On sparse hard sets for counting classes

On Sparse Hard Sets for Counting Classes

Sparse Hard Sets for

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexi...

Sparse Hard Sets for P

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexi...

On Counting Independent Sets in Sparse Graphs

We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree ∆. The first implies that the Markov chain Monte Carlo technique is likely to fail if ∆ ≥ 6. The second shows that no fully polynomial randomized approximation scheme can exist for ∆ ≥ 25, unless RP = NP.

Counting Complexity Classes for Numeric Computations. III: Complex Projective Sets

In [7] counting complexity classes #PR and #PC in the BlumShub-Smale setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [7] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FPR R . In this paper, we prove that the corresponding result is true over C, namely that the computati...