On the typical case complexity of graph optimization

In combinatorial optimization problems that exhibit phase transition it is a frequently observed phenomenon that the algorithmically hard instances are concentrated around the phase transition region. The location, the size and sometimes the mere existence of this critical region, however, may depend on several factors: on the choice of an “order parameter”, on the solving algorithm or on the probabilistic model. We investigate a large class of graph optimization problems and show that this concentration of hardness is in fact a more general phenomenon, if we focus on the complexity of finding or approximating the optimal value (such as the size of a maximum clique), rather than finding a witness (an actual maximum clique). Specifically, we prove that for a general class of graph optimization problems there is always a critical region of input instances in which the hardness is sharply concentrated in the following sense: (1) if the inputs that fall in the critical region are excluded, then the remaining task cannot be NP-hard, unless unlikely complexity collapses happen; (2) the critical region is a small, vanishing subset of all inputs. Thus, in this sense, the hardness of the overall task is necessarily caused by a small, exponentially vanishing critical region of the possible inputs. This concentration of hardness is invariant in the sense that it does not depend on the choice of any order parameter, or on a specific solving algorithm or on the choice of a particular probabilistic model within the considered broad family. Since a random input, drawn by any probability distribution in the family, falls almost surely outside the critical region, therefore, it is justified in a rigorous sense that the typical case complexity of these problems is easier than their worst case complexity and this phenomenon remains invariant for a broad class of models.