Home » Explaining computational complexity theory

Explaining computational complexity theory

Solutons:


Hoooo, doctoral comp flashback. Okay, here goes.

We start with the idea of a decision problem, a problem for which an algorithm can always answer “yes” or “no.” We also need the idea of two models of computer (Turing machine, really): deterministic and non-deterministic. A deterministic computer is the regular computer we always thinking of; a non-deterministic computer is one that is just like we’re used to except that is has unlimited parallelism, so that any time you come to a branch, you spawn a new “process” and examine both sides. Like Yogi Berra said, when you come to a fork in the road, you should take it.

A decision problem is in P if there is a known polynomial-time algorithm to get that answer. A decision problem is in NP if there is a known polynomial-time algorithm for a non-deterministic machine to get the answer.

Problems known to be in P are trivially in NP — the nondeterministic machine just never troubles itself to fork another process, and acts just like a deterministic one. There are problems that are known to be neither in P nor NP; a simple example is to enumerate all the bit vectors of length n. No matter what, that takes 2n steps.

(Strictly, a decision problem is in NP if a nodeterministic machine can arrive at an answer in poly-time, and a deterministic machine can verify that the solution is correct in poly time.)

But there are some problems which are known to be in NP for which no poly-time deterministic algorithm is known; in other words, we know they’re in NP, but don’t know if they’re in P. The traditional example is the decision-problem version of the Traveling Salesman Problem (decision-TSP): given the cities and distances, is there a route that covers all the cities, returning to the starting point, in less than x distance? It’s easy in a nondeterministic machine, because every time the nondeterministic traveling salesman comes to a fork in the road, he takes it: his clones head on to the next city they haven’t visited, and at the end they compare notes and see if any of the clones took less than x distance.

(Then, the exponentially many clones get to fight it out for which ones must be killed.)

It’s not known whether decision-TSP is in P: there’s no known poly-time solution, but there’s no proof such a solution doesn’t exist.

Now, one more concept: given decision problems P and Q, if an algorithm can transform a solution for P into a solution for Q in polynomial time, it’s said that Q is poly-time reducible (or just reducible) to P.

A problem is NP-complete if you can prove that (1) it’s in NP, and (2) show that it’s poly-time reducible to a problem already known to be NP-complete. (The hard part of that was provie the first example of an NP-complete problem: that was done by Steve Cook in Cook’s Theorem.)

So really, what it says is that if anyone ever finds a poly-time solution to one NP-complete problem, they’ve automatically got one for all the NP-complete problems; that will also mean that P=NP.

A problem is NP-hard if and only if it’s “at least as” hard as an NP-complete problem. The more conventional Traveling Salesman Problem of finding the shortest route is NP-hard, not strictly NP-complete.

Michael Sipser’s Introduction to the Theory of Computation is a great book, and is very readable. Another great resource is Scott Aaronson’s Great Ideas in Theoretical Computer Science course.

The formalism that is used is to look at decision problems (problems with a Yes/No answer, e.g. “does this graph have a Hamiltonian cycle”) as “languages” — sets of strings — inputs for which the answer is Yes. There is a formal notion of what a “computer” is (Turing machine), and a problem is in P if there is a polynomial time algorithm for deciding that problem (given an input string, say Yes or No) on a Turing machine.

A problem is in NP if it is checkable in polynomial time, i.e. if, for inputs where the answer is Yes, there is a (polynomial-size) certificate given which you can check that the answer is Yes in polynomial time. [E.g. given a Hamiltonian cycle as certificate, you can obviously check that it is one.]

It doesn’t say anything about how to find that certificate. Obviously, you can try “all possible certificates” but that can take exponential time; it is not clear whether you will always have to take more than polynomial time to decide Yes or No; this is the P vs NP question.

A problem is NP-hard if being able to solve that problem means being able to solve all problems in NP.

Also see this question:
What is an NP-complete in computer science?

But really, all these are probably only vague to you; it is worth taking the time to read e.g. Sipser’s book. It is a beautiful theory.

This is a comment on Charlie’s post.

A problem is NP-complete if you can prove that (1) it’s in NP, and
(2) show that it’s poly-time reducible to a problem already known to
be NP-complete.

There is a subtle error with the second condition. Actually, what you need to prove is that a known NP-complete problem (say Y) is polynomial-time reducible to this problem (let’s call it problem X).

The reasoning behind this manner of proof is that if you could reduce an NP-Complete problem to this problem and somehow manage to solve this problem in poly-time, then you’ve also succeeded in finding a poly-time solution to the NP-complete problem, which would be a remarkable (if not impossible) thing, since then you’ll have succeeded to resolve the long-standing P = NP problem.

Another way to look at this proof is consider it as using the the contra-positive proof technique, which essentially states that if Y –> X, then ~X –> ~Y. In other words, not being able to solve Y in polynomial time isn’t possible means not being to solve X in poly-time either. On the other hand, if you could solve X in poly-time, then you could solve Y in poly-time as well. Further, you could solve all problems that reduce to Y in poly-time as well by transitivity.

I hope my explanation above is clear enough. A good source is Chapter 8 of Algorithm Design by Kleinberg and Tardos or Chapter 34 of Cormen et al.

Related Solutions

Extract file from docker image?

You can extract files from an image with the following commands: docker create $image # returns container ID docker cp $container_id:$source_path $destination_path docker rm $container_id According to the docker create documentation, this doesn't run the...

Transfer files using scp: permission denied

Your commands are trying to put the new Document to the root (/) of your machine. What you want to do is to transfer them to your home directory (since you have no permissions to write to /). If path to your home is something like /home/erez try the following:...

What’s the purpose of DH Parameters?

What exactly is the purpose of these DH Parameters? These parameters define how OpenSSL performs the Diffie-Hellman (DH) key-exchange. As you stated correctly they include a field prime p and a generator g. The purpose of the availability to customize these...

How to rsync multiple source folders

You can pass multiple source arguments. rsync -a /etc/fstab /home/user/download bkp This creates bkp/fstab and bkp/download, like the separate commands you gave. It may be desirable to preserve the source structure instead. To do this, use / as the source and...

Benefits of Structured Logging vs basic logging

There are two fundamental advances with the structured approach that can't be emulated using text logs without (sometimes extreme levels of) additional effort. Event Types When you write two events with log4net like: log.Debug("Disk quota {0} exceeded by user...

Interfaces vs Types in TypeScript

2019 Update The current answers and the official documentation are outdated. And for those new to TypeScript, the terminology used isn't clear without examples. Below is a list of up-to-date differences. 1. Objects / Functions Both can be used to describe the...

Get total as you type with added column (append) using jQuery

One issue if that the newly-added column id's are missing the id number. If you look at the id, it only shows "price-", when it should probably be "price-2-1", since the original ones are "price-1", and the original ones should probably be something like...

Determining if a file is a hard link or symbolic link?

Jim's answer explains how to test for a symlink: by using test's -L test. But testing for a "hard link" is, well, strictly speaking not what you want. Hard links work because of how Unix handles files: each file is represented by a single inode. Then a single...

How to restrict a Google search to results of a specific language?

You can do that using the advanced search options: http://www.googleguide.com/sharpening_queries.html I also found this, which might work for you: http://www.searchenginejournal.com/how-to-see-google-search-results-for-other-locations/25203/ Just wanted to add...

Random map generation

Among the many other related questions on the site, there's an often linked article for map generation: Polygonal Map Generation for Games you can glean some good strategies from that article, but it can't really be used as is. While not a tutorial, there's an...

How to prettyprint a JSON file?

The json module already implements some basic pretty printing in the dump and dumps functions, with the indent parameter that specifies how many spaces to indent by: >>> import json >>> >>> your_json = '["foo", {"bar":["baz", null,...

How can I avoid the battery charging when connected via USB?

I have an Android 4.0.3 phone without root access so can't test any of this but let me point you to /sys/class/power_supply/battery/ which gives some info/control over charging issues. In particular there is charging_enabled which gives the current state (0 not...

How to transform given dataset in python? [closed]

From your expected result, it appears that each "group" is based on contiguous id values. For this, you can use the compare-cumsum-groupby pattern, and then use agg to get the min and max values. # Sample data. df = pd.DataFrame( {'id': [1, 2, 2, 2, 2, 2, 1, 1,...

Output of the following C++ Program [closed]

It works exactly like this non-recursive translation: int func_0() { return 2; } int func_1() { return 3; } int func_2() { return func_1() + func_0(); } // Returns 3 + 2 = 5 int func_3() { return func_2() + func_1(); } // Returns 5 + 3 = 8 int func_4() { return...

Making a circle out of . (periods) [closed]

Here's the maths and even an example program in C: http://pixwiki.bafsoft.com/mags/5/articles/circle/sincos.htm (link no longer exists). And position: absolute, left and top will let you draw: http://www.w3.org/TR/CSS2/visuren.html#choose-position Any further...

Should I use a code converter (Python to C++)?

Generally it's an awful way to write code, and does not guarantee that it will be any faster. Things which are simple and fast in one language can be complex and slow in another. You're better off either learning how to write fast Python code or learning C++...

tkinter: cannot concatenate ‘str’ and ‘float’ objects

This one line is more than enough to cause the problem: text="რეგულარი >> "+2.23+ 'GEL' 2.23 is a floating-point value; 'GEL' is a string. What does it mean to add an arithmetic value and a string of letters? If you want the string label 'რეგულარი...