Approximating P-Complete Problems (2024)

FAQs

Is it possible to solve NP-complete problems? ›

(i) All NP-complete problems are solvable in polynomial time: Yes. Every problem in NP is polynomially reducible to SAT, and SAT is reducible to every NP-hard problem. Therefore, a polynomial time solution to any NP-hard problem (such as 3Col) implies that every problem in NP can be solved in polynomial time.

The answer is complexity. It's much more difficult to quickly find a solution to an NP problem than a P problem. Computers can easily check solutions to NP problems, but devising an algorithm that can propose solutions to NP problems in a reasonable time is much more difficult.

A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it even though it may not be in NP. A problem is NP-complete if it is both in NP and NP-hard. The NP-complete problems represent the hardest problems in NP.

Hence, P equals NP would mean that proving mathematical theorems can be done by a simple computer program. “P equals NP would transform mathematics by allowing a computer to find a formal proof of any theorem which has a proof of a reasonable length, since formal proofs can easily be recognized in polynomial time.”

Some are, some are not. Boolean Satisfiability (SAT) is solvable and NP-hard (in fact, NP-complete). The Halting Problem is NP-hard and not solvable. NP hardnes and solvability is two distinct categories defined on different types of criteria.

NP-hard problems commonly come up as human-solvable puzzles. Like Sudoku... or perhaps a more applicable problem... layout and routing of electronic components on a PCB and/or chip. Or even assembly-language register allocation (coloring and packing problem).

While the P versus NP problem is generally considered unsolved, many amateur and some professional researchers have claimed solutions.

AI, with its advanced pattern recognition and data processing capabilities, is ideally positioned to tackle the P vs NP problem. Machine learning algorithms can process and analyze vast datasets much faster than humans, identifying patterns that could lead to a solution.

Is Chess NP complete or NP hard? “Real” chess is in P because it's of finite size so all positions can be (in a theoretical, computational-complexity sense) looked up in a table. “Generalized” chess is harder than NP, but you have to define how you generalize it to larger boards.

Why NP-hard problems are hardest to solve? ›

A “P problem” takes a computer “polynomial time” to complete, while an “NP-Hard problem” takes exponential time to solve because there is no known algorithm that can solve it in polynomial time.

It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.

The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré ...

'P' stands for 'Polynomial Time', problems that can be solved relatively quickly by a computer. 'NP' stands for 'Nondeterministic Polynomial Time', problems for which a solution can be checked quickly. The question (P vs NP) is whether every problem whose solution can be checked quickly can also be solved quickly.

Thus, P problems are said to be easy, or tractable. A problem is called NP if its solution can be guessed and verified in polynomial time, and nondeterministic means that no particular rule is followed to make the guess.

So far we've seen a lot of good news: such-and-such a problem can be solved quickly (in close to linear time, or at least a time that is some small polynomial function of the input size).

It was recently proved mathematically that memcomputing machines have the same computational power of non-deterministic Turing machines. Therefore, they can solve NP-complete problems in polynomial time and, using the appropriate architecture, with resources that only grow polynomially with the input size.

Importantly, these problems don't have an algorithm that can solve them in polynomial time. They are called NP-hard because they are at least as hard as the hardest problems in NP class. Can np hard problems be solved? Yes, NP-hard problems can be solved, but there's a catch.

It is widely suspected that NP complete problems require exponential time. However, this has not been proved and it's still possible that there is some clever way to solve them in polynomial time.