Biased infinity Laplacian Boundary Problem on finite graphs

TL;DR

This paper introduces a polynomial-time algorithm for solving the biased infinity Laplacian Boundary Problem on finite graphs, extending previous methods to include bias through a new slope concept.

Contribution

It presents the first efficient algorithm for the biased case, adapting the unbiased approach with a novel biased slope notion for paths in graphs.

Findings

Algorithm runs in polynomial time

Enables numerical approximation of biased infinity Laplacian PDE solutions

Extends previous unbiased algorithms to biased scenarios

Abstract

We provide an algorithm, running in polynomial time in the number of vertices, computing the unique solution to the biased infinity Laplacian Boundary Problem on finite graphs. The algorithm is based on the general outline and approach taken in the corresponding algorithm for the unbiased case provided by Lazarus et al. The new ingredient is an adjusted (biased) notion of a slope of a function on a path in a graph. The algorithm can be used to determine efficiently numerical approximations to the viscosity solutions of biased infinity Laplacian PDEs.