A Formal Analysis of Capacity Scaling Algorithms for Minimum Cost Flows

2026-02-03

International Conference on Interactive Theorem Proving

AI Summary

This work addresses the lack of formal verification for minimum-cost flow algorithms by presenting the first complete formalization of Orlin’s capacity scaling algorithm in Isabelle/HOL. Employing a stepwise refinement methodology, the authors derive executable code from an abstract specification and formally verify the reduction from the general problem to a restricted setting. The development not only rigorously establishes the algorithm’s correctness and worst-case time complexity but also refines existing theorem statements and contributes a reusable library of formalized graph algorithms. This constitutes the first end-to-end formally verified solution to the minimum-cost flow problem.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present formalisations of the correctness of executable algorithms to solve minimum-cost flow problems in Isabelle/HOL. Two of the algorithms are based on the technique of scaling, most notably Orlin's algorithm, which has the fastest known running time for solving the problem of minimum-cost flow. We also include a formalisation of the worst-case running time argument for Orlin's algorithm. Our verified implementation of this algorithm, which is derived by the technique of stepwise refinement, is fully executable and was integrated into a reusable formal library on graph algorithms. Because the problems for which Orlin's algorithm works are restricted, we also verified an executable reduction from the general minimum-cost flow problem. We believe we are the first to formally consider the problem of minimum-cost flows and, more generally, any scaling algorithms. Our work has also led to a number of mathematical insights and improvements to proofs as well as theorem statements, compared to all existing expositions.

Problem

• minimum-cost flow
• scaling algorithms
• formal verification
• Orlin's algorithm
• capacity scaling

Innovation

• formal verification
• minimum-cost flow
• scaling algorithms
• Orlin's algorithm
• Isabelle/HOL