The max-out min-in problem: A tool for data analysis

Consider a graph with vertex set V and non-negative weights on the edges. For every subset of vertices S , define phi(S) to be the sum of the weights of edges with one vertex in S and the other in V backslash S , minus the sum of the weights of the edges with both vertices in S. We consider the problem of finding S subset of C V for which phi(S) is maximized. We call this combinatorial optimization problem the max-out min-in problem (MOMIP). In this paper we (i) present a linear 0/1 formulation and a quadratic unconstrained binary optimization formulation for MOMIP; (ii) prove that the problem is NP-hard; (iii) report results of computational experiments on simulated data to compare the performances of the two models; (iv) illustrate the applicability of MOMIP for two different topics in the context of data analysis, namely in the selection of variables in exploratory data analysis and in the identification of clusters in the context of cluster analysis; and (v) introduce a generalization of MOMIP that includes, as particular cases, the well-known weighted maximum cut problem and a novel problem related to independent dominant sets in graphs.

Automated summary

This paper presents methods for solving the "max-out min-in problem (MOMIP)" and provides a linear 0/1 formulation and a quadratic unconstrained binary optimization formulation for it. The paper proves that the problem is NP-hard and compares the performances of the two models through computational experiments. Additionally, the applicability of MOMIP in data analysis is demonstrated through two different scenarios.