Constrained Cuts, Flows, and Lattice-Linearity

2025-12-19

AI Summary

This paper studies the minimum cut problem in directed graphs under additional constraints. While the set of all minimum cuts forms a distributive lattice, imposing constraints typically renders the problem NP-hard. To address this, we model constraints as lattice-linear predicates and—novelty—integrate them with max-flow preprocessing, introducing *k-transfer predicates* and a *strong push mechanism*. We design a parallel polynomial-time algorithm to efficiently compute sublattice-irreducible elements satisfying regular constraints; provide a succinct representation and enumeration scheme for the feasible sublattice; and, for non-lattice-linear constraints, propose an exact algorithm based on *poset slicing* and *predicate propagation*, outperforming brute-force enumeration. Our main contributions are: (i) establishing a unified lattice-theoretic framework for constrained minimum cuts; (ii) enabling efficient parallel computation; (iii) achieving succinct representation of solution sublattices; and (iv) supporting scalable, structured enumeration of feasible cuts.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Parallel algorithms for min-cuts with lattice-linear constraints
• Enumerating sublattices of min-cuts using Birkhoff's theorem
• Exact algorithms for NP-hard constrained min-cut problems

Innovation

• Parallel min-cut algorithms using lattice-linear predicates
• Sublattice enumeration via Birkhoff's theorem representation
• Poset slicing for exact NP-hard constrained min-cuts