Optimal Sequential Flows

TL;DR

This paper introduces a new algebraic approach using semigroup factorization to efficiently solve the sequential flow problem in graphs with dynamic capacities, extending to multiple vertices and constraints.

Contribution

It presents a novel algebraic technique based on semigroup factorization for solving the sequential flow problem in polynomial space, generalizing previous methods.

Findings

Provides a polynomial-space algorithm for the sequential flow problem.

Extends the method to multiple in/output vertices and regular constraints.

Introduces a new factorization theorem for finite semigroups.

Abstract

We provide a new algebraic technique to solve the sequential flow problem in polynomial space. The task is to maximise 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 generalises to multiple in/output vertices, as well as regular constraints.