Discretely Beyond $1/e$: Guided Combinatorial Algorithms for Submodular Maximization

2024-05-08

AI Summary

This paper addresses the discrete maximization of non-monotone submodular functions under cardinality and matroid constraints, breaking the long-standing $1/e approx 0.367$ approximation barrier for combinatorial algorithms. We propose the **guided randomized greedy framework**, integrating fast local search while avoiding costly continuous extensions. We further design **deterministic and nearly-linear-time variants** that preserve the approximation guarantees. Under cardinality constraints, our algorithm achieves a $0.385$ approximation ratio—improving upon the previous best $0.367$; under matroid constraints, it attains $0.305$, surpassing $0.281$. The deterministic variant achieves $0.377$ with nearly-linear time complexity. To our knowledge, this is the first purely combinatorial algorithm—requiring no continuous optimization—that strictly exceeds the $1/e$ barrier, significantly enhancing scalability and practical applicability.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than $1/e$ require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: $1/e approx 0.367$ for size constraint and $0.281$ for the matroid constraint in $mathcal O (kn)$ queries, where $k$ is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the $1/e$ barrier: we obtain approximation ratio of $0.385$ in $mathcal O (kn)$ queries to the submodular set function for size constraint, and $0.305$ for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio $0.377$.

Problem

• Submodular maximization algorithms
• Breaking 1/e approximation barrier
• Combinatorial vs continuous methods

Innovation

• Guided randomized greedy algorithm
• Fast local search integration
• Deterministic linear time algorithm