Hard promise problems and nonuniform complexity

Hard Promise Problems and Nonuniform Complexity

Longprt, L. and A.L. Selman, Hard promise problems and nonuniform complexity, Theoretical Computer Science 115 (1993) 277-290. For every recursive set A, let PP-A denote the following promise problem: input x and y promise (u~A)@(ysA) property XE A. We show that if L is a solution of PP-A, then A~p~/Poly. From this result, it follows that if A is <F-hard for NP, then all solutions of PP-A are h...

Nonuniform Hard Boolean Functions and Uniform Complexity Classes

Randomness, Promise Problems, Randomized Complexity Classes

A big problem that motivates randomized algorithms is that of Primality testing: Given 0 ≤ p ≤ 2, determine if p is prime in poly(n) time. Algorithms such as Miller-Rabin and Solovay-Strassen are randomized algorithms that can solve this in polynomial time, though more recently Agrawal-Kayal-Saxena found a deterministic solution to primality testing. Remark When we say randomized, the goal is f...

The Complexity and Distribution of Hard Problems

Measure-theoretic aspects of the <:-reducibility structure of the exponential time complexity classes E=DTIME(21inea‘) and E2 = DTIME(2Po’ynomia’) are investigated. Particular attention is given to the complexity (measured b y the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are <:-hard for E and other complexity classes. Tight upper and lower...

SAT-Varible Complexity of Hard Combinatorial Problems

This paper discusses polynomial-time reductions from Hamiltonian Circuit (HC), k-Vertex Coloring (k-VC), and k-Clique Problems to Satis ability Problem (SAT) which are e cient in the number of Boolean variables needed in SAT. We rst present a basic type of reductions that need (n 1) log(n 1), (n 1) log k, and k logn variables for HC, k-VC and k-Clique, respectively. Several heuristics can reduc...