Questions related to the (computational) complexity of solving problems

learn more… | top users | synonyms (3)

3
votes
1answer
42 views

Negligible functions in definitions of statistical closeness and computational indistinguishability

Statistical closeness implies computational indistinguishability. Is there any (simple) relationship between negligible function that is used in definition of statistical closeness and negligible ...
-1
votes
2answers
56 views

Find the optimal way [on hold]

We consider the TSP in Grid-City. The roads in Grid-City have the form of a grid, so that the intersection points can be described by an integer coordinate system. The distance of $2$ points $C=(x,...
3
votes
0answers
33 views

Quantum circuits for multiply-accumulation

Classically, multiplication can be done in $O(n \ \lg(n) \ 8^{\lg^* n})$ steps on a multi-tape Turing machine via Fürer's algorithm. Using that algorithm, combined with uncomputing, you can make a ...
2
votes
1answer
34 views

Is there a similar categorization to Elementary Cellular Automata?

Elementary cellular automata show different systems and fractals like rule 30, rule 90, and rule 110. I want to know if there is another classification or type which shows similar structures just like ...
0
votes
0answers
8 views

effect on storage of Tertiary language [duplicate]

Binary language or machine language is the basic language of computer system and it consists of zeros and ones. As it is a universal fact; that modern digital computers understand and store data in ...
1
vote
1answer
23 views

Checking membership in DFA with fixed length using AC1 circuit?

I'm supposed to find circuits , which can solve the question of membership in a regular language A with fixed length. The depth is limited by O(log(n)) and the size by O(n). Divide and Conquer should ...
2
votes
1answer
46 views

Finding integer square root for large integers [find asymptotic time complexity]

So I found this tasks in one book I am practicing from where it says: "Find a divide-and-conquer algorithm for finding square roots for large integers and along this, find its asymptotic time ...
0
votes
2answers
95 views

AC-3 Algorithms on CSP problem, What is happened when enocunter to an empty domain variable?

Suppose We Applying Arc-Consistency (AC3) algorithms on one Constraint Satisfaction Problem, if domain of one variable be empty, what is the next step of this algorithm? According to This Link and ...
3
votes
1answer
25 views

Proving complexity of computing product of matrices

If $A$ is a non-singular $n\times n$ matrix, $B$ is an $n\times p$ matrix, and $C$ is a $p\times n$ matrix (where $1\le p \ll n$), how does one prove that the complexity of $$D=A^{-1}(BC)$$ is $\frac{...
0
votes
1answer
47 views

How to prove intersection between languages L1 (belongs to NP) and L2 (belongs to P) actually belongs to NP?

I have to prove that if L <=p L1 intersection L2, where L1 and L2 are described as above, L belongs to NP. I thought about the definitions of P and NP and built a DTM D that decides L2 and a NTM N ...
0
votes
0answers
27 views

Computational Complexity of Exponential Expression Divisibility?

Garey and Johnson mentions the problem of 'Exponential Expression Divisibility" as "Not known to be in NP or co-NP, but solvable in pseudo-polynomial time using standard GCD Algorithms". Pseudo-...
2
votes
0answers
66 views

Is there a philosophical counterpart question to P != NP?

Gödels motivation to prove his incompleteness theorems was the philosophical statement "This sentence is wrong.". Is there a philosophical counterpart to the statement P != NP? For example such ...
2
votes
2answers
121 views

How can I efficiently find the optimal order to apply special offers to a shopping cart?

Given a list of items which represent items in a shopping cart, and a list of available special offers which replace one or more regular items to lower the cost of those items, how can I decide the ...
7
votes
2answers
89 views

Why is first-order logic (without arithmetic) VALIDITY only recursively enumerable, and not recursive?

Papadimitriou's "Computational Complexity" states that VALIDITY, the problem of deciding whether a first-order logic (without arithmetic) formula is valid, is recursively enumerable. This follows from ...
0
votes
1answer
59 views

Does the fact that there exists a polynomial time quantum algorithm for integer factorization suggest that integer factorization is in P?

Just as the title says: Does the fact that there exists a polynomial time quantum algorithm for integer factorization suggest that integer factorization is in P? Additionally, if one could show that a ...
4
votes
1answer
36 views

Question regarding Karp-Lipton theorem

In the proof in wikipedia, it goes like this: Let $L \in \Pi_{2}$, so we can describe membership in $L$ as a formula: $\forall_{y}\exists_{z} V(x,y,z)=1 \iff x \in L$ (where V is polynomial ...
-1
votes
0answers
33 views

Prove that all P problems except {} and {a,b}* are complete [duplicate]

It is easy to say that {} and {a,b}* are not P complete because other problems in P can't be reduced to these because ...
6
votes
1answer
81 views

Why do we reject turing machines that use space less than the log of the length of the input?

In Computational complexity: Modern Approach by Arora and Barak, it's mentioned that We will require however that $S(n)> \log n$ since the work tape has length $n$, and we would like the ...
2
votes
2answers
36 views

Does 'subexponential algorithm' refer to input or number of bits used to represent input?

When an algorithm is said to be subexponential - does this refer to the input N or the number of bits used to represent N? Consider the following: trial division for integer factorization (i.e. try ...
1
vote
1answer
39 views

Label coloring to maximize number of “balanced” triangles (NP-hardness)

Define a triangle in undirected graph $G$ is balanced if the edge labels in the triangle are $(+1, +1, +1)$, $(-1, -1, +1)$, $(+1, -1, -1)$ or $(-1, +1, -1)$ (social balance theory). Problem ...
3
votes
1answer
30 views

The class of languages that can be certified in a small amount of space

NP can be characterized in two different ways, one of them is that it's the class of languages that can be certified by a witness in a polynomial time. I wonder, if we consider the same notion but ...
0
votes
1answer
25 views

General number field sieve is slower then exhaustive search for 'small' numbers?

In an attempt to understand the efficiency of the GNFS, I've been looking at runtimes. The calculations seem to indicate the GNFS runs slower than exhaustive search for smallish n. For example: ...
0
votes
1answer
36 views

Why does using unary in subset sum problem result polynomial time complexity?

From my understanding, the complexity of the algorithm is O(number of inputs * number of bits for input). The number of bits in binary notation is obviously less ...
2
votes
1answer
25 views

Is the runtime for the general number sieve given in base 10, e or 2?

When the runtime of the GNFS is given as e^(64/9*b(log b)^2)^1/3, what base is the log? I'm assuming its e, but other options would obviously be 10 and 2.
1
vote
1answer
24 views

Hardness amplification of PRFs by increasing key length

I was reading the GGM construction for PRFs and wondering the relation between key length and hardness. GGM construction does not seem to yield any significant improvements. Are there any PRF ...
3
votes
1answer
16 views

Using oracle machine to speed up tree solution search

Let $P$ be a boolean problem of size $n$, thus the complete solution search space tree is of size $2^n$. Applying simple tree search for the solution will have take $O(2^n)$ operations, (for ...
0
votes
0answers
25 views

Time complexity of fixed point program

Suppose $\mathcal{M}_f$ is a Turing machine that computes the total function $f(x)$ in time $T_{\mathcal{M}_f}(|x|)$. Also suppose $M_H$ is a Turing machine that computes the total function $H(n,x)=\...
0
votes
0answers
52 views

What's the complexity of the problem of optimally distribute n balls in m boxes?

Assuming that there is a function $f(x)$ (non linear, non convex) where $x$ is the vector $[n_1,n_2,\dots n_m]$ where $n_i$ is the number of balls in the box $i \in \{1,\dots m\}$, and $\sum \limits_i^...
5
votes
5answers
186 views

Why does restricting size of input for NP complete problem imply a runtime of O(1)?

I've seen this statement mentioned a few times here on cs.stackexchange and have not been able to follow the logic. The statement is 'If you restrict the input size of the problem then solving that ...
0
votes
1answer
135 views

A version of the longest simple cycle problem - NP-completeness reduction proof

I've been learning about proving NP-completeness via reduction, and came across the following problem: Prove via reduction the following: whether a graph $G = (V, E)$ contains a simple cycle using $\...
3
votes
1answer
83 views

Show Recognizing two Regular Expressions as equal is in PSPACE

If I have $EQ_{REX} = \{\langle R,S \rangle|\text{ $R$ and $S$ are equivalent regular expressions}\}$, how do I show that $EQ_{REX}\in PSPACE$ ? What I know so far is that there are decidable ...
0
votes
0answers
20 views

Why does not the result in this notes show P is not NP in BSS model?

As far as I know (not understand that well though) BSS model is a real computation model and $P_{\Bbb R}\neq NP_{\Bbb R}$ is a real analog of $P\neq NP$ problem in BSS model. The lecture notes http://...
0
votes
0answers
30 views

Consequence of P=NP in BSS model

We know that in the Valiant model the result VP=VNP would imply NP is in P/poly. Do we know any consequences in case P=NP holds in BSS model?
2
votes
3answers
205 views

Asymptotic equivalent of the recurrence T(n)=3⋅T(n/2)+n

The questions is to find the running time $T(n)$ of the following function: $$T(n)=3\cdot T(n/2) + n \tag{1}$$ I know how to solve it using Master theorem for Divide and Conquer but I am trying to ...
1
vote
1answer
34 views

How do I find running time for the following divide and conquer problem?

Question is to find the runtime $T(n)$ of following problem by solving the recurrence. $T(n)=16\cdot T(\frac{n}{4}) + n!$. I went through the following theory. If the recurrence relation is of the ...
2
votes
2answers
142 views

Scheduling a sequence of queue operations that push and pop items at specified times

What is the time complexity of the following problem? Definitions A FIFO is a queue functional unit with the commands: PUSH (data to back of queue), POP (the head of the queue), PNP (POP the head of ...
2
votes
0answers
14 views

Uses of unary or sparse languages in other models

In the turing model we have the statements that if there is an unary or sparse language that is NP complete then P=NP and if there is a Turing reduction from an NP complete problem to an unary or ...
1
vote
0answers
27 views

Single-tape Universal Turing Machine time complexity

When studying the time-hierarchy and space-hierarchy theorems, the main idea is to use a simulation by the universal TM. It is mentioned that the time bound is increased by a logarithmic factor while ...
0
votes
1answer
37 views

Relating memory complexity and decidablity

Given a language $L_u$, about which we know that there exists a non-deterministic turing machine which accepts it (as in, implying $L_u \in RE$) with memory complexity of $c^{p(n)}$, where $c$ is a ...
-1
votes
0answers
85 views

Complexity of Polynomial Division

Given two polynomials $A(x)$ and $B(x)$. What is a fast way to find $Q(x)$ and $R(x)$ such that $A(x) = Q(x) \times B(x) + R(x)$ (and the degree of $R(x)$ is less than degree of $B(x)$) I know that ...
1
vote
1answer
86 views

Problems that are easy on bipartite but hard on general graphs

Are there any problems that are easy for bipartite graphs, but hard for general graphs? I am asking because some classical problems are formulated in reference to people looking for a spouse, such as ...
2
votes
0answers
74 views

Understanding Levin's Universal Search

I am having troubles understanding Levin's universal search method. In Scholarpedia, http://www.scholarpedia.org/article/Universal_search, it is claimed that “If there exists a program $p$, of length $...
1
vote
1answer
30 views

Is the prime factorization problem not an instance of the change making problem?

When using as the set of coins all logarithms of the prime numbers or numbers in general, and when using the logarithm of the number to be factored. The problem is just finding the logarithms that can ...
5
votes
3answers
357 views

Is Green's the best 16-input sorting network so far?

Every paper says that Green's construction is the best 16-input sorting network as for now. But why does Wikipedia says: "Size, lower bound: 53"? I thought "lower bound" meant:"If there exists at ...
4
votes
1answer
90 views

is the problem of parallelising any program, NP-complete?

Consider a program written in a common language such as C. Assume that it does not have any explicit parallel constructs. Then, once it is compiled to an executable program, it will be run serially, ...
1
vote
1answer
99 views

Variants of the 3-Partition problem

The 3-Partition problem (wiki) is a $\text{NP}$-complete problem which is to decide whether a given multiset of integers can be partitioned into triples that all have the same sum. It is well-known ...
0
votes
0answers
28 views

Are there any methods to quantify complexity of finite problems?

On page 348 of "Sipser M. Introduction to the Theory of Computation. Cengage Learning; 2012 Jun 27", it says Perhaps at some time in the future, methods that can quantify the complexity of finite ...
0
votes
1answer
54 views

Is this argument wrong “since DOM is special kind of RDOM, then RDOM is NP-hard”?

The domination problem $DOM$ is defined as $$ DOM = \{ \langle G,k \rangle\ | \ G \text{ has a domination of size } k, K \in \mathbb{N} \}, $$ and the rainbow domination problem $RDOM$ is defined as $$...
0
votes
0answers
18 views

Find reduction from Hamiltonian Cycle to Double Hamiltonian Cycle

$$DoubleHC=\{G\,| \text{G has at least two Hamiltonian Cycles}\}$$ I think about take a graph with HC and add to it two vertexes and edges to two randomally vertexes, but without success. Is my try ...
23
votes
1answer
865 views

Is Logical Min-Cut NP-Complete?

Logical Min Cut (LMC) problem definition Suppose that $G = (V, E)$ is an unweighted digraph, $s$ and $t$ are two vertices of $V$, and $t$ is reachable from $s$. The LMC Problem studies how we can ...