# Questions tagged [complexity-theory]

Questions related to the (computational) complexity of solving problems

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### What is the definition of P, NP, NP-complete and NP-hard?

I'm in a course about computing and complexity, and am unable to understand what these terms mean. All I know is that NP is a subset of NP-complete, which is a subset of NP-hard, but I have no idea ...
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### Why hasn't there been an encryption algorithm that is based on the known NP-Hard problems?

Most of today's encryption, such as the RSA, relies on the integer factorization, which is not believed to be a NP-hard problem, but it belongs to BQP, which makes it vulnerable to quantum computers. ...
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### How not to solve P=NP?

There are lots of attempts at proving either $\mathsf{P} = \mathsf{NP}$ or $\mathsf{P} \neq \mathsf{NP}$, and naturally many people think about the question, having ideas for proving either direction....
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### How can we assume that basic operations on numbers take constant time?

Normally in algorithms we do not care about comparison, addition, or subtraction of numbers -- we assume they run in time $O(1)$. For example, we assume this when we say that comparison-based sorting ...
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### Knapsack problem -- NP-complete despite dynamic programming solution?

Knapsack problems are easily solved by dynamic programming. Dynamic programming runs in polynomial time; that is why we do it, right? I have read it is actually an NP-complete problem, though, which ...
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### Is legislation NP-complete?

I would like to know if there has been any work relating legal code to complexity. In particular, suppose we have the decision problem "Given this law book and this particular set of circumstances, is ...
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### What would be the real-world implications of a constructive $P=NP$ proof?

I have a high-level understanding of the $P=NP$ problem and I understand that if it were absolutely "proven" to be true with a provided solution, it would open the door for solving numerous problems ...
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### Are there subexponential-time algorithms for NP-complete problems?

Are there NP-complete problems which have proven subexponential-time algorithms? I am asking for the general case inputs, I am not talking about tractable special cases here. By sub-exponential, I ...
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### Algorithmic intuition for logarithmic complexity

I believe I have a reasonable grasp of complexities like $\mathcal{O}(1)$, $\Theta(n)$ and $\Theta(n^2)$. In terms of a list, $\mathcal{O}(1)$ is a constant lookup, so it's just getting the head of ...
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### If everyone believes P ≠ NP, why is everyone sceptical of proof attempts for P ≠ NP?

Many seem to believe that $P\ne NP$, but many also believe it to be very unlikely that this will ever be proven. Is there not some inconsistency to this? If you hold that such a proof is unlikely, ...
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### Why polynomial time is called "efficient"?

Why in computer science any complexity which is at most polynomial is considered efficient? For any practical application(a), algorithms with complexity $n^{\log n}$ are way faster than algorithms ...
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### Why are some games np-complete?

I read the Wikipedia entry about "List of NP-complete problems" and found that games like super mario, pokemon, tetris or candy crush saga are np-complete. How can I imagine np-completeness of a game? ...
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### Why did Google not use an NP problem for their quantum supremacy experiment?

Reading discussions of the recent quantum supremacy experiment by Google I noticed that a lot of time and effort (in the experiment itself, but also in the excellent blog posts by Scott Aaronson and ...
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### What is the difference between an algorithm, a language and a problem?

It seems that on this site, people will often correct others for confusing "algorithms" and "problems." What are the difference between these? How do I know when I should be considering algorithms and ...
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### Can a computer determine whether a mathematical statement is true or not?

I was reading Introduction to the Theory of Computation by Michael Sipser and I found the following paragraph quite interesting: During the first half of the twentieth century, mathematicians such as ...
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### What are common techniques for reducing problems to each other?

In computability and complexity theory (and maybe other fields), reductions are ubiquitous. There are many kinds, but the principle remains the same: show that one problem $L_1$ is at least as hard as ...
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### Are there NP problems, not in P and not NP Complete?

Are there any known problems in $\mathsf{NP}$ (and not in $\mathsf{P}$) that aren't $\mathsf{NP}$ Complete? My understanding is that there are no currently known problems where this is the case, but ...
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### Why do we believe that PSPACE ≠ EXPTIME?

I'm having trouble intuitively understanding why PSPACE is generally believed to be different from EXPTIME. If PSPACE is the set of problems solvable in space polynomial in the input size $f(n)$, ...
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### Decision problems vs "real" problems that aren't yes-or-no

I read in many places that some problems are difficult to approximate (it is NP-hard to approximate them). But approximation is not a decision problem: the answer is a real number and not Yes or No. ...
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### Why is linear programming in P but integer programming NP-hard?

Linear programming (LP) is in P and integer programming (IP) is NP-hard. But since computers can only manipulate numbers with finite precision, in practice a computer is using integers for linear ...
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### Explaining the relevance of asymptotic complexity of algorithms to practice of designing algorithms

In algorithms and complexity we focus on the asymptotic complexity of algorithms, i.e. the amount of resources an algorithm uses as the size of the input goes to infinity. In practice, what is ...
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### How can I verify a solution to Travelling Salesman Problem in polynomial time?

So, TSP (Travelling salesman problem) decision problem is NP complete. But I do not understand how I can verify that a given solution to TSP is in fact optimal in polynomial time, given that there is ...
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### How do I construct reductions between problems to prove a problem is NP-complete?

I am taking a complexity course and I am having trouble with coming up with reductions between NPC problems. How can I find reductions between problems? Is there a general trick that I can use? How ...
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### Generalised 3SUM (k-SUM) problem?

The 3SUM problem tries to identify 3 integers $a,b,c$ from a set $S$ of size $n$ such that $a + b + c = 0$. It is conjectured that there is not better solution than quadratic, i.e. $\mathcal{o}(n^2)$....
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### Is it NP-hard to fill up bins with minimum moves?

There are $n$ bins and $m$ type of balls. The $i$th bin has labels $a_{i,j}$ for $1\leq j\leq m$, it is the expected number of balls of type $j$. You start with $b_j$ balls of type $j$. Each ball of ...
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### Would the P vs. NP problem become trivial as a result of the development of universal quantum computers?

If someone were to build a universal quantum computer, would that have any implications on the problem of P vs. NP?
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### Why is Relativization a barrier?

When I was explaining the Baker-Gill-Solovay proof that there exists an oracle with which we can have, $\mathsf{P} = \mathsf{NP}$, and an oracle with which we can have $\mathsf{P} \neq \mathsf{NP}$ to ...
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### NP-Hard problems that are not in NP but decidable

I'm wondering if there is a good example for an easy to understand NP-Hard problem that is not NP-Complete and not undecidable? For example, the halting problem is NP-Hard, not NP-Complete, but is ...
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### Is there a task that is solvable in polynomial time but not verifiable in polynomial time?

A colleague of mine and I have just hit some notes of one of our professors. The notes state that there are tasks that are possible to solve in polynomial time (are in the class of PF) but that are ...
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It is known that each optimization/search problem has an equivalent decision problem. For example the shortest path problem optimization/search version: Given an undirected unweighted graph $G ... 32 votes 2 answers 3k views ### Problems that are polynomially "hard" to compute but "easy" to verify In the (unlikely) event that$P=NP$with a constructive proof of a polynomial time algorithm that solves 3SAT, obviously things will be very different. However, practically, it could happen that the ... • 462 31 votes 1 answer 11k views ### Does space complexity analysis usually include output space? Since most examples of complexity analysis I've seen involve functions that return either nothing (e.g. in-place sort) or a single value (e.g. computation, lookup), I haven't been able to figure this ... • 413 31 votes 3 answers 10k views ### Teaching NP-completeness - Turing reductions vs Karp reductions I'm interested in the question of how best to teach NP-completeness to computer science majors. In particular, should we teach it using Karp reductions or using Turing reductions? I feel that the ... • 162k 31 votes 5 answers 14k views ### "NP-complete" optimization problems I am slightly confused by some terminology I have encountered regarding the complexity of optimization problems. In an algorithms class, I had the large parsimony problem described as NP-complete. ... 30 votes 3 answers 1k views ### Measuring the difficulty of SAT instances Given an instance of SAT, I would like to be able to estimate how difficult it will be to solve the instance. One way is to run existing solvers, but that kind of defeats the purpose of estimating ... • 4,862 30 votes 5 answers 62k views ### How can I reduce Subset Sum to Partition? Maybe this is quite simple but I have some trouble to get this reduction. I want to reduce Subset Sum to Partition but at this time I don't see the relation! Is it possible to reduce this problem ... • 517 30 votes 1 answer 13k views ### How hard is counting the number of simple paths between two nodes in a directed graph? There is an easy polynomial algorithm to decide whether there is a path between two nodes in a directed graph (just do a routine graph traversal with, say, depth-first-search). However it seems that, ... • 1,379 30 votes 1 answer 2k views ### Is regex golf NP-Complete? As seen in this recent XKCD strip and this recent blog post from Peter Norvig (and a Slashdot story featuring the latter), "regex golf" (which might better be called the regular expression separation ... • 2,529 30 votes 1 answer 993 views ### Graph problem known to be$NP$-complete only under Cook reduction The theory of NP-completeness was initially built on Cook (polynomial-time Turing) reductions. Later, Karp introduced polynomial-time many-to-one reductions. A Cook reduction is more powerful than a ... 29 votes 3 answers 4k views ### Are there any problems that are easy to compute but hard to verify? Assuming P$\neq$NP, NP-complete problems are "hard to solve, but have answers that are easy to check." Does it make any sense to consider the opposite, that is, problems for which it's easy to ... • 1,624 29 votes 1 answer 20k views ### Is the k-clique problem NP-complete? In this Wikipedia article about the Clique problem in graph theory it states in the beginning that the problem of finding a clique of size K, in a graph G is NP-complete: Cliques have also been ... 29 votes 1 answer 744 views ### Subset Sum problem with many divisibility conditions How does the computational complexity of the Subset Sum problem depend on the parameter$\alpha(S)$of the input$(S, t)$, defined as follows? Considering$S$under the divisibility partial order, i.e.... • 3,083 29 votes 2 answers 2k views ### Longest Repeated (Scattered) Subsequence in a String Informal Problem Statement: Given a string, e.g.$ACCABBAB$, we want to colour some letters red and some letters blue (and some not at all), such that reading only the red letters from left to right ... • 393 28 votes 4 answers 6k views ### Why is SAT so important in theoretical computer science? In my Computability and Complexity class, we are focusing on P, NP, NP-complete, and NP-hard problems and the one thing that keeps coming up is the SAT problem, in the context of reduction from one ... • 391 28 votes 2 answers 11k views ### Rule of thumb to know if a problem could be NP-complete This question was inspired by a comment on StackOverflow. Apart from knowing NP-complete problems of the Garey Johnson book, and many others; is there a rule of thumb to know if a problem looks like ... • 1,218 27 votes 5 answers 14k views ### Why isn't this undecidable problem in NP? Clearly there aren't any undecidable problems in NP. However, according to Wikipedia: NP is the set of all decision problems for which the instances where the answer is "yes" have [.. proofs that ... 27 votes 5 answers 18k views ### What is meant by "solvable by non deterministic algorithm in polynomial time" [duplicate] In many textbooks NP problems are defined as: Set of all decision problems solvable by non deterministic algorithms in polynomial time I couldn't understand the part "solvable by non deterministic ... • 2,191 27 votes 3 answers 9k views ### Is it really possible to prove lower bounds? Given any computational problem, is the task of finding lower bounds for such computation really possible? I suppose it boils down to how a single computational step is defined and what model we use ... • 733 27 votes 3 answers 846 views ### NP-complete problems not "obviously" in NP It occurred to many that in all the$\textbf{NP}$-completeness proofs I've read (that I can remember), it's always trivial to show that a problem is in$\textbf{NP}$, and showing that it is$\textbf{...
Dominosa is a relatively new puzzle game. It is played on an $(n+1)\times(n+2)$ grid. Before the game begins, the domino bones $\left(0,0\right),\left(0,1\right),\ldots,\left(n,n\right)$ are ...