Approximation Algorithms for Matroidal Prerequisite Systems

2026-07-09

AI Summary

This work addresses optimal decision-making under functional prerequisite constraints by introducing the Matroidal Prerequisite System (MPS)—a novel model jointly defined by a poset and a matroid, where feasible sequences correspond to matroid-independent sets and dependencies can be satisfied via functional substitution. The study establishes, for the first time, an isomorphism between MPS and strongly polyhedral greedy systems, enabling the design of both deterministic and randomized approximation algorithms. For additive objectives, the algorithms achieve approximation ratios of Δ and (1 + λ_max), respectively. For submodular objectives, they yield a deterministic (2 + λ_max)-approximation and a randomized Δ²·(1 − 1/e − δ)⁻¹-approximation. Furthermore, under the Gap-ETH assumption, the paper proves that no algorithm can attain a min{Δ, λ_max}^{o(1)}-approximation.

Abstract

Optimal selections in a decision process are often constrained by prerequisites. However, such prerequisites can encode functional rather than literal dependencies, so a required dependency may be supplied by one or several interacting alternatives. We introduce matroidal prerequisite systems (MPS), a constraint structure where a poset specifies prerequisites while a matroid determines when those prerequisites have been satisfied by its span. This creates an order-sensitive notion of feasibility over words, where feasible words are associated with independent sets, while dependencies may be fulfilled through substitutable functionality. Our main contribution is approximation algorithms for additive maximization and submodular maximization over the feasible words of an MPS. The guarantees are determined by two structural parameters: the maximum matroid rank $Δ$ of a principal ideal in the poset and the maximum matroid connectivity $λ_\mathrm{max}$. These measure the distance an MPS is from encoding a matroid or a poset antimatroid, respectively, both of which are generalized by an MPS. For additive maximization, we obtain efficient deterministic $Δ$- and $(1+λ_\mathrm{max})$-approximation algorithms. By extending these techniques, we obtain efficient deterministic $(2+λ_\mathrm{max})$-approximation and randomized $(Δ^2\cdot(1 - 1/e - δ)^{-1})$-approximation algorithms for all $δ>0$ for submodular maximization. The algorithm design and analysis use the theory of polymatroid greedoids, via cryptomorphism we prove between an MPS and a strong polymatroid greedoid. Finally, an approximation-preserving reduction from densest $k$-subgraph shows it is not possible to efficiently compute a $\min\{Δ,λ_\mathrm{max}\}^{o(1)}$-approximation to additive maximization over the feasible words of an MPS under the Gap Exponential Time Hypothesis.

Problem

• matroidal prerequisite systems
• submodular maximization
• additive maximization
• feasible words
• approximation algorithms

Innovation

• matroidal prerequisite systems
• submodular maximization
• approximation algorithms
• polymatroid greedoids
• structural parameters