Bicriteria Submodular Maximization

TL;DR

This paper studies bicriteria approximation algorithms for constrained submodular maximization, allowing constraint violations to improve optimization results across various constraints and submodular function classes.

Contribution

It provides a comprehensive analysis of bicriteria algorithms for multiple constraints and submodular classes, achieving optimal or improved results over existing methods.

Findings

Optimal results for many constraint-function combinations

Improved approximation ratios over state-of-the-art methods

Relaxing constraints offers new insights even for feasible solutions

Abstract

Submodular functions and their optimization have found applications in diverse settings ranging from machine learning and data mining to game theory and economics. In this work, we consider the constrained maximization of a submodular function, for which we conduct a principled study of bicriteria approximation algorithms -- algorithms which can violate the constraint, but only up to a bounded factor. Bicrteria optimization allows constrained submodular maximization to capture additional important settings, such as the well-studied submodular cover problem and optimization under soft constraints. We provide results that span both multiple types of constraints (cardinality, knapsack, matroid and convex set) and multiple classes of submodular functions (monotone, symmetric and general). For many of the cases considered, we provide optimal results. In other cases, our results improve over the state-of-the-art, sometimes even over the state-of-the-art for the special case of single-criterion (standard) optimization. Results of the last kind demonstrate that relaxing the feasibility constraint may give a perspective about the problem that is useful even if one only desires feasible solutions.