Divisors on graphs, orientations, syzygies, and system reliability

TL;DR

This paper explores the algebraic and combinatorial structures underlying system reliability, using graph divisors, orientations, and matroids to explicitly describe minimal free resolutions and compute system reliability.

Contribution

It introduces a novel combinatorial approach to describe minimal free resolutions of ideals in system reliability using graph theory and algebraic geometry.

Findings

Explicit combinatorial descriptions of syzygy modules.

Betti numbers are characteristic-independent.

Applications to computing system reliability.

Abstract

We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations and matroids on graphs to describe the minimal polyhedral cellular free resolutions of these ideals. In each case we give an explicit combinatorial description of the minimal generating set for each higher syzygy module in terms of the acyclic orientations of the graph, the $q$-reduced divisors and the bounded regions of the graphic hyperplane arrangement. The resolutions of all these ideals are closely related, and their Betti numbers are independent of the characteristic of the base field. We apply these results to compute the reliability of their associated coherent systems.