# Tagged Questions

**1**

vote

**2**answers

102 views

### Combinatorial optimization problem involving infinite spin system

In material science research, I am developing an algorithm to solve an infinite combinatorial optimization problem which I believe is the most natural problem when the system size goes to infinity.
...

**2**

votes

**1**answer

87 views

### enumeration of connected blocks in finite size square

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected?
By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we ...

**1**

vote

**0**answers

39 views

### Complexity of an algorithm to solve linear diophantine equations

A friend of mine ask me yesterday a problem, however, the interesting part for me it is not his problem, but what I will ask here.
I want to know the optimal complexity of an algorithm (I mean the ...

**8**

votes

**2**answers

172 views

### Is this problem on weighted bipartite graph solvable in polynomial time or it is NP-Complete

I encounter this problem recently and I want to know whether it is NP-Complete or solvable in polynomial time:
Given a undirected weighted bipartite graph $G = (V, E)$ where $V$ can be partitioned ...

**2**

votes

**0**answers

166 views

### Computing the chromatic polynomial of graph modulo $x-3$

The chromatic polynomial of graph $P(G,x)$ is univariate
polynomial which counts the number of colorings of $G$
with $x$ colors for natural $x$.
Graph is not $k$ colorable iff $P(G,k)=0$.
The ...

**1**

vote

**0**answers

140 views

### Is finding a single vector in the null space as difficult as discovering the whole null space?

Let $A \in \mathbb R^{k\times n}$ be a matrix of rank $k$, where $k \ll n$. One can use Gaussian eliminations to discover $\operatorname{null}(A)$ at the cost of $O(nk^2)$. My question is:
Is the ...

**7**

votes

**1**answer

550 views

### Can the Legendre symbol be calculated in polynomial time?

Is there an algorithm which on input "$(a,p)$" (where $0\leq a<p$ are integers) takes time polynomial in $\log p$ and outputs "NOT PRIME" if $p$ is not prime and otherwise outputs the Legendre ...

**1**

vote

**1**answer

35 views

### How to select a subset of points from a universal to minimize the distance from outside to inside?

Here is the detailed problem.
I have a set of N points in K-dimension space, called U, and I want select M points of them, called S. For each point p in U, we define the distance from p to S as
$$ ...

**1**

vote

**1**answer

81 views

### Is undirected short-simple-path-through-3-vertices decidable in polynomial time?

Consider the following language:
$L=\{\langle G=(V,E),s,v,t,l\rangle\;|\;s,v,t\in V, l\in \mathbb{N} \wedge $ There exists a simple path from $s$ to $t$, going through $v$ of length $\leq l\}$.
($G$ ...

**2**

votes

**1**answer

509 views

### Efficient algorithms to determine the roots of: $p(x) = r^x $ in the finite field $GF(q)$, where $r$ a primitive root of the field

I need to make sure that no efficient (i.e., polynomial time) algorithm exists for the following problem:
Exponentiating Polynomial Root Problem (EPRP)
Let $p(x)$ be a polynomial with $\deg(p) \geq ...

**12**

votes

**2**answers

516 views

### Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product
$$ p=b_1 b_2 \cdots b_n$$
where each $b_i\in A$.
Clearly $n-1$ multiplications suffice to compute $p$; ...

**1**

vote

**1**answer

205 views

### Does this algorithm terminate in all scenarios?

Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x_1, \dots, x_n\}$ and $B = \{x_{n+1}, \dots, x_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in ...

**6**

votes

**4**answers

494 views

### How long does it take to compute a class number?

I was wondering if there are any known (upper and lower) bounds for the complexity of computing the class-number of a finite extension of the rationals. (A general bound should be in function of the ...

**3**

votes

**2**answers

210 views

### Complexity of a problem remotely related to the discrete logarithm $A=x g^x$

Let $x,g \in \mathbb{F}_p^\ast$.
Given $g$ and either
$$ A = x g^ x$$
or
$$ A = x g^{x^2-1}$$
find $x$.
What is the complexity of solving this?
Is there a reduction to the discrete ...

**5**

votes

**5**answers

302 views

### Efficient Hamiltonian cycle algorithms for graph classes

Generally speaking finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $G$ then we can reduce the finding of a Hamiltonian cycle in $G$ to a Eurler your of $H$ ...

**2**

votes

**1**answer

123 views

### Complexity of numerically solving systems over the reals

Basically I am interested in
What is the complexity of numerically solving systems over $\mathbb{R}$?
By solving I mean finding at least one numeric solution with given
precision.
Probably the ...

**1**

vote

**0**answers

108 views

### Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...

**1**

vote

**1**answer

107 views

### Separation of Anti-Hole Inequality

Given an undirected graph $G=(V,E)$ with no loops or multiple edges, a stable set is a set of vertices for which no two vertices are adjacent.
An induced subgraph $H$ of $G$ is called an odd-antihole ...

**6**

votes

**2**answers

731 views

### Approximate number of primes below a given integer?

The problem of the complexity of the exact counting problem for primes is interesting. The best result we have about primes is that it is hard for TC0. But counting the number of witnesses to a TC0 ...

**6**

votes

**1**answer

243 views

### How do you compute the primes of bad reduction?

Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some ...

**4**

votes

**1**answer

391 views

### Exact arithmetic for real algebraic numbers

There was a reply to a question (that I can't find) which mentioned SARAG (Some Algorithms
in Real Algebraic Geometry) see http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted2.html. This ...

**2**

votes

**1**answer

225 views

### Complexity of establishing finite groups (non)-isomorphism ?

Question Given two finite groups G and H of the same order N what are the algorithms and what is their complexity (in terms of N) to check is G isomorphic to H or not ? Is there polynomial in N ...

**2**

votes

**2**answers

250 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**0**

votes

**1**answer

93 views

### Provided a list of sets, $L$, computing an array where each entry $q_i \in Q$ is the family of sets in $L$ that have intersection $k$ with $l_i \in L$

I have a set of $(l_1, ..., l_N) \in L$ smaller sets, each with $(r_1, ..., r_M) \in R$ integer elements. I would like create an ordered array of $(q_1, ..., q_N) \in Q$ sets s.t.:
(1) Each $q_i \in ...

**1**

vote

**0**answers

90 views

### Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?

Did anybody come across a computational problem which is related to the notion of cartesian product and is at least NP-hard?
Equally interesting would be to learn about such problems with a ...

**2**

votes

**0**answers

176 views

### A natural problem on “cartesian union” of set families (hypergraphs). Does anybody know NP-complete problems related to the notion of cartesian product?

I'm curious whether the problem below is NP-complete.
I provide two simple definitions and one example at first.
Definition 1.
Let $\langle {\cal{S}}_i\rangle\substack{i\in I}$ and $\langle ...

**1**

vote

**0**answers

136 views

### Randomized alternative to Buchberger's algorithm

Richard Lipton's blog describes a A New Way To Solve Linear Equations by Prasad Raghavendra.
Can the ideas in this algorithm be generalized to systems of polynomial equations to provide a randomized ...

**0**

votes

**1**answer

237 views

### Algorithm for vector space

I have $n$ vectors $e_1 \in (\mathbb Z/2 \mathbb Z)^m,\dots,e_n \in (\mathbb Z/2 \mathbb Z)^m $
and a vector $ v \in (\mathbb Z/2 \mathbb Z)^m $
I need to find the better algorithm which answers ...

**3**

votes

**2**answers

1k views

### Generation of All Path in a Directed Acyclic Graph

I am working on a very large dataset of a single DAG whose vertices have a low branching factor. I need to generate all possible (simple) paths starting from the source and write them to a file.
My ...

**12**

votes

**2**answers

465 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**10**

votes

**1**answer

634 views

### A generalization of the triangle counting problem for simple weighted graphs

One nice identity is $$tr(A^3)/6$$ which counts the number of triangles of a graph represented with adjacency matrix $A.$ It also implies that triangle counting can be performed in subcubic time.
...

**3**

votes

**2**answers

164 views

### Determination of rationality and computing a rational parametrization

Suppose I have a hypersurface in $\mathbb{C}P^n$ given by some $f(z_1, \dots, z_{n+1}) = 0.$ Is there an algorithm which returns a rational parametrization if there is one, and "not rational" ...

**18**

votes

**1**answer

2k views

### How fast can we *really* multiply matrices?

Background: The Strassen Algorithm, described here, has a computational complexity of $\text{O}(n^{2.8})$ for the multiplication of two $n \times n$ matrices. However, the constant is so large that ...

**1**

vote

**0**answers

76 views

### Deciding / Approximating Parity of Small Depth Decision Trees

Let C be a circuit such that:
C: $\{0,1\}^n$ to $\{0,1\}$
the top most gate is a parity gate
all the inputs to the parity gate are small depth decision trees
there is a total of $2^{ log^k n}$ ...

**1**

vote

**3**answers

1k views

### How to get the largest subset of a set of sets of intervals with no overlapping intervals

Given: Set of Set of Intervals. Example {{(1,2), (3,4)}, {(1, 3)}, {(13,14)}}
Wanted Result: Largest Subset, in which no Interval overlaps with another from every Set pairwise.
Example:
Input ...

**2**

votes

**3**answers

275 views

### Generating a set of integer passwords that can be securely authenticated

First, apologies for the title. This is an odd question, and I couldn't come up with a simple title for it.
My question is as follows.
Given a positive integer $k$, determine a set of properties ...

**4**

votes

**1**answer

550 views

### Lovász $\delta$ condition for LLL Algorithm

http://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm
What is the importance of the $\delta$ parameter for LLL bases called Lovász condition?
...

**1**

vote

**0**answers

608 views

### How to solve simple bilinear equations under extra linear constraints

Hello,
This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy:
$\lambda_i^T ...

**7**

votes

**2**answers

863 views

### Distribution of the computable numbers on the real number line

If we order all the positive computable real numbers $r_1,r_2,r_3...$ by their Kolmogorov complexity in some language $L$, then make a histogram plot of the $r_i$ on the real line, and we scale it ...

**5**

votes

**2**answers

417 views

### Complexity of Membership-Testing for finite abelian groups

Consider the following abelian-subgroup membership-testing problem.
Inputs:
A finite abelian group $G=\mathbb{Z}_{d_1}\times\mathbb{Z}_{d_1}\ldots\times\mathbb{Z}_{d_m}$ with ...

**6**

votes

**2**answers

376 views

### Complexity of detecting a convex body in $\mathbb{R}^n$?

Let $K_0$ be a bounded convex set in $\mathbb{R}^n$ within which lie two sets $K_1$ and $K_2$. $K_0,K_1,K_2$ have nonempty interior. Assume that,
$K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$.
The ...

**10**

votes

**1**answer

522 views

### Complexity of a weirdo two-dimensional sorting problem

Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...

**12**

votes

**1**answer

2k views

### Evidence for integer factorization is in $P$

Peter Sarnak believes that integer factorization is in $P$. It is a well-known open problem in TCS to identify the real complexity class of integer factorization. Take a look at this link for Peter ...

**4**

votes

**1**answer

825 views

### Finding a vertex of least distance to all other vertices in a graph

Some background to my question: if $G$ is a (simple) graph of $N$ vertices, labelled by integers $0,1,...,N-1$, the closeness centrality of a vertex $i$, denoted by $C(i)$, is defined to be the ...

**0**

votes

**2**answers

210 views

### Formal verification in complexity theory

Reading books and papers on complexity theory, I am struck by the extreme degree to which proofs are stated in an intuitive, hand-wavy way. The alternative is to give a lot of details about the coding ...

**8**

votes

**3**answers

688 views

### Definition of relativization of complexity class

Is there any general definition, for a class $C$ of languages, what is the relativized class $C^A$ for an oracle $A$?
Usually, these classes and their relativizations seem to be defined in an ad-hoc ...

**2**

votes

**0**answers

339 views

### Hamiltonian paths in subgraphs of rectangular lattice graphs

Is following decision problem NP-hard / NP-complete:
Having vertex-induced subgraph of rectangular lattice graph determine if any Hamiltonian path exists
Having vertex-induced subgraph of ...

**3**

votes

**0**answers

278 views

### Amortized analysis of data structure via potential function

One common method for proving that a data structure supports an operation in $O(f(n))$ amortized time is to construct a potential function $\Phi: \mathcal S \rightarrow \mathbf R^{+}$, which ...

**4**

votes

**1**answer

263 views

### Computing Simultaneous Hamming Neighborhood for a Set of Strings

Let $S = \lbrace s_1, s_2 \ldots s_n \rbrace$ be a set of strings each of length $k$ from an alphabet $\Sigma$, $h(s_i, s_j)$ denote the hamming distance between two strings. The simultaneous hamming ...

**5**

votes

**3**answers

652 views

### SDP Feasibility

I have a decision problem that I have formulated as a feasibility SDP. The answer to the decision problem depends on whether the SDP is feasible or not. It is known that a SDP can be solved to ...