Efficient Computations of Interesting Paths
Marc Raphael Vicuna · 2024
Given a high dimensional dataset and a function of interest defined on all points, the Mapper algorithm outputs a topologically accurate summary of any numerical dataset.This summary helps identify subpopulations with interesting properties.These subpopulations typically appear as cycles, disconnected components and flares (i.e., branching paths).The discovery of these subpopulations is unique to the Mapper algorithm.This motivates its use for data mining and dataset identification.The interestingness score of a path is defined as a sum of its edge weights multiplied by a nonlinear function of the edge ranks.Continuing the work on interesting paths by Kalyanaraman, Kamruzzaman and Krishnamoorthy, we consider the graph form of the output of the Mapper algorithm and study three optimization problems to maximize the total interestingness score of the flares: the Max-IP problem, the k-IP problem, and the IP problem.The solution to these problems leads to automatic detection of subpopulations of interest, which greatly facilitates the use of the Mapper algorithm to practitioners.The Max-IP problem is solved in directed acyclic graphs, but we extend the solution to a special class of graphs which is common for the Mapper algorithm.For the k-IP problem, where the number of edges in each path is fixed to k, we show the NP-completeness proof given by Kalyanaraman, Kamruzzaman and Krishnamoorthy has gaps.We give a new NPcompleteness proof of the k-IP problem for k g 4. We design three approximation algorithms, where the best approximation bound is 3 k+1+ϵ , where ϵ > 0. We also design three exact algorithms with varying assumptions, namely: no assumptions, limited graph diameter and constant output size.For the IP problem, where the number of edges in each path is not fixed, we give a proof of the NP-completeness of the IP problem.We design exact algorithms and 9/20-approximate algorithms, using various heuristics.iii Topology is the science of fundamental pattern and structural relationships of event constellations.-R. Buckminster