All Questions
598 questions with no upvoted or accepted answers
3
votes
0
answers
145
views
2-ball billiards in a circle
Consider a 2D circular billiards table with diameter 1m containing two
balls with diameter 0.25m. Let each ball start with a speed of 1m/s.
In general, this speed could change after the balls hit ...
- 1,547
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781
3
votes
0
answers
50
views
Modular counting of integral points under sparse non-negativity
Given a polyhedron
$$Ax\geq b$$
where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
- 13.9k
3
votes
0
answers
186
views
Decidable equality for computable functions $\mathbb{N}\to\mathbb{N}$
Suppose we have two computable functions $f, g:\mathbb{N}\to\mathbb{N}$. When is $f=g$ algorithmically decidable?
For example it is decidable if $f$ and $g$ are polynomials of a priori known degree.
- 31
3
votes
0
answers
87
views
Additional symmetries of the Traveling Salesman Polytope
Given the complete graph $K_n=(V,E)$, the Traveling Salesman Polytope is a convex polytope in $\Bbb R^E$ obtained as the convex hull of the indicator vectors of (edge-sets of) Hamiltonian cycles in $...
- 13.6k
3
votes
0
answers
132
views
Can equality of chromatic symmetric functions of two trees be checked in polynomial time?
Stanley defined chromatic symmetric functions (CSF) in 1995 (Advances in Math) where he conjectured that trees can be distinguished by their CSF. However, tree isomorphism is decidable in P (...
- 31
3
votes
0
answers
73
views
Reference: Asymptotic bit-complexity of algebraic operations and transcendental functions
This question is a reference request. Does anyone know of a reference that lists the asymptotic bit-complexity of algebraic operations and transcendental functions implemented on a Turing machine that ...
- 131
3
votes
0
answers
122
views
Convex optimization upper bound for a non-linear optimization
Is there any good convex optimization problem based upper-bound for the following non-linear optimization problem?
\begin{align}
\max_{x_1,\ldots,x_N}&\quad \sum_{n=1}^{N} \log(1+\frac{x_n}{1+\...
- 287
3
votes
0
answers
125
views
Does Barvinok's algorithm apply to convex integer program?
Barvinok provided a counting algorithm to count number of integer solutions to integer linear program that runs in polynomial time if the number of integer variables is fixed.
If we have convex ...
- 1,826
3
votes
0
answers
163
views
A new "adversarial" Wasserstein distance?
Let us consider $\mu_1, \mu_2$ and $\mu_3$ three probability measures living on $[0,1]^{k_1}, [0,1]^{k_2}$ and $[0,1]^k$respectively, with $k_1 +k_2=k$. Let us denote by $\Gamma(\mu,\nu)$ the set of ...
- 555
3
votes
0
answers
173
views
The complexity of cutting hackers in a computer network
Let $l_1,l_2,\dots,l_m$ be parallel lines in the plane, say $l_k=\mathbb R\times\{k\}$. On the $k$th line fix a set $V_k$ consisting of $n_k$ points.
Let $(V,E)$ be a directed graph whose set of ...
- 4,535
3
votes
0
answers
155
views
Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?
All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
- 13.9k
3
votes
0
answers
178
views
Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
- 131
3
votes
0
answers
75
views
Functional characterization of local correlation matrices?
Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
- 1,826
3
votes
0
answers
56
views
Karp hardness of two cycles which lengths differ by one
Our problem is as follows:
NEARLY-EQUAL-CYCLE-PAIR
Input: An undirected graph $G(V,E)$
Output: YES if there exists $2$ (simple) cycles in $G$ which lengths differ by $1$, otherwise NO
Is it $NP$-...
- 121
3
votes
0
answers
75
views
Quantum versus classical communication complexity
Problem. Is it true that any 2-party communication problem $f(x,y)$ of poly-logarithmic complexity in the quantum simultaneous message passing model ($Q''$) has complexity $n^{o(1)}$ (i.e., strongly ...
- 4,535
3
votes
0
answers
165
views
$\mathrm{NP}$-complete problems in graph theory: undirected vs. directed
Is it true that it is much easier to establish $\mathrm{NP}$-complete on undirected graphs than digraphs (directed graph)?
Academic articles proving $\mathrm{NP}$-completeness of problems on ...
user128192
3
votes
0
answers
130
views
Algorithm detecting all distinct k-th powers in a string for all k ≥ 3
In string theory, the $k$-th power of a string $w$ is named as $w^k$, where $w^0$ is the empty string $\epsilon$ and $w^n$ is the concatenation of $w$ and $w^{n - 1}$ $(n \in \mathbb{N}^{+})$.
The ...
- 123
3
votes
0
answers
186
views
Factoring problem similar to $RSA$ structure that is possibly not $NP$ complete and not $coNP$ also?
Standard factoring problem $\Pi_1$ is 'Given integers $N$ and $M$ is there a factor $d\in[1,M]$ of $N$?'. This is in $NP$ since such a factor is the witness and in $coNP$ since one can check all the ...
- 13.9k
3
votes
0
answers
98
views
Deterministic procedure to find irreducible polynomials
In $\Bbb F_q[x_1,\dots,x_n]$ given $d_1,\dots,d_n\in\Bbb N$ is there a deterministic $O(poly(nd\log q))$ algorithm to find an irreducible polynomial with $d=\max_{i\in\{1,\dots,n\}}d_i$ and $d_i=deg(...
- 13.9k
3
votes
0
answers
120
views
Is the Evasiveness Conjecture strong enough to constructively imply the negation of the Pizzazz-conjecture?
Question. Assume the truth of the (notoriously open) Evasiveness Conjecture. Does this constructively imply the negation of the Pizzazz-conjecture?
Remarks.
The relevant statements are, by ...
- 6,051
3
votes
0
answers
105
views
Are there scenarios under which feasibility bilinear programming is easy?
Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know ...
- 13.9k
3
votes
0
answers
81
views
Computing distribution of non-identical coin flips
Suppose I have $N$ coins, where coin $i$ has probability $p_i$ of coming up heads. I flip all $N$ coins and let $S_N$ be the number of heads. How can I compute the distribution of $S_N$ efficiently?
...
- 3,979
3
votes
1
answer
368
views
Lot sizing problem: how to add these cuts efficiently
Consider the set of constraints of the uncapacitated lot sizing problem:
$$
\{(x,s,y)\in \mathbb{R}^n_+ \times \mathbb{R}^n_+ \times \mathbb{B}^n \;|\;s_{t-1}+x_t = d_t+s_t,\; x_t \le My_t,\; t=1,\...
- 225
3
votes
0
answers
71
views
Dependence of optimization problem on the linear constraints
Let $I=\{x_1,\cdots, x_n\}\subset \mathbb R$ be fixed. Given two probability distributions $\alpha=(\alpha_i)_{1\le i\le n}$ and $\beta=(\beta_i)_{1\le i\le n}$ on $I$, and a matrix $c=(c_{i,j})_{1\le ...
- 1,835
3
votes
0
answers
98
views
Is the Graph Isomorphism problem in βP class?
βP is the limited nondeterminism NP, cf. https://complexityzoo.uwaterloo.ca/Complexity_Zoo:B#betap
Last year Laslo Babai proved that the GI problem can be solved in (deterministic) time
$\exp(\log^c(...
- 193
3
votes
0
answers
63
views
Equational theory for resolution proof system
Is there any equational theory $T$ like $PV$ with following properties:
If $T\vdash f=g$ for terms $f$ and $g$, translation of $f=g$ to propositional formulas has polynomial resolution proof.(like $...
- 1,689
3
votes
0
answers
230
views
On weight enumerators of codes
Are there $[n,k]_q$ constant rate $\frac kn$ and constant alphabet linear code families with automorphism group of size $\Omega((n-n^\beta)!)$ that have minimum distance $d=O(n^\alpha)$ and number of ...
- 13.9k
3
votes
0
answers
181
views
Hypergraph edge colouring
I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
- 903
3
votes
0
answers
148
views
Average nastiness of a Newton polytope
Given a Newton polytope $P$ inside the $d$-scaled simplex ( i.e $\alpha \in P \implies | \alpha |=d $ ), then we define the following quantity:
$$ P(x)= \left( \sum_{\alpha \in P} \binom{d}{\alpha}...
- 351
3
votes
0
answers
105
views
State of the art for univariate complex polynomials factorization with algebraic coefficients
Let $\mathbb{K}:=\overline{\mathbb{Q}}$ be the field of algebraic numbers. We choose to represent an element of $\mathbb{K}$ as its minimal monic polynomial, which is a vector in some $\mathbb{Q}^n$.
...
- 5,428
3
votes
0
answers
50
views
Complexity of OBDD isomorphism (representing same function after permutation of variables)?
According to wikipedia Ordered Binary Decision Diagarams (OBDD) are a data structure that is used to represent a Boolean function.
OBDD is a DAG with two sinks $0,1$.
The size of the BDD is number ...
- 25.4k
3
votes
0
answers
240
views
Derandomization barriers in complexity theory applicable as barriers to constructive arguments replacing probabilistic method
The probabilistic method as first pioneered by Erdős (although others used this before) shows existence of a certain object while finding that object may take exponential time.
(1) Is there any ...
- 13.9k
3
votes
0
answers
573
views
Complexity of $\mathsf{gcd}(a,b)\bmod N$
Given $a,b\in\Bbb N$ where each $a,b$ is $n$-bits, we can compute $\mathsf{gcd}(a,b)$ in $cn^{1+\epsilon}$ bit operations for some fixed $c\geq1$.
My query is given $N,a,b$ where $a,b$ is $n$-bits ...
- 13.9k
3
votes
0
answers
970
views
Testing if a point is inside a convex polytope formed by halfspaces in n-dimension
Assume we have a convex polytope that is formed by the intersection of $k$-halfspaces in $\mathbb{R}^{n}$.
$$
a_{0,0}x^{n-1} + {a}_{0,1}x^{n-2} + ... a_{0,n-1} \leq 0
$$
$$
a_{1,0}x^{n-1} + {a}_{1,...
- 31
3
votes
0
answers
91
views
Existence of universal witness set and efficient sampling of coNP sets
Inspired by this answer given by Noam, which (I think) implies that a set $A \in NP$ if and only if there is polynomial-time computable function $f$ from random strings to elements of $A$ such that $A$...
- 2,993
3
votes
0
answers
240
views
A factorial related statement
Is the following promise problem in $\mathsf{NP}$ or $\mathsf{coNP}$ or even in $\mathsf{PH}$?
$$\Pi:\mathsf{Given}\mbox{ }p,a,s\in\Bbb N,\mbox{ }\mathsf{with}\mbox{ }p\mbox{ }\mathsf{a}\mbox{ }\...
user76479
3
votes
0
answers
66
views
$n!$ computation in $\mathsf{BSS}$ model
It is well known that if $n!$ cannot be computed in $\mathsf{polylog}(n)$ ring ($\Bbb Z$) operations, then $\mathsf P\neq\mathsf{NP}$ in $\mathsf{BSS}$ model.
Suppose if we assume $\mathsf P=\mathsf{...
- 13.9k
3
votes
0
answers
243
views
Polynomial-time algorithm solving approximately a generalization of the travelling salesman problem
Take a graph $G$ and a number of sets of nodes of $G$. The problem is to find the shortest path passing through at least one node in each node set. If each node set consists of only one node, the ...
- 367
3
votes
0
answers
262
views
About the small set expansion conjecture
Given a graph $G=(V,E)$ and a $\delta > 0$ one wants to calculate $h(G,\delta)=min_{\vert S\vert \leq \delta \vert V \vert } \phi(S)$. ($\phi(S) = \frac{ E(S,\bar{S}) }{d min \{\vert S \vert , n - \...
- 1,893
3
votes
0
answers
79
views
Are all $k$th-longest-tour problems equally hard?
It is well known, that determining the shortest and, the longest Hamilton Cycle of a complete graph with real edge weights are algorithmically two sides of the same medal: one transforms to the other ...
- 13.2k
3
votes
0
answers
125
views
counting irreducible factors
In How hard is it to compute the number of prime factors of a given integer? a question was asked on computing number of prime factors of an integer.
Suppose we have a polynomial $f(X)\in \Bbb Z[X]$ ...
- 13.9k
3
votes
0
answers
260
views
Partitioning a cubic graph into two induced cycles of equal order
I am aware that deciding the existence of a partition of the vertices of a connected graph $G(V, E)$ into two induced cycles is $NP$-complete(Theorem 2). Induced cycle is a cycle without any chord (...
- 2,993
3
votes
0
answers
264
views
Nesting big-O with big-Omega $O(g(\Omega(h(n))))$: is it $O$ for all $\Omega$ or for one $\Omega$?
I want to express the following statement about a function $f(n)$: there exists $f_\Omega\in\Omega(h(n))$ such that $f\in O(g(f_\Omega(n))$. What's the correct notation for this? Is it $f\in O(g(\...
- 163
3
votes
0
answers
706
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 ...
- 167
3
votes
0
answers
244
views
Intermediate $\mathsf{NP}$-complete problems?
Partition problem is weakly NP-complete since it has polynomial (pseudo-polynomial) time algorithm if input integers are bounded by some polynomial. However, 3-Partition problem is strongly NP-...
- 2,993
3
votes
0
answers
445
views
Gaussian Elimination in terms of Group Action
Gaussian elimination makes determinant of a matrix polynomial-time computable. The reduction of complexity in computing the determinant, which is otherwise sum of exponential terms, is due to presence ...
- 245
3
votes
0
answers
3k
views
0,1 solution to system of linear integer equations
I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
- 149
3
votes
0
answers
220
views
Could SVD be used to optimize the partial inner-products?
Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...
- 131
3
votes
0
answers
458
views
Connected Sum Decomposition of a Knot
Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?
- 405