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