Research Challenge: Combinatorial linear optimization with comparison queries
Source paper: Combinatorial Optimization using Comparison Oracles
Reference: arXiv: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
Your task: Prove the statement above, using only the definitions and assumptions provided. State any additional standard background results you invoke.