is deliberately omitted.

## Overview

For any family of subsets with unknown element weights, the minimum-weight feasible set can be found using about O(n log^2 n · log|F|) comparison queries between sets, with no knowledge of the numeric weights.

## Definitions and setup

The following definitions and notation (quoted from the source paper) are referenced by the theorem.

### comparison oracle  *(— Abstract)*

> we consider the weaker and more robust comparison oracle, which for any two feasible sets $S, T \in F$, reveals only if $w(S)$ is less than/equal to/greater than $w(T)$.

### \langle w, \mathbbm{1}_S \rangle / w(S) (linear objective value)  *(— Abstract / Introduction)*

> our goal is to find $S^* = \arg\min_{S \in F} \langle w,\mathbbm{1}_S \rangle$. Traditionally, we are either given the weight vector up-front, or else we are given a value oracle which allows us to evaluate $w(S) := \langle w, \mathbbm{1}_S \rangle$ for any $S \in F$.

### \widetilde{O}(\cdot) (tilde-O notation)  *(— Preamble macro definition (used throughout, e.g. Theorem BLO))*

> \newcommand{\tO}{\widetilde{O}}


## The challenge

**Hypotheses / assumptions:**

- $F \subseteq 2^U$
- unknown $w^*: U \to \RR$
- $n = |U|$

**Statement to prove (Theorem boolean.linear.optimization):**

> For any family $F \sse 2^U$ and unknown weight function $w^*: U \rightarrow \RR$, we can solve $\arg\min_{S\in \cF} \sum_{e \in S} w^*_e$ using $O(n \log^2 n \cdot \log |F|) = \widetilde{O}(n^2)$ comparison queries, where $n = |U|$.

*Free variables in the statement:* F, U, w^*, n

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**Your task:** Prove the statement above, using only the definitions and assumptions provided. State any additional standard background results you invoke.