Chasing Submodular Objectives, and Submodular Maximization via Cutting Planes
Niv Buchbinder, Joseph, Naor, David Wajc
TL;DR
This paper studies dynamic constrained submodular maximization where both the ground set and objective evolve over time, formalizing the Submodular Objectives Chasing Problem and introducing competitive recourse as a key performance metric. It develops a fractional approach built on the Wolsey extension and multilinear extension, then harnesses a novel Approximate-or-Separate meta-algorithm to implement a cutting-plane-based maximization under separable polytopes, achieving a $(1-1/e- ext{ε})$-approximation with $O( ext{ε}^{-1}\, ext{log}( ext{max}_t |E_t| ext{ε}^{-1}))$-competitive recourse for cardinality and partition matroid constraints. The work further provides an efficient randomized rounding framework that preserves recourse guarantees via negative association, extends to partition matroids, and demonstrates curvature-sensitive refinements and communication-efficient protocols. By connecting submodular chasing to chasing positive bodies and leveraging cutting-plane methods, it yields a versatile meta-framework with broad applicability to static, dynamic, and distributed computation settings. Overall, the results advance understanding of how to maintain high-quality submodular solutions under evolving inputs while controlling movement costs, with implications for dynamic welfare, coverage, and related optimization problems.
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 $(1-1/e-ε)$-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 $(1-1/e-ε)$-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 [Vondrák, PhD~Thesis'07]. The algorithm, whose guarantees are similar to the influential {\em continuous greedy} algorithm [Calinescu-Chekuri-Pál-Vondrák 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 [Grötschel/Lovász/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.