闭包系统中极大相容闭集的计数
Enumerating maximal consistent closed sets in closure systems
摘要：
给定有限集上的蕴涵基、闭包系统的一个著名表示和一个不相容二元关系，我们感兴趣的是枚举所有极大相容闭集（简称MCCEnum）的问题.我们表明，MCCEnum不能解决在输出多项式时间，除非$\textsf{P} = \textsf{NP}$，即使是下界格。给出了求解常Carathéodory数闭包系统MCCEnum的增量多项式时间算法。最后，我们证明了在双原子原子闭包系统中，如果最小生成元遵守原子模格中成立的独立条件，则MCCEnum可以在输出准多项式时间内求解。对于在管接头下闭合的闭合系统（即，分布式），MCCEnum先前已通过多项式延迟算法求解。
英文摘要：
Given an implicational base, a well-known representation for a closure system, an inconsistency binary relation over a finite set, we are interested in the problem of enumerating all maximal consistent closed sets (denoted by MCCEnum for short). We show that MCCEnum cannot be solved in output-polynomial time unless $\textsf{P} = \textsf{NP}$, even for lower bounded lattices. We give an incremental-polynomial time algorithm to solve MCCEnum for closure systems with constant Carathéodory number. Finally we prove that in biatomic atomistic closure systems MCCEnum can be solved in output-quasipolynomial time if minimal generators obey an independence condition, which holds in atomistic modular lattices. For closure systems closed under union (i.e., distributive), MCCEnum has been previously solved by a polynomial delay algorithm.
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