Finding maximal independent sets in an independence system

Question

An independence system is a collection $I$ of subsets of $\Omega$ such that if $A\in I$, then any subset of $A$ is in $I$. These sets are called independent.

Suppose I have an oracle for testing independence. In particular I have an ambient set $\Omega$ and an independence system $I$ of subsets of $\Omega$. I pass a subset $A$ of $\Omega$ to the oracle, and it tells me if $A\in I$. Is there an efficient algorithm for finding all maximal independent sets? I especially care about minimizing the number of calls to the oracle, which is expensive.

I implemented a brute force method (just looping over the powerset) which chokes for inputs sizes above about $n=6$. On the other hand, the Bron-Kerbosch algorithm solves exactly this problem in a special case, and my implementation of it runs quite happily up to around $n=30$. That would be plenty satisfactory to me.

Answer 1

This looks equivalent to finding all maximal zeroes of a monotone function, given ability to make oracle queries to the function. Let $f:\{0,1\}^n \to \{0,1\}$ be a monotone Boolean function, and call $x$ a maximal zero if $f(x)=0$ and $f(y)=1$ for all $y\ge x$ (i.e., $y_i \ge x_i$ for all $i$).

This problem has been studied extensively. I believe there are no known polynomial time algorithms. See https://cstheory.stackexchange.com/q/14772/5038, Maximal Elements in a Lower Set, https://cstheory.stackexchange.com/q/18047/5038, How to enumerate minimal covers of a set, and https://cstheory.stackexchange.com/q/14772/5038. I think the concepts of learning DNF representations for monotone boolean functions and monotone dualization will be relevant. I confess I don't fully understand the literature and don't know if there are any algorithms that are good-enough-in-practice.

The relationship: Suppose $\Omega=\{1,2,\dots,n\}$. We can identify the set $A$ with its characteristic vector, which is an element of $\{0,1\}^n$. Now an independence system $I$ determines a function $f$ given by $f(A)=0$ if $A \in I$, otherwise $f(A)=1$. You have oracle access to $f$, and want to find all maximal sets $A$ such that $f(A)=0$.