Questions related to the (computational) complexity of solving problems

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1answer
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Are regex crosswords NP-hard?

I was fooling around the other day on this website: http://regexcrossword.com/ and it got me wondering what the best way to solve it was. Can you solve the following problem in polynomial time or is ...
0
votes
2answers
67 views

Finding an exactly weighted st-path in a digraph

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
1
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1answer
117 views

Is finding if a graph has k isolated nodes a NP-Complete problem?

I was wondering if finding if a graph has k or more isolated nodes is a NP-Complete problem. I found the following problem: Prove that the following problem is NP-Complete. Given a set of T ...
0
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0answers
14 views

Polynomial time optimisation algorithm for a poly-time computable function with bounded number of maxima?

Suppose we have a polynomial time algorithm for computing a function (we think of as existing on rational numbers between $0$ and $1$ of limited binary length n). We know that this function is made up ...
7
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2answers
135 views

Are there established complexity classes with real numbers?

A student recently asked me to check an NP-hardness proof for them. They performed a reduction along the lines of: I reduce this problem $P'$ that is known to be NP-complete to my problem $P$ ...
2
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1answer
35 views

Lower space bound on a turing machine accepting palindromes

Let $$ PAL = \lbrace x \in \lbrace 0, 1, \# \rbrace^* | x = rev(x) \rbrace $$ How do I show that a turing machine deciding $PAL$ must use space $\Omega(\log n)$? I have a feeling that I need to use ...
0
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0answers
36 views

Do poly-computable differentiable functions on [0,1] with bounded number of turning points have poly-time computable inverse?

Given a polynomially computable continuous function which is a composite of m strictly monotone functions, can we guarantee the existence of polynomially computable inverse? The function I have in ...
-1
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1answer
25 views

Relationship between an NP-hard problems with the subsets of them (part 2)?

I asked two questions about NP-hard problems here Relationship between an NP-hard problems with the subsets of them? and here Does this manner of proof for being NP-hard is true? but unfortunately ...
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0answers
38 views

Does this manner of proof for being NP-hard is true?

I have a problem and I want to prove the problem is NP-hard. Thus, I considered a subset of the problem that has a minimum answer (NP-complete problem). Afterward, I proved there is not any solution ...
0
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1answer
83 views

NP-hardness proof, what is wrong with it?

My question is the following: If we have a problem divided into two versions, weighted and unweighted. Can we prove that the unweighted problem is NP-hard from the fact that the weighted problem is ...
4
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1answer
38 views

Understanding the Sipser-Gacs-Lautemann theorem

The class $BPP$ contains all the languages decided by a probabilistic Turing machine in polynomial time with probability of success more that 2/3 for every input. The class $\Sigma^p_2$ contains all ...
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1answer
42 views

What do we know about covering the edges of a graph by disjoint paths?

Two related things I have heard/know of are, (1) That there exists a polynomial algorithm to find a cover of the vertices by $k$ vertex disjoint cycles. (Can someone give a reference for this?) ...
6
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0answers
87 views

Complexity class for probabilistic approximation algorithms with bounded error

What's the name of a complexity class of optimization problems that have "bounded error probabilistic approximation algorithms"? Bounded error probabilistic version of APX (as BPP is bounded error ...
8
votes
1answer
70 views

What do complexity classes look like, if we use Turing reductions?

For reasoning about things like NP-completeness, we typically use many-one reductions (i.e., Karp reductions). This leads to pictures like this: (under standard conjectures). I'm sure we're all ...
10
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3answers
653 views

Decision problems in $\mathsf{P}$ without fast algorithms

What are some examples of difficult decision problems that can be solved in polynomial time? I'm looking for problems for which the optimal algorithm is "slow", or problems for which the fastest known ...
12
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2answers
260 views

Does every NP problem have a poly-sized ILP formulation?

Since Integer Linear Programming is NP-complete, there is a Karp reduction from any problem in NP to it. I thought this implied that there is always a polynomial-sized ILP formulation for any problem ...
-1
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1answer
36 views

How to reduce bin-packing problems? [duplicate]

This is my first time with reductions and I can't figure out how to do them. I have read the few standard examples that are given in the standard books. For example, given $n$ numbers $\{ 0 < ...
0
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0answers
53 views

Homomorphism erasing information

I would be grateful if anyone could help me with the tricky exerciese *7.52 from Sipser's Introduction to the Theory of Computation 3rd ed. I got stuck in proving that, if P is closed under ...
2
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2answers
101 views

Known lower bounds on halting for finite machines?

It's possible to determine whether a deterministic machine with finite memory will halt in O(n) time if the machine has n possible states. You simply run the machine until it halts or visits the same ...
5
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1answer
78 views

Complexity of Pythagorean triples

We define a Pythagorean triple as a triple $\langle a,b,c\rangle$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $\langle a,b,c\rangle$ is legit ...
6
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0answers
80 views

Problems with Θ(n³) complexity on TMs with lower bounds by communication complexity arguments

One of the most used simple examples of application of Communication Complexity is the $\Omega(n^2)$ lower bound for recognizing palindromes of length $2n$ on a single tape Turing machine. Is ...
14
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1answer
186 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 ...
11
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3answers
361 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 ...
67
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5answers
16k views

In basic terms, 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 ...
6
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3answers
929 views

How fast can we find all Four-Square combinations that sum to N?

A question was asked at Stack Overflow (here): Given an integer $N$, print out all possible combinations of integer values of $A,B,C$ and $D$ which solve the equation $A^2+B^2+C^2+D^2 = N$. ...
5
votes
2answers
197 views

Direct NP-Complete proofs

I'm just starting to learn about NP-completeness. While I understand that reducibility plays a key role in this, I'm astonished how few problems I've been able to find who's proof that they are ...
2
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1answer
90 views

Is finding a solution of a satisfiability problem harder than deciding satisfiability?

Is the problem of determining whether or not a given Boolean expression is satisfiable computationally distinct from actually finding a solution to the expression? In other words, is there another ...
4
votes
1answer
127 views

Complexity of the decision version of determining a min-cut

I was wondering what the complexity of the following problem is: Given: A flow network $N$ with a source $s$, sink $t$ and a number $k$. Question: Is there an $s$-$t$ cut of capacity at most ...
6
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1answer
83 views

Computing the number of bits of a large power of integer

Given two integers $x$ and $n$ in binary representation, what is the complexity of computing the bit-size of $x^n$? One way to do so is to compute $1+\lfloor \log_2(x^n)\rfloor=1+\lfloor ...
5
votes
1answer
152 views

Why does the solution of an NP problem have to be polynomial size?

I've read in "Introduction to Algorithms" (CLRS) that formal language $L$ is NP-language if and only if there is a polynomial verification algorithm $A(x, y)$ and a constant $c$ such that ...
1
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2answers
66 views

NP-hardness of an optimization problem with real value

I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$. ...
4
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1answer
20 views

Probabilistic hardness of approximation or solution of NP-hard optimization problems under a probabilistic generative model for input data

So in biology (DNA sequences), sequence alignment is a generalization of longest common subsequence where an alignment of two sequences is scored typically with a linear function of how many spaces ...
4
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1answer
82 views

Complexity of Linear Diophantine equations

My question is simply, can linear Diophantine equations be solved in polynomial time? Specifically, I am looking at equations of the form $a_1 x_1+a_2 x_2 + ... + a_n x_n = k$, where $a_i,x_i,k$ are ...
5
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2answers
76 views

Exponential-size numbers in NP completeness reduction

In the proof of Theorem 4 in [GS'12], the authors reduce an instance of PARTITION to their problem. Therefore, they create for each element $a_i$ in the instance of PARTITION a number $2^{c \cdot ...
14
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1answer
418 views

Proving the (in)tractability of this Nth prime recurrence

As follows from my previous question, I've been playing with the Riemann hypothesis as a matter of recreational mathematics. In the process, I've come to a rather interesting recurrence, and I'm ...
14
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4answers
2k views

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. ...
0
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1answer
37 views

tightest upper bound on binary search tree insertion? [closed]

The upper bound on the runtime of binary search tree insertion algorithm is O(n) which is if it is not balanced What will be the tighter upper bound on this,will it become O(logn) I have read that ...
0
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2answers
65 views

Execution time of NP and NP-Complete algorithms

For P algorithms, we say that the execution time can be logarithmic O(log n), lineal O(n), quadratic O(n^2), etc. For NP and NP-Complete algorithms is there a way to represent the execution time? Or ...
4
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1answer
104 views

How to challenge Hutter's algorithm?

For a given sufficiently strong formal axiomatic system $\mathsf{F}$ (like $\mathsf{PA}$ or $\mathsf{ZFC}$) and any given function $p^*(x)$ that can be specified within the formal system $\mathsf{F}$, ...
0
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1answer
68 views

Problem with my proof that NP = coNP?

Is there a problem with this proof that NP = coNP? It suffices to show that Satisfiability can be solved efficiently with at most a polynomial number of queries to an oracle for Tautology. The ...
0
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2answers
86 views

Why does binary search take $O(\log n)$ time?

My question seems like an elementary question, but it's really not. Suppose I have one million cars in a line sorted in alphabetical order by license plate. I am standing at the top of the line with ...
7
votes
1answer
153 views

Is this classic puzzle book game NP-complete?

There is a classic puzzle book game very similar to a crossword puzzle, except a list of words is given and then a $N \times N$ square board made up of unit squares is given, with some squares blacked ...
11
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2answers
90 views

Does #$P$-Completeness imply approximation hardness?

Let $\Pi$ be some counting problem which is known to be #$P$-Complete. Does it imply that $\Pi$ is $APX$-hard (i.e. no PTAS for the problem exists unless $P=NP$)?
0
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1answer
41 views

Connection of “modern” runtimes and number of steps on a Turing machine

Why an evaluation of Turing machine efficiency is equal to the algorithm which is implemented by this machine and vise versa? For example, we can say that efficiency of merge sorting algorithm is ...
0
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1answer
33 views

Descriptive complexity: 3-colorability example

So in Neil Immerman's book http://books.google.co.kr/books?id=kWSZ0OWnupkC&pg=PA113&lpg=PA113#v=onepage&q&f=false, 3-colorability problem in descriptive complexity fashion is expressed ...
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3answers
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A graph in descriptive complexity - is $x$ already a vertex?

So suppose that there is an undirected graph with edge connections known. Now in first-order logic there is quantifier $\forall x$. Then does this automatically refer to vertexes, or can we use ...
2
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1answer
76 views

Is FACTORIZATION or PRIMES known to be in LOGSPACE

Are the integer factorization and PRIMES known to be in LOGSPACE? Recently, it has been shown by researchers that PRIMES is in P. But this does not say anything about LOGSPACE since it is not known ...
2
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1answer
74 views

Proof technique in complexity theory

I have a (stupid ?) question about complexity theory. It's about a "proof technique". I want to compare 2 models of computation. I want to prove that for each langage recognized in polynomial time by ...
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1answer
65 views

Reduction to Maximum Independent Set

Suppose you had a set $P$ of people. Every person $p_j \in P$ is familiar with atleast one other person $p_i$ (familiarity is symmetric). Is there a subset $S$ of people such that for $|S| \ge k$, no ...
1
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1answer
58 views

Name for class of algorithms preserving accuracy/confidence

I am considering the following class of algorithms: The algorithm has access to some probabilistic oracle (procedure) $f$ in addition to input. The answer of procedure $f$ (we may assume it is ...