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

TL;DR

The paper addresses constrained, not-necessarily monotone submodular maximization under size and matroid constraints in the value oracle model, where standard combinatorial methods are stuck at a $$1/e$$-approximation. It introduces a novel framework that guides a fast local search-based signal into a randomized greedy procedure, achieving a $$0.385-\\varepsilon$$-approximation for size and $$0.305-\\varepsilon$$ for general matroids with $$\mathcal{O}(kn/\\varepsilon)$$ queries; it also provides deterministic variants with the same ratios and a nearly linear-time deterministic algorithm achieving $$0.377-\\varepsilon$$. The approach hinges on a Fast Local Search (FastLS) to produce a guiding set and a Guided RandomGreedy (GuidedRG) that uses this guidance to improve the recurrences governing progress toward the optimum. Empirically, the methods outperform classical Greedy and RandomGreedy baselines on video summarization and Maximum Cut problems, illustrating practical gains despite higher per-instance query costs, and establishing a path toward more scalable combinatorial algorithms for non-monotone submodular maximization.

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 $0.377$.

Figures (7)

• Figure 1: (a): The evolution of $\mathbb{E} \left[ f \left( O\cup A_i \right) \right]$ and $\mathbb{E} \left[ f \left( A_i \right) \right]$ in the worst case of the analysis of RandomGreedy, as the partial solution size increases to $k$. (b): Illustration of how the degradation of $\mathbb{E} \left[ f \left( O\cup A_i \right) \right]$ changes as we introduce an $(0.385+ \varepsilon , 0.385)$-guidance set. (c): The updated degradation with a switch point $tk$, where the algorithm starts with guidance and then switches to running without guidance. The dashed curved lines depict the unguided values from (a).
• Figure 2: Depiction of how analysis of InterpolatedGreedy changes if there is no $(0.377, 0.46)$-guidance set.
• Figure 3: The objective value (higher is better) and the number of queries (log scale, lower is better) are normalized by those of StandardGreedy. Our algorithm (blue star) outperforms every baseline on at least one of these two metrics.
• Figure 4: This set of figures indicates how guiding benefits RandomGreedy under matroid constraints. The figure (a) depicts the evolution of $f \left( O\cup A_i \right)$ and $f \left( A_i \right)$ with RandomGreedy. The figure (b) illustrates how the degradation of $f \left( O\cup A_i \right)$ changes as we introduce an $(0.305+ \varepsilon , 0.305)$-guidance set. Additionally, we also need to consider the degradation of $f \left( (O\setminus Z)\cup A_i \right)$, which is the value that the solution approaches with the guidance. The figure (c) shows the updated degradation with a switch point $tk$, where the algorithm starts with guidance and then switches to running without guidance. It demonstrates that even though the value of $A_i$ decreases initially when the selection starts outside of $Z$, it benefits from the improved degradation of $f \left( O\cup A_i \right)$ upon switching back to the original algorithm.
• Figure 5: Frames selected for Video Summarization

Theorems & Definitions (52)

• theorem 2.1
• definition 2.1
• lemma 2.1
• lemma 2.1
• proof : Proof of Theorem \ref{['thm:irg']} under size constraint
• theorem 2.2
• theorem 3.1
• definition A.1
• lemma A.1
• proof