Questions related to the (computational) complexity of solving problems

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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
16 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
69 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 ...
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2answers
67 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 ...
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1answer
415 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 ...
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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. ...
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1answer
24 views

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

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 ...
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2answers
61 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 ...
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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 $\mathsf{F}$, ...
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1answer
62 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 ...
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2answers
74 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 ...
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1answer
140 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 ...
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0answers
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Prove that VC (Vertex Cover) is in NP [duplicate]

I have this question in which it states that I have to prove whether Vertex Cover is in NP. I'm trying to understand my notes but the problem is that I do not know whether this is actually valid I ...
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2answers
85 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 $P=NP$)?
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1answer
37 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
31 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
35 views

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 ...
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1answer
75 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 ...
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1answer
70 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
62 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 ...
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1answer
57 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 ...
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1answer
51 views

are there any decidable problems not verifiable in polynomial time?

As I understand it NP requires a solution to be verifiable in polynomial time. Can you provide examples of solvable problems not verifiable in polynomial time ?
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1answer
27 views

Poly-time computability of inversion of poly-time real functions

At pp. 7-8 of Ker-I Ko's Computational Complexity of Real Functions (1991), the following is stated for one dimensional cases: Let $INV_1$ be the operator that maps a one-to-one function ...
4
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1answer
69 views

Universal One-Way Functions

The Berman-Hartmanis conjecture discusses one-way functions (functions with hard to compute inverse functions). As a step to solving the conjecture, if one-way functions could be reduced to a ...
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1answer
31 views

Why does reduction from vertex cover to subset sum use base-4? [closed]

Why does reduction from vertex cover to subset sum use base-4? 30.13 Subset Sum (from Vertex Cover)
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2answers
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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 ...
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1answer
108 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 ...
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1answer
66 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 ...
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1answer
40 views

Is there a limit on the length of queries for the oracle?

Consider the class $L^A$ which contains all the languages that are decided by a deterministic Turing Machine that uses $O(log(n))$ space and that can make queries to an oracle that decides $A$. My ...
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0answers
30 views

Does $\#W$[1]-hardness imply approximation hardness?

Let $\Pi$ be a parametrized counting problem, where the parameter is the solution cost, e.g. counting the number of $k$-sized vertex cover in a graph, parametrized by $k$. Assume that $\Pi$ is ...
5
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0answers
103 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 ...
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4answers
202 views

Recovering a point embedding from a graph with edges weighted by point distance

Suppose I give you an undirected graph with weighted edges, and tell you that each node corresponds to a point in 3d space. Whenever there's an edge between two nodes, the weight of the edge is the ...
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1answer
66 views

Reduce Set problem to SAT

So the problem is, given some set $M = \{x_1,x_2,\ldots,x_n\}$ and a set of subsets $S = \{S_1, S_2, \ldots, S_m\}$ where $S_i \subseteq M$. We want to find some set $X \subseteq M$ such that $|X| \le ...
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1answer
33 views

Is finding negative cycle vertices NP complete?

I was trying to find all the negative cycle vertices using the Bellman–Ford algorithm using this paper solution 7.1(b) in $O(V)$ by tracing back the predecessor subgraph.It is also stated in ...
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1answer
47 views

What are the current known implications of the complexity of Integer Factorization?

According to my limited knowledge we know that since Integer Factorization lies in the intersection of NP and co-NP it cannot be NP-complete unless NP=co-NP. However, do we know any other ...
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1answer
22 views

What is an oracle separating IP and PSPACE?

I saw this link on cstheory: http://cstheory.stackexchange.com/questions/6634/is-there-an-oracle-that-separates-two-complexity-classes-known-to-be-equal but it did not provide specific details. Can ...
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2answers
153 views

Generating a set of minimal-length strings that, together, invoke every production of a context free language

Problem (tl;dr) Given a context free grammar, $G$, find a set of strings that take $G$ through every production it has at least once. How and how fast can it be done? Background I'm working on a ...
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1answer
106 views

Can an NP-hard problem be polynomial on average?

I'm wondering if there are any $NP$-hard problems which are ``polynomial" in the average case. I think there are two ways to interpret this? If $P \neq NP$, can there be an algorithm solving an ...
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1answer
134 views

Are there are problems in NP that have been shown to be not NP-complete but it is still not known if they are in P or not?

I am not talking about NP-indeterminate class because those problems have to be shown to not exist either in P or NP-complete class and existence of such problems proves P!=NP. I am interested to know ...
3
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1answer
59 views

A Reduction from XORSAT to 2-SAT

Does anyone know of a non-trivial reduction from XORSAT to 2-sat since they are both in P? (By non-trivial I mean one that does not just solve the instance of XORSAT and map it to a fixed instance of ...
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1answer
29 views

Is random access allowed in the Bit Complexity model, or is it just expensive?

In the RAM model, you're allowed to do unbounded indirect access (pointers can be arbitrarily large and still fit in a single machine word). In the Bit Complexity model (no wiki article, sorry), ...
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1answer
73 views

How hard is this constrained $n$-rooks problem?

I asked this over on math.stackexchange.com, then I found out about this forum. Suppose you have an $(n\times n)$-chessboard, together with a constraining function $C : n \times n \to 2$ where ...
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2answers
343 views

Proof that Hamiltonian cycle/circuit with a specified edge is NP-complete

I'm a little stuck on this question, any help would be appreciated! Given that the Hamiltonian Path (HP) and the Hamiltonian Circuit/Cycles (HC) problems are known to be NP-complete, show that HCE is ...
3
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1answer
58 views

What happens to quantum algorithms such as BB84 if P=NP

Under the hypothesis that P=NP, many cryptographic protocols are no longer secure (i.e. attacks are feasible). The BB84 algorithm (http://en.wikipedia.org/wiki/BB84) is based on the idea that by ...
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1answer
29 views

Why is there only a polynomial number of provers in multi-prover interactive protocols?

The paper On The Power of Multi-prover Interactive Protocols by Fortnow, Rompel, Sipser states the following: There are provers $P_1, P_2, \dots, P_k$ in a multi-prover interactive proof system such ...
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1answer
36 views

A detail on variant of Mahaney's theorem about reductions of sparse languages vs P/NP

Wikipedia states on sparse languages that There is a Turing reduction (as opposed to the Karp reduction from Mahaney's theorem) from a NP-complete language to a sparse language iff NP $\subseteq$ ...
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1answer
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Easy infinite subclass of cubic graphs for Hamiltonian cycle problem

I know that Hamiltonian cycle problem is $NP$-complete for 2-connected planar bipartite cubic graphs. I'm interested in non-trivial infinite subclass of cubic graphs where the Hamiltonian cycle ...
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2answers
128 views

Does this mean $P = NP$

I am not a formally trained guy on Complexity theory, but due to interest I am learning it. Based on different feedbacks, I have started my journey with Micheal Sipser's "Theory of Computation" (2013 ...
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1answer
35 views

First Order interpretation of arbitrary structures as a graph

I am currently trying to get some intuition on the concept of First Order reductions, and have come across this exercise question by Immerman, dubbed "Everything is a Graph". Given some arbitrary ...
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5answers
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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 ...