Domination and Multistate Systems
Arne Bang Huseby
TL;DR
This work extends domination theory from binary monotone systems to multistate systems by representing multistate structure functions $\phi_k$ through signed domination functions $\delta_k$ via Möbius inversion. A central result is that the signed domination of an MMS equals the signed domination of an associated binary monotone system $\psi_k$, allowing all binary-domain techniques to apply to MMS. The authors provide explicit Möbius-inversion formulas, discuss coherence and relevance, and connect the theory to matroid and oriented matroid frameworks, yielding closed-form expressions in several canonical cases (e.g., $k$-out-of-$n$, undirected flow networks). Overall, the approach promises computationally efficient algorithms for MMS reliability and deepens the link between algebraic structures (posets, Möbius functions, Hilbert series) and multistate reliability metrics.
Abstract
Domination theory has been studied extensively in the context of binary monotone systems, where the structure function is a sum of products of the component state variables, and with coefficients given by the signed domination function. Using e.g., matroid theory, many useful properties of the signed domination function has been derived. In this paper we show how some of these results can be extended to multistate systems. In particular, we show how the signed domination function can be extended to such systems. Using Möbius inversion we show how the signed domination function can be expressed in terms of a multistate structure function. Moreover, using this expression we show how calculating the signed domination function of a multistate system can be reduced to calculating the signed domination function of an associated binary system. This way many results from binary theory can easily be extended to multistate theory.