Questions tagged [complexity-theory]

Questions related to the (computational) complexity of solving problems

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5
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2answers
336 views

What is co-something?

What does the notation co- mean when prefixing co-NP, co-RE (recursively enumerable), or <...
4
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1answer
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Complexity of 3SAT variants

This question is motivated by my answer to another question in which I stated the fact that both Betweeness and Non-Betweeness problems are $NP$-complete. In the former problem there is a total order ...
2
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1answer
4k views

NP-completeness of a spanning tree problem

I was reviewing some NP-complete problems on this site, and I meet one interesting problem from NP completeness proof of a spanning tree problem In this problem, I am interested in the original ...
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3answers
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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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5answers
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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 ...
37
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2answers
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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 ...
7
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2answers
389 views

Complexity of Special Case Problems

Often I see a sentence like this while reading texts on Computational Complexity: "For this special case of $\text{TSP}$" or "This is a special case of $\text{SAT}$" or "$k$-$\text{PARTITION}$ is ...
21
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1answer
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Classification of intractable/tractable satisfiability problem variants

Recently I found in a paper [1] a special symmetric version of SAT called the 2/2/4-SAT. But there are many $\text{NP}$-complete variants out there, for example: MONOTONE NAE-3SAT, MONOTONE 1-IN-3-SAT,...
16
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1answer
357 views

Are asymptotic lower bounds relevant to cryptography?

An asymptotic lower bound such as exponential-hardness is generally thought to imply that a problem is "inherently difficult". Encryption that is "inherently difficult" to break is thought to be ...
9
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1answer
343 views

Lower bounds of calculating a function of a set

Having a set $A$ of $n$ elements, let's say I want to calculate a function $f(A)$ that is sensitive to all parts of the input, i.e. depends on very member of $A$ (i.e. it is possible to change any ...
21
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3answers
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HALF CLIQUE - NP Complete Problem

Let me start off by noting this is a homework problem, please provide only advice and related observations, NO DIRECT ANSWERS please. With that said, here is the problem I am looking at: Let HALF-...
15
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2answers
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Complexity of computing matrix powers

I am interested in calculating the $n$'th power of a $n\times n$ matrix $A$. Suppose we have an algorithm for matrix multiplication which runs in $\mathcal{O}(M(n))$ time. Then, one can easily ...
14
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1answer
727 views

Finding a 5-Pointed Star in polynomial time

I want to establish that this is part of my homework for a course I am currently taking. I am looking for some assistance in proceeding, NOT AN ANSWER. This is the question in question: A 5-...
11
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1answer
610 views

A continuous optimization problem that reduces to TSP

Suppose I am given a finite set of points $p_1,p_2,..p_n$ in the plane, and asked to draw a twice-differentiable curve $C(P)$ through the $p_i$'s, such that its perimeter is as small as possible. ...
19
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3answers
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For every computable function $f$ does there exist a problem that can be solved at best in $\Theta(f(n))$ time?

For every computable function $f$ does there exist a problem that can be solved at best in $\Theta(f(n))$ time or is there a computable function $f$ such that every problem that can be solved in $O(f(...
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2answers
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What is the difference between "Decision" and "Verification" in complexity theory?

In Michael Sipser's Theory of Computation on page 270 he writes: P = the class of languages for which membership can be decided quickly. NP = the class of languages for which membership can be ...
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3answers
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Subset-sum and 3SAT

Two things (this may be naive): Does anyone believe there is a sub-exponential time algorithm for the Subset-sum problem? It seems obvious to me that you would have to look through all possible ...
7
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3answers
381 views

Subset Sum Requirements

Consider the following problem. Given a set $S$ of integers, a function $f : \mathbb{Z} \to \mathbb{Z}$ and $k \in \mathbb{Z}$, decide wether there is $X \subseteq S$ such that $f\left(\sum_{x\in ...
18
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4answers
571 views

Showing that a problem in X is not X-Complete

The Existential Theory of the Reals is in PSPACE, but I don't know whether it is PSPACE-Complete. If I believe that it is not the case, how could I prove it? More generally, given a problem in some ...
20
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1answer
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Complexity of Towers of Hanoi

I ran into the following doubts on the complexity of Towers of Hanoi, on which I would like your comments. Is it in NP? Attempted answer: Suppose Peggy (prover) solves the problem & submits it ...
30
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5answers
10k 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
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2answers
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Optimization version of decision problems

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 ...
62
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3answers
29k views

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 ...
4
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2answers
225 views

Is the number of coin tosses of a probabilistic Turing machine a Blum complexity measure?

I read that the number of coin tosses of a probabilistic Turing machine (PTM) is not a Blum complexity measure. Why? Clarification: Note that since the execution of the machine is not deterministic, ...
11
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1answer
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Proving that directed graph diagnosis is NP-hard

I have a homework assignment that I've been bashing my head against for some time, and I'd appreciate any hints. It is about choosing a known problem, the NP-completeness of which is proven, and ...
23
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2answers
6k views

NP completeness proof of a spanning tree problem

I am looking for some hints in a question asked by my instructor. So I just figured out this decision problem is $\sf{NP\text{-}complete}$: In a graph $G$, is there a spanning tree in $G$ that ...
6
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1answer
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Reducing directed hamiltonian cycle to graph coloring

The 3-SAT problem can be reduced to both the graph coloring and the directed hamiltonian cycle problem, but is there any chain of reductions which reduce directed hamiltonian cycle to graph coloring ...
14
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2answers
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Reduction from 3-Partition problem to Balanced Partition problem

The 3-Partition problem asks whether a set of $3n$ integers can be partitioned into $n$ sets of three integers such that each set sums up to some given integer $B$. The Balanced Partition problem asks ...
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2answers
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Decidable restrictions of the Post Correspondence Problem

The Post Correspondence Problem (PCP) is undecidable. The bounded version of the PCP is $\mathrm{NP}$-complete and the marked version of the PCP (the words of one of the two lists are required to ...
6
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1answer
761 views

Reduction to equipartition problem from the partition problem?

Equipartition Problem: Instance: $2n$ positive integers $x_1,\dots,x_{2n}$ such that their sum is even. Let $B$ denote half their sum, so that $\sum x_{i} = 2B$. Query: Is there a subset $I \...
2
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2answers
189 views

Non-trivial tractable properties of triples

Many intractable $NP$-complete problems can be modeled as deciding whether a set of triples, $F=${$t_1, t_2, ..., t_n$} where each triple $t_i$ is a subset of three elements over base set $U=${$a_1, ...
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3answers
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Logarithmic vs double logarithmic time complexity

In real world applications is there a concrete benefit when using $\mathcal{O}(\log(\log(n))$ instead of $\mathcal{O}(\log(n))$ algorithms ? This is the case when one use for instance van Emde Boas ...
61
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8answers
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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 ...
11
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3answers
459 views

Notions of efficient computation

A polynomial-time Turing machine algorithm is considered efficient if its run-time, in the worst-case, is bounded by a polynomial function in the input size. I'm aware of the strong Church-Turing ...
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3answers
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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. ...
30
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1answer
12k 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, ...
14
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1answer
300 views

Approximation of minimum bandwidth on binary trees

Minimum bandwidth problem is to a find an ordering of graph nodes on integer line that minimizes the largest distance between any two adjacent nodes. The decision problem is NP-complete even for ...
29
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3answers
909 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 ...
121
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6answers
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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. ...
19
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1answer
239 views

Decision problem such that any algorithm admits an exponentially faster algorithm

In Hromkovič's Algorithmics for Hard Problems (2nd edition) there is this theorem (2.3.3.3, page 117): There is a (decidable) decision problem $P$ such that for every algorithm $A$ that solves $P$ ...
15
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3answers
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Decidable non-context-sensitive languages

It is arguable that most languages created to describe everyday problems are context-sensitives. In the other hand, it is possible and not hard to find some languages that are not recursive or even ...
14
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3answers
277 views

Is there an abstract machine that can capture power consumption?

When reporting algorithmic complexity of an algorithm, one assumes the underlying computations are performed on some abstract machine (e.g. RAM) that approximates a modern CPU. Such models allow us to ...
9
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1answer
189 views

Influence of the dimension of cellular automata on complexity classes

Let's take as an example the 3d → 2d reduction: What's the cost of simulating a 3d cellular automaton by a 2d cellular automaton? Here is a bunch of more specific questions: What kind of algorithms ...
21
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1answer
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Easy reduction from 3SAT to Hamiltonian path problem

There is a reduction in Sipser's book "Introduction to the theory of computation" on page 286 from 3SAT to Hamiltonian path problem. Is there a simpler reduction? By simpler I mean a reduction ...
52
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4answers
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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 ...
20
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3answers
331 views

Extension of SQL capturing $\mathsf{P}$

According to Immerman, the complexity class associated with SQL queries is exactly the class of safe queries in $\mathsf{Q(FO(COUNT))}$ (first-order queries plus counting operator): SQL captures safe ...
22
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1answer
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Natural candidates for the hierarchy inside NPI

Let's assume that $\mathsf{P} \neq \mathsf{NP}$. $\mathsf{NPI}$ is the class of problems in $\mathsf{NP}$ which are neither in $\mathsf{P}$ nor in $\mathsf{NP}$-hard. You can find a list of problems ...
24
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2answers
326 views

Is Smoothed Analysis used outside academia?

Did the smoothed analysis find its way into main stream analysis of algorithms? Is it common for algorithm designers to apply smoothed analysis to their algorithms?
12
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3answers
11k views

All NP problems reduce to NP-complete problems: so how can NP problems not be NP-complete?

My book states this If a decision problem B is in P and A reduces to B, then decision problem A is in P. A decision problem B is NP-complete if B is in NP and for every problem in A in ...
20
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1answer
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Proving that the conversion from CNF to DNF is NP-Hard

How can I prove that the conversion from CNF to DNF is NP-Hard? I'm not asking for an answer, just some suggestions about how to go about proving it.

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