On the monotonization of the training set

TL;DR

This paper addresses the challenge of minimally correcting training data to satisfy monotonic constraints, proving NP-hardness and developing an approximate polynomial algorithm for specific cases.

Contribution

It establishes the NP-hardness of the monotonic correction problem and introduces an approximate convex optimization-based algorithm for certain cases.

Findings

NP-hardness of the general problem

Polynomial approximation algorithm for total order cases

Reduction to quadratic convex optimization in dimension 2 cases

Abstract

We consider the problem of minimal correction of the training set to make it consistent with monotonic constraints. This problem arises during analysis of data sets via techniques that require monotone data. We show that this problem is NP-hard in general and is equivalent to finding a maximal independent set in special orgraphs. Practically important cases of that problem considered in detail. These are the cases when a partial order given on the replies set is a total order or has a dimension 2. We show that the second case can be reduced to maximization of a quadratic convex function on a convex set. For this case we construct an approximate polynomial algorithm based on convex optimization.