Chasing Submodular Objectives, and Submodular Maximization via Cutting Planes

TL;DR

This paper introduces a new framework for dynamically maintaining high-quality solutions to evolving submodular maximization problems with changing constraints, using cutting plane methods and a novel meta-algorithm.

Contribution

It presents the submodular objectives chasing problem, along with polynomial-time algorithms achieving near-optimal approximation and recourse, and introduces a new meta-algorithm for multilinear extension maximization.

Findings

Achieves optimal approximation and recourse for cardinality and matroid constraints.

Develops a new meta-algorithm called approximate-or-separate for multilinear extension maximization.

Extends cutting plane methods to constrained submodular maximization.

Abstract

We introduce the \emph{submodular objectives chasing problem}, which generalizes many natural and previously-studied problems: a sequence of constrained submodular maximization problems is revealed over time, with both the objective and available ground set changing at each step. The goal is to maintain solutions of high approximation and low total \emph{recourse} (number of changes), compared with exact offline algorithms for the same input sequence. For the central cardinality constraint and partition matroid constraints we provide polynomial-time algorithms achieving both optimal $(1-1/e-\epsilon)$-approximation and optimal competitive recourse for \emph{any} constant-approximation. Key to our algorithm's polynomial time, and of possible independent interest, is a new meta-algorithm for $(1-1/e-\epsilon)$-approximately maximizing the multilinear extension under general constraints, which we call {\em approximate-or-separate}. Our algorithm relies on an improvement of the round-and-separate method [Gupta-Levin SODA'20], inspired by an earlier proof by [Vondr\'ak, PhD~Thesis'07]. The algorithm, whose guarantees are similar to the influential {\em continuous greedy} algorithm [Calinescu-Chekuri-P\'al-Vondr\'ak SICOMP'11], can use any cutting plane method and separation oracle for the constraints. This allows us to introduce cutting plane methods, used for exact unconstrained submodular minimization since the '80s [Gr\"otschel/Lov\'asz/Schrijver Combinatorica'81], as a useful method for (optimal approximate) constrained submodular maximization. We show further applications of this approach to static algorithms with curvature-sensitive approximation, and to communication complexity protocols.