Computational Complexity

ZPP

RP ∩ coRP; the randomized poly-time class with zero error probability

Translation

Relations between complexity classes translate upwards (padding argument)

Church-Turing hypothesis

The intuitive notion of an algorithm is a halting Turing machine

Approximation for optimization problems

There exist problems for which solutions sufficiently close to optimum can be found (and there exist problems without this property)

Randomized Algorithms

There exist problems for which the use of a random number generator allows for finding the exact solution most of the time

log-space uniform (Ubc-uniform)

a circuit family C for which there exists a DLOG machine to construct every circuit in C

Boolean expression

a combination of variables, logical connectives, and parenthesis

Independent Set

a discrete subgraph; a set of nodes from a graph such that no two nodes are adajcent

Clause

a disjunction (or) of literals

Parallel RAM (PRAM)

a network of RAMs running concurrently on a global, shared memory (CRCW, CREW, EREW)

Random Turing Machine

a restricted NTM with a clock such that the computation tree is a full binary tree with either YES (accepting) or NO (rejecting) in each leaf node. For every input, there must be either no accepting path or at least ε (1/2) of the paths are accepting

Uniform circuit family

a set of Boolean circuit C = {C0, C1, ...} such that (the encoding of) each Cn can be constructed from n by using an algorithm

Literal

a variable or its negation

Boolean circuit

acyclic directed graph with 3 types of nodes (input, gates, output)

Alternating TM (ATM)

an extension of an NTM in which every state is either existential (∃) or universal (∀)

Randomized poly-time class (RP)

class of problems defined by poly-time RTMs

Conjunctive Normal Form (CNF)

conjunctions of clauses

Counting argument

counts all possible configs

Inherently sequential problems

problems for which no significant speedup can be achieved by parallel computation

Parallelizable problems

problems that can be decomposed into smaller subproblems which can be executed by using multiple processors

Complete problem

represents a complexity class in that it is at least as hard as any other problem in that same class

The class NP

the class of all languages that can be recognized by polynomial time bounded non-deterministic TMs

Time-constructible function

A function for which there exists a TM that halts after exactly f(|x|) steps for every input x

Intractable Problem

A problem that cannot be solved by any polynomial time bounded algorithm

CLASS-complete problem

A problem that is CLASS-hard and an element of CLASS

Decision problem

A problem that only has answers "yes" and "no"

Padding argument

Adds padding to input string

Strongly NP-complete

An NP-complete problem with no pseudo-poly time algorithm (assumes NP is not P)

Probabalistic TM

An NTM that accepts its input iff more than half of all paths terminate in YES (accepts by majority vote, not existence of one path; repeating this machine does not increase the error probability)

Pseudo-polynomial time algorithm

An algorithm for a number problem that is poly-time bounded if the involved numbers are polynomially bounded in the input size

Boolean circuit family

C = {C0, C1, ... } can compute f if each Cn computes f restricted to inputs of size n

Space Hierarchy Theorem

DSPACE(o(f(n))) is a proper subset of DSPACE(f(n))

Time Hierarchy Theorem

DTIME(o(f(n)/log(f(n))) is a proper subset of DTIME(f(n))

Partition Problem

Given a finite set A and its size set S, decide if there exists a partition such that the sum of the sizes of the two sets are equal

Relativity Concept

If a class is closed under a certain resource-bounded reduction, any larger class is also closed under that reduction

Las Vegas Algorithms

Iterating random algorithms until a definite answer is reached (zero error probability, but may never answer; expected poly-time) A ∈ ZPP so A ∈ RP and ∼A ∈ RP. Check A and ∼A interleaved and answer if one accepts

Component Design

Match components of two problems to relate them

CLASS-hard problem with respect to reduction R

every problem in CLASS reduces by R to this problem

Monte Carlo Algorithms

iterating an RTM with a RNG on the same input and say YES if any iterations accepts; else say NO. P(M(x) = 1 | x ∈ L) = (1-ε)^i P(M(x) = 0 | x ∈ L) = 1 - (1-ε)^i P(M(x) = 1 | x ∉ L) = 0 P(M(x) = 0 | x ∉ L) = 1

Bounded error PP class (BPP)

languages accepted by poly-time PTM's accepting (rejecting) iff more than half the paths terminate in YES (NO) state Repeating a BPP machine rapidly increases the probability of accuracy L ∈ BPP iff ∃ a PTM M M runs in poly-time ∀ x ∀ x∈L, P(M(x) = 1) ≥ 2/3 (1-ε) ∀ x∉L, P(M(x) = 1) ≤ 1/3 (ε)

Probabilistic poly-time class (PP = coPP)

languages defined by poly-time PTMs