拟阵先决条件系统的近似算法
Approximation Algorithms for Matroidal Prerequisite Systems
AI总结：
研究拟阵先决条件系统中最优选择受先决条件约束的问题，通过引入MPS，给出针对其可行词加法和次模最大化的近似算法，算法保证由结构参数决定，还证明特定近似计算的不可行性。
摘要：
决策过程中的最优选择常受先决条件约束，而这些先决条件可能编码功能依赖而非文字依赖。本文引入拟阵先决条件系统（MPS），其由偏序集指定先决条件，拟阵确定先决条件何时通过其跨度得到满足。主要贡献是针对MPS可行词的加法最大化和次模最大化的近似算法，保证由两个结构参数决定。还通过扩展技术得到次模最大化的相关近似算法，且证明在特定假设下无法高效计算某近似值。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
