### 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
Keywords: machine learning, supervised learning, monotonic constraints
### introduction ###
Requirements to a classifying rule in supervised learning problems consist of two parts
The first part is induced by a set of precedents, called the training set
Each element in the training set is a pair of "object--reply" type
A classifying rule which is a mapping from objects set to the replies set should map objects from the training set pairs to the consistent replies
And the second part of requirements express our common knowledge of a classifying rule
One of the popular types of such requirements is the monotonicity which is considered in that paper
In some cases these two parts of requirements can not be satisfied both and then we have a problem of a minimal correction of the training set
Let us see what that problem is
Suppose the sets  SYMBOL  are given and on this sets we have partial orders  SYMBOL  consistently
We assume more that the partial order  SYMBOL  is a lattice
For any given mapping  SYMBOL  where  SYMBOL  we pose a problem of finding a function  SYMBOL  which is monotone due to partial orders  SYMBOL  and minimizes the following functional:  SYMBOL
Let us denote the set of monotonic functions from  SYMBOL  to  SYMBOL  by  SYMBOL
Then for a given mapping  SYMBOL  our task is the following:  SYMBOL   Every mapping  SYMBOL  which is monotone on the subset  SYMBOL  can be extended to the mapping monotone on the whole set  SYMBOL  because  SYMBOL  is a lattice
Actually on every finite subset of the lattice  SYMBOL  the operation  SYMBOL  is defined and the function  SYMBOL  is both monotone and satisfies  SYMBOL
From this we see that in the posed problem we can imply that  SYMBOL
From the above said we conclude more that this problem is equivalent to finding a maximal subset  SYMBOL  such that the function  SYMBOL  restricted on the subset  SYMBOL  is monotone
So let us consider the following generalization of our problem which we will call MaxCMS(Maximal Consistent with Monotonicity Set) {MaxCMS } The finite sets  SYMBOL  where  SYMBOL  are given; on each of them partial orders  SYMBOL  are defined consistently and the function  SYMBOL  is given
Then every element  SYMBOL  is assigned by a positive integer weight  SYMBOL
Our task is to find a maximal by weight subset  SYMBOL  such that the function  SYMBOL  restricted on  SYMBOL  is monotone i e SYMBOL {Definition 1 } The set  SYMBOL  is called acceptable iff the function  SYMBOL  restricted on  SYMBOL  is monotone {Definition 2 } A set which  is acceptable and maximal by weight is denoted by  SYMBOL (in some cases we use this notation to mean the weight of this set)
In the remainder of the paper we will consider that problem
