A Practical Algorithm for Max-Norm Optimal Binary Labeling of Graphs

Abstract

This paper concerns the efficient implementation of a method for optimal binary labeling of graph vertices, originally proposed by Malmberg and Ciesielski (2020). This method finds, in quadratic time with respect to graph size, a labeling that globally minimizes an objective function based on the $L_\infty$-norm. The method enables global optimization for a novel class of optimization problems, with high relevance in application areas such as image processing and computer vision. In the original formulation, the Malmberg-Ciesielski algorithm is unfortunately very computationally expensive, limiting its utility in practical applications. Here, we present a modified version of the algorithm that exploits redundancies in the original method to reduce computation time. While our proposed method has the same theoretical asymptotic time complexity, we demonstrate that is substantially more efficient in practice. Even for small problems, we observe a speedup of 4–5 orders of magnitude. This reduction in computation time makes the Malmberg-Ciesielski method a viable option for many practical applications.