Abstract

Combinatorial optimisation (CO) is a sub-area of discrete mathematics. Basic examples for CO problems are finding a shortest path or a minimum spanning tree in a graph. So-called network flows or variations of matching would be more advanced problems. There are also abstract concepts like matroids that offer an algebraic point of view and a uniform foundation for some of the more concrete problems.

Since the considered structures are finite, it is a natural aim to compute a solution efficiently. That implies an overlap with the theory of algorithms, especially running time analysis.

This talk is mainly about the Isabelle/HOL formalisation of a specific CO problem, namely, minimum cost flows, which are a subtype of network flows. Among others, this includes Orlin’s Algorithm, which is a most efficient method to compute a minimum cost flow in general networks. Also, the running time argument for this advanced algorithm and some reductions among flow problems were formalised.

- The Isabelle proof scripts can be found in this GitHub repo: https://github.com/mabdula/Isabelle-Graph-Library

- The formalisation is described in this paper: A Formal Analysis of Capacity Scaling Algorithms for Minimum Cost Flows by Mohammad Abdulaziz and Thomas Ammer, ITP 2024

=== Hybrid talk ===

Join Zoom Meeting https://cam-ac-uk.zoom.us/j/87143365195?pwd=SELTNkOcfVrIE1IppYCsbooOVqenzI.1

Meeting ID: 871 4336 5195

Passcode: 541180