Subgraph Domatic Problem and Writing Capacity of Memory Devises with Restricted State Transitions

TL;DR

This paper formulates the subgraph domatic partition problem for memory devices with restricted state transitions, proves its NP-completeness, and analyzes the asymptotic behavior of the writing capacity for dense graphs using probabilistic methods.

Contribution

It introduces the subDP problem for memory coding, proves its NP-completeness, and derives the asymptotic behavior of writing capacity for dense graphs using the Lovasz local lemma.

Findings

SubDP problem is NP-complete.

Asymptotic writing capacity C(G) = Omega(n/ln n) for dense bidirectional graphs.

Probabilistic methods are used to analyze graph properties.

Abstract

A code design problem for memory devises with restricted state transitions is formulated as a combinatorial optimization problem that is called a subgraph domatic partition (subDP) problem. If any neighbor set of a given state transition graph contains all the colors, then the coloring is said to be valid. The goal of a subDP problem is to find a valid coloring with the largest number of colors for a subgraph of a given directed graph. The number of colors in an optimal valid coloring gives the writing capacity of a given state transition graph. The subDP problems are computationally hard; it is proved to be NP-complete in this paper. One of our main contributions in this paper is to show the asymptotic behavior of the writing capacity $C(G)$ for sequences of dense bidirectional graphs, that is given by C(G)=Omega(n/ln n) where n is the number of nodes. A probabilistic method called Lovasz local lemma (LLL) plays an essential role to derive the asymptotic expression.