Constrained Cuts, Flows, and Lattice-Linearity

TL;DR

This paper develops parallel algorithms for constrained min-cut problems in directed graphs using lattice-linear predicates, providing efficient computation methods, representations, and enumeration techniques, despite some problems being NP-hard.

Contribution

It introduces lattice-linear predicate frameworks for parallel min-cut algorithms, including new methods for enumeration, complexity improvements, and the concept of $k$-transition predicates.

Findings

Parallel algorithms for min-cuts with lattice-linear constraints.

Succinct representations of constrained min-cuts via sublattice irreducibles.

Enhanced complexity bounds using poset slicing and $k$-transition predicates.

Abstract

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.