## 3 Materials and Methods
### Methods of Comparison by Solving a Multi-Objective Optimization Problem  
The problem formulated in the previous subsection should be solved using the methods of solving multi-objective optimization problems. The simplest method to reduce a multi-criteria problem to a single-criteria problem involves scalarization of the criteria by using a weighted average. In this way, instead of several original criteria, a single maximized substitute criterion is formulated:  
\[\sum_{i=1}^{N}w_{i}\cdot\overline{K_{i}}(x)\rightarrow\max, \tag{24}\]  
where \(\overline{K_{i}}(x)\)--normalized \(i\)-th criterion function and, additionally, the weights meet the following conditions:  
\[ \begin{array}{c}\sum_{i=1}^{N}w_{i}=1\\ w_{i}\in[0,\ \ 1],\ \ i=\overline{1,N}\end{array}  \tag{25}\]  
and the problem can be solved using methods of classical single-criterion optimization (with minimization of the meta-criterion formulated in this way). In particular, providing equal weights for all criteria makes each of them equally important in regard to the impact on the value of the objective function. Another way to solve the problem is goal programming. This approach does not involve calculating the extreme value (minimum/maximum of the objective function on the set of admissible solutions) but instead finding a solution that minimizes the distance of the function from the given vector of values constituting the objectives for individual criteria or the given value of the meta-criterion calculated as in (24). This problem is therefore designed to seek the assignment of the vertices of one graph to another, so that the similarity between them is as close as possible to this set of \"reference\" values or to the given value of the meta-criterion, respectively. Formally speaking, the problem in the first variant (minimization of deviations from \(c_{i}\)--the vector of the given values) can be described as follows:  
\[\sum\limits_{i=1}^{N}\left|\left(w_{i}\cdot\overline{K_{i}}(x)\right)-c_{i} \right|\rightarrow\min, \tag{26}\]  
with constraints as before--i.e., \(\overline{K_{i}}(x)\)-- normalized \(i\)-th criterion function; \(c_{i}\)--vector of goals set by the decision-maker; and with the following constraints:  
\[ \begin{array}{l}\sum\limits_{i=1}^{N}w_{i}=1\\ w_{i}\in[0,\ \ 1],\ \ i=\overline{1,N}\\ c_{i}\in[0,\ \ 1],\ \ i=\overline{1,N}\end{array}  \tag{27}\]  
or, alternatively, in the second variant (minimizing the deviation from \(c\)--given single meta-criterion value),  
\[\left|c-\sum\limits_{i=1}^{N}\left(w_{i}\cdot\overline{K_{i}}(x)\right)\right| \rightarrow\min, \tag{28}\]  
with constraints  
\[ \begin{array}{l}\sum\limits_{i=1}^{N}w_{i}=1\\ w_{i}\in[0,\ \ 1],\ \ i=\overline{1,N}\\ c\in[0,\ \ 1]\end{array}  \tag{29}\]  
If vector values are selected appropriately, this can be used to select an assignment that results in a relatively high similarity between the graphs, but at the same time renders it impossible to detect \"plagiarism\" in the musical arrangement built on the basis of the graph representation of the piece. The third method of reducing the problem to a single-criterion problem involves the use of threshold criteria. In this approach, we choose one leading criterion (and single-criteria optimization will be performed against it), whereas for the remaining criteria, we define the so-called satisfaction thresholds (namely, we add constraints). Doing so narrows the space of solutions and makes the problem solved by single-criterion methods. At the same time, the solution selected in this way is acceptable to the decision-maker because exceeding the threshold values of all other criteria means that the selected solution can be considered \"good enough\" in relation to them. \[ \begin{array}{l}\overline{K_{j}}(x)\rightarrow\max\\ K_{i}(x)\geq p_{i},i\
eq j,i=\overline{1,N}\\ x\in[0,\ \ 1]\end{array}  \tag{30}\]  
One should note, however, that the above methods are not an exhaustive list.