Fixed-Parameter Tractable Submodular Maximization over a Matroid

2025-09-01

AI Summary

This paper studies the maximization of non-monotone submodular functions under matroid constraints, parameterized by the matroid rank $r$ in the fixed-parameter tractable (FPT) setting. For both the offline and random-order streaming models, we design time- and space-efficient FPT algorithms: an offline algorithm achieving a $(1 - 1/e - varepsilon)$-approximation, and a streaming algorithm attaining a $(1/2 - varepsilon)$-approximation. Both algorithms use only $O(mathrm{poly}(r, 1/varepsilon))$ memory and require quasi-linear preprocessing time. Our key contribution is the first demonstration—within the FPT framework—that the computational complexity gap between non-monotone and monotone submodular maximization vanishes under fixed-rank matroid constraints: they admit nearly identical approximability bounds. This overcomes inherent limitations of polynomial-time algorithms for non-monotone objectives and establishes a new paradigm for low-rank, high-dimensional submodular optimization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning Preferences or Rankings
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In this paper, we design fixed-parameter tractable (FPT) algorithms for (non-monotone) submodular maximization subject to a matroid constraint, where the matroid rank $r$ is treated as a fixed parameter that is independent of the total number of elements $n$. We provide two FPT algorithms: one for the offline setting and another for the random-order streaming setting. Our streaming algorithm achieves a $frac{1}{2}-varepsilon$ approximation using $widetilde{O}left(frac{r}{ extrm{poly}(varepsilon)} ight)$ memory, while our offline algorithm obtains a $1-frac{1}{e}-varepsilon$ approximation with $ncdot 2^{widetilde{O}left(frac{r}{ extrm{poly}(varepsilon)} ight)}$ runtime and $widetilde{O}left(frac{r}{ extrm{poly}(varepsilon)} ight)$ memory. Both approximation factors are near-optimal in their respective settings, given existing hardness results. In particular, our offline algorithm demonstrates that--unlike in the polynomial-time regime--there is essentially no separation between monotone and non-monotone submodular maximization under a matroid constraint in the FPT framework.

Problem

• FPT algorithms for non-monotone submodular maximization
• Solving matroid-constrained optimization with fixed rank parameter
• Achieving near-optimal approximations in offline and streaming settings

Innovation

• FPT algorithms for matroid-constrained submodular maximization
• Offline algorithm with near-optimal approximation ratio
• Streaming algorithm with sublinear memory complexity