In a capacitated directed graph, it is known that the set of all min-cuts forms a distributive lattice [1], [2]. Here, we describe this lattice as a regular predicate whose forbidden elements can be advanced in constant parallel time after precomputing a max-flow, so as to obtain parallel algorithms for min-cut problems with additional constraints encoded by lattice-linear predicates [3]. Some nice algorithmic applications follow. First, we use these methods to compute the irreducibles of the sublattice of min-cuts satisfying a regular predicate. By Birkhoff's theorem [4] this gives a succinct representation of such cuts, and so we also obtain a general algorithm for enumerating this sublattice. Finally, though we prove computing min-cuts satisfying additional constraints is NP-hard in general, we use poset slicing [5], [6] for exact algorithms with constraints not necessarily encoded by lattice-linear predicates) with better complexity than exhaustive search. We also introduce $k$-transition predicates and strong advancement for improved complexity analyses of lattice-linear predicate algorithms in parallel settings, which is of independent interest.

翻译：在有容量的有向图中，已知所有最小割构成一个分配格[1][2]。本文将该格描述为一个正则谓词，其禁止元素在预计算最大流后可在常数并行时间内推进，从而得到用于解决由格线性谓词编码附加约束的最小割问题的并行算法[3]。由此衍生出若干优美的算法应用。首先，我们利用这些方法计算满足正则谓词的最小割子格的不可约元。根据Birkhoff定理[4]，这为此类割提供了简洁表示，因此我们也获得了枚举该子格的通用算法。最后，尽管我们证明了在一般情况下计算满足附加约束的最小割是NP难的，但通过使用偏序集切片技术[5][6]，我们针对约束（不一定由格线性谓词编码）设计了精确算法，其复杂度优于穷举搜索。我们还引入了$k$-转移谓词与强推进概念，以改进并行环境下格线性谓词算法的复杂度分析，这本身具有独立的研究价值。