# The minimum cost query problem on matroids with uncertainty areas

## TL;DR

This paper addresses the problem of identifying a minimal set of element queries in a matroid with uncertain weights to guarantee a uniformly optimal basis, providing polynomial algorithms and characterizations.

## Contribution

It introduces a polynomial-time algorithm for selecting minimal query sets in uncertain matroids to ensure a uniformly optimal basis, along with combinatorial characterizations.

## Key findings

- Polynomial-time algorithm for query set selection
- Characterizations of uniformly optimal bases
- Optimal query sets guarantee basis optimality under uncertainty

## Abstract

We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements. Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base.