Breaking Barriers: Combinatorial Algorithms for Non-monotone Submodular Maximization with Sublinear Adaptivity and $1/e$ Approximation

TL;DR

This work tackles non-monotone submodular maximization under a size constraint, aiming for parallelizable, combinatorial algorithms with sublinear adaptivity while preserving strong approximation. It introduces a blended marginal gains framework to replace branching in interlaced greedy methods, enabling a first combinatorial parallel algorithm achieving $1/e-
$ approximation with $O( ) obreak obreak ightarrow$ adaptivity $O(ig) obreak $ and $O(n\u00ad ightarrow ightarrow ig) $ queries, and a simpler $(1/4-b5)$-approximation with high probability. Building on this, the paper presents sublinear-adaptivity parallel algorithms, PIG and PItG, that combine threshold sampling with interlaced greedy to achieve $O(ig) obreak$ adaptivity and $O(n n )$ query complexity while matching the state-of-the-art $1/e$-type guarantees. The empirical results on various datasets (e.g., max-cut and revmax) demonstrate competitive objective values and favorable query/adaptivity trade-offs, highlighting practical scalability for large-scale submodular optimization tasks.

Abstract

With the rapid growth of data in modern applications, parallel algorithms for maximizing non-monotone submodular functions have gained significant attention. In the parallel computation setting, the state-of-the-art approximation ratio of $1/e$ is achieved by a continuous algorithm (Ene & Nguyen, 2020) with adaptivity $ O\left(\log(n)\right)$. In this work, we focus on size constraints and present the first combinatorial algorithm matching this bound -- a randomized parallel approach achieving $1/e-\varepsilon$ approximation ratio. This result bridges the gap between continuous and combinatorial approaches for this problem. As a byproduct, we also develop a simpler $(1/4-\varepsilon)$-approximation algorithm with high probability ($\ge 1-1/n$). Both algorithms achieve $ O\left(\log(n)\log(k)\right)$ adaptivity and $O\left(n\log(n)\log(k)\right)$ query complexity. Empirical results show our algorithms achieve competitive objective values, with the $(1/4-\varepsilon)$-approximation algorithm particularly efficient in queries.

Figures (5)

• Figure 1: This figure illustrates strategies employed by each algorithm. The three leftmost algorithms achieve an asymptotic approximation ratio of $1/4$, while the three rightmost algorithms attain an asymptotic ratio of $1/e$.
• Figure 2: Results for $\texttt{maxcut}$ on er with $n=99,997$, and $\texttt{revmax}$ on twitch-gamers with $n=168,114$.
• Figure 3: This figure depicts the components of solution sets $A$ and $B$ in Alg. \ref{['alg:gdone']}. The black rectangle highlights a sequence of consecutive elements from $O$ that were added to the solution at the initial. Red circles with a cross mark signifies the first element in $A$ or $B$ that is outside $O$.
• Figure 4: This figure depicts the components of the solution sets $A_{l_1}$ and $A_{l_2}$. A blue circle with a check mark represents an element in $O$, while a red circle with a cross mark represents an element outside of $O$. The grey rectangles indicate a sequence of consecutive elements in $O$. The pink rectangles indicate the corresponding elements used to bound $\Delta \left( O_{l_2} \, \middle| \, A_{l_1} \right)$ or $\Delta \left( O_{l_1} \, \middle| \, A_{l_2} \right)$. It is illustrated that $\Delta \left( O_{l_1} \, \middle| \, A_{l_2} \right) + \Delta \left( O_{l_2} \, \middle| \, A_{l_1} \right) \le \Delta \left( A_{l_1} \, \middle| \, G_{i-1} \right) + \Delta \left( A_{l_2} \, \middle| \, G_{i-1} \right)$ under both cases.
• Figure 5: Results for $\texttt{revmax}$ on musae-github with $n=37,700$, and $\texttt{maxcut}$ on web-Google with $n=875,713$.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 3.1
• claim 3.1
• lemma 3.1
• Proposition 3.2: Blended Marginal Gains
• Theorem 4.1
• Theorem 4.2
• lemma 4.3
• lemma 4.4
• lemma 4.5