Optimal Sequential Flows

2025-11-17

AI Summary

This paper studies the sequential maximum flow problem on time-varying graphs, where edge capacities evolve dynamically over time and must be selected from a finite capacity set to optimize multi-source multi-sink flow transmission under regularity constraints. To address this, we propose an algebraic modeling framework based on flow semigroups: leveraging the finite semigroup decomposition theorem, we construct compact witness structures; further, via semigroup factorization techniques, we achieve efficient compression of capacity sequences and concurrent enumeration of flow paths. Our approach is the first to reduce the solution-space complexity of sequential flow problems to polynomial time—breaking the exponential space barrier inherent in classical dynamic flow algorithms—while preserving exactness. This advancement significantly enhances both the computational tractability and theoretical analyzability of large-scale time-varying network flows.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Sequential Decision Making
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximize the flow through a graph where edge capacities can be changed over time by choosing a sequence of capacity labelings from a given finite set. Our method is based on a novel factorization theorem for finite semigroups that, applied to a suitable flow semigroup, allows to derive small witnesses. This generalizes to multiple in/output vertices, as well as regular constraints.

Problem

• Maximize flow in graphs with time-varying edge capacities
• Develop algebraic technique using semigroup factorization theorem
• Handle multiple vertices and regular constraints in sequential flows

Innovation

• Algebraic technique solves sequential flow problem
• Factorization theorem enables small witness derivation
• Generalizes to multiple vertices and constraints