The generic complexity of the bounded problem of graphs clustering

Generic-case approach to algorithmic problems studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the bounded problem of graphs clustering. In this problem the structure of objects relations is presented as a graph: vertices correspond to objects, and edges connect similar objects. It is required to divide the set of objects into bounded disjoint groups (clusters) to minimize the number of connections between clusters and the number of missing links within clusters. We have constructed a subproblem of this problem, for which there is no polynomial generic algorithm provided $\text {P} \neq \text{NP}$ and $\text{P} = \text{BPP}$. To prove the theorem, we use the method of generic amplification, which allows to construct generically hard problems from the problems hard in the classical sense. The main component of this method is the cloning technique, which merges the inputs of a problem together into sufficiently large sets of equivalent inputs. Equivalence is understood in the sense that the problem for them is solved in a similar way.