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The hardest problems in PSPACE. PTAS: Polynomial-time approximation scheme (a subclass of APX). QIP: Solvable in polynomial time by a quantum interactive proof system. QMA: Quantum analog of NP. R: Solvable in a finite amount of time. RE: Problems to which we can answer "YES" in a finite amount of time, but a "NO" answer might never come. RL
A problem is hard for a class of problems C if every problem in C can be polynomial-time reduced to . Thus no problem in C is harder than , since an algorithm for allows us to solve any problem in C with at most polynomial slowdown. Of particular importance, the set of problems that are hard for NP is called the set of NP-hard problems.
In fact, significant progress (by den Boer, Maurer, Wolf, Boneh and Lipton) has been made towards showing that over many groups the DHP is almost as hard as the DLP. There is no proof to date that either the DHP or the DLP is a hard problem, except in generic groups (by Nechaev and Shoup). A proof that either problem is hard implies that P ≠ NP.
The notion of hard problems depends on the type of reduction being used. For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems. If a problem is in and hard for , then is said to be complete for .
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be ...
Get ready for all of today's NYT 'Connections’ hints and answers for #553 on Sunday, December 15, 2024. Today's NYT Connections puzzle for Sunday, December 15, 2024The New York Times.
Today's spangram answer on Wednesday, December 11, 2024, is WOODWORKING. What Are Today’s NYT Strands Answers, Word List for Wednesday, December 11? SANDER. PLANE. CHISEL. JIGSAW. LATHE.
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