Comparing Apples and Oranges: Query Tradeoff in Submodular Maximization

TL;DR

This paper introduces a new algorithm for monotone submodular maximization under matroid constraints that offers a flexible tradeoff between the number of value oracle queries and matroid independence queries, maintaining optimal approximation guarantees.

Contribution

It presents a novel algorithm establishing a tradeoff between query types, improving efficiency in specific cases, and providing faster solutions for common constraints.

Findings

The algorithm maintains the best approximation guarantee while balancing query costs.

It reduces total oracle queries when the matroid rank is small.

Faster algorithms are provided for cardinality and partition matroid constraints.

Abstract

Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently Badanidiyuru and Vondr\'{a}k (2014) were the first to explicitly look for such algorithms in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The algorithm of Badanidiyuru and Vondr\'{a}k matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm performs. Our main result is a new algorithm for this general case which establishes a surprising tradeoff between two seemingly unrelated quantities: the number of value oracle queries and the number of matroid independence queries performed by the algorithm. Specifically, one can decrease the former by increasing the latter and vice versa, while maintaining the best possible approximation guarantee. Such a tradeoff is very useful since various applications might incur significantly different costs in querying the value and matroid independence oracles. Furthermore, in case the rank of the matroid is $O(n^c)$, where $n$ is the size of the ground set and $c$ is an absolute constant smaller than $1$, the total number of oracle queries our algorithm uses can be made to have a smaller magnitude compared to that needed by Badanidiyuru and Vondr\'{a}k. We also provide even faster algorithms for the well studied special cases of a cardinality constraint and a partition matroid independence constraint, both of which capture many real-world applications and have been widely studied both theorically and in practice.