A Note on the Boltzmann Distribution and the Linear Ordering Problem

Abstract

The Boltzmann distribution plays a key role in the field of optimization as it directly connects this field with that of probability. Basically, given a function to optimize, the Boltzmann distribution associated to this function assigns higher probability to the candidate solutions with better quality. Therefore, an efficient sampling of the Boltzmann distribution would turn optimization into an easy task. However, inference tasks on this distribution imply performing operations over an exponential number of terms, which hinders its applicability. As a result, the scientific community has investigated how the structure of objective functions is translated to probabilistic properties in order to simplify the corresponding Boltzmann distribution. In this paper, we elaborate on the properties induced in the Boltzmann distribution associated to permutation-based combinatorial optimization problems. Particularly, we prove that certain characteristics of the linear ordering problem are translated as conditional independence relations to the Boltzmann distribution in the form of $L-decomposability$.