8
votes
2answers
163 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
0answers
162 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
0answers
139 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
1answer
528 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
1answer
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
1answer
79 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
1answer
475 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 ...
11
votes
1answer
410 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
1answer
203 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
4answers
490 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
2answers
208 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
5answers
289 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
1answer
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
0answers
104 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
1answer
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
2answers
727 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
1answer
240 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
1answer
375 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
1answer
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
2answers
248 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
1answer
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
0answers
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
0answers
172 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
0answers
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
1answer
236 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
2answers
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
2answers
444 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
1answer
607 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
2answers
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" ...
17
votes
1answer
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
0answers
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
3answers
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
3answers
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
1answer
542 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
0answers
595 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
2answers
860 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
2answers
415 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
2answers
375 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
1answer
520 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
1answer
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
1answer
812 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
2answers
208 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
3answers
675 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
0answers
338 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
0answers
274 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
1answer
262 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
3answers
647 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 ...
1
vote
2answers
1k views

Greedy approach to 0-1 Knapsack problem in specific instances

The 0-1 knapsack problem is known to be NP-complete, and the greedy approach by Dantzig (based on choosing on the basis of density or value/weight) can be shown to be suboptimal using counterexamples. ...
2
votes
3answers
2k views

Worst known algorithm in terms of Big-O (more precisely Big-theta)?

Hello, I have been trying to find the worst algorithm in terms of it's Big-O function. By worst I mean n! is worse than n^2, n^n is worse than n!, etc. Essentially the worst algorithm would be the ...
1
vote
1answer
280 views

Computation for composition of polynomials

Let $R$ be a ring, $f(X)$ be a polynomial with coefficients in $R$ of degree $n$. It's known that for any $\alpha \in R$, one can evaluate $f$ at $\alpha$, i.e compute $f( \alpha) $ in $O(n)$ ...