Guess Free Maximization of Submodular and Linear Sums

TL;DR

This paper introduces a new, cleaner algorithm for maximizing the sum of a monotone submodular and linear function under polytope constraints, matching the best approximation guarantees without guessing steps.

Contribution

It presents a novel weighting technique that simplifies the algorithm while maintaining optimal approximation guarantees for the problem.

Findings

Achieves the same approximation guarantee as previous algorithms

Removes the need for guessing steps, improving simplicity and efficiency

Provides a more elegant and practical algorithm for submodular and linear sum maximization

Abstract

We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al.