Nonuniform Complexity Classes Specified by Lower and Upper Bounds

Upper and lower bounds of symmetric division deg index

Symmetric Division Deg index is one of the 148 discrete Adriatic indices that showed good predictive properties on the testing sets provided by International Academy of Mathematical Chemistry. Symmetric Division Deg index is defined by $$ SDD(G) = sumE left( frac{min{d_u,d_v}}{max{d_u,d_v}} + frac{max{d_u,d_v}}{min{d_u,d_v}} right), $$ where $d_i$ is the degree of vertex $i$ in graph $G$. In th...

Almost Everywhere High Nonuniform Complexity

We investigate the distribution of nonuniform complexities in uniform complexity classes We prove that almost every problem decidable in exponential space has essentially maximum circuit size and space bounded Kolmogorov complexity almost everywhere The circuit size lower bound actually exceeds and thereby strengthens the Shannon n n lower bound for almost every problem with no computability co...

Nonuniform Lower Bounds for Exponential Time Classes

Nonuniform Reductions and NP-Completeness

Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for nonuniform completeness vs un...

On the Complexity of Rank and Rigidity

Given a matrix M over a ring K, a target rank r and a bound k, we want to decide whether the rank of M can be brought down to below r by changing at most k entries of M . This is a decision version of the well-studied notion of matrix rigidity. We examine the complexity of the following related problems and obtain completeness results for small (counting logspace or smaller) classes: (a) comput...