Questions tagged [permutations]
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76
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Decomposition of a permutation into increasing subsequences
Given a permutation $P$, the goal is to decompose this permutation into $k$ increasing subsequences $L_1,L_2,\ldots,L_k$, such that every element in $P$ appears exactly once in some increasing ...
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What does width $4$ permutation branching program correspond to?
$L$ can be computed by a family of programs over $S_3$
of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of
polynomial size.
$L$ can be computed by a family ...
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Inverting Kronecker product on vectors is in P?
Problem: Given a vector V of positive integers, find two vectors v1 and v2 such that the Kronecker product of v1 and v2 is equal to p(V) (where p(V) is a suitable permutation of V).
Example:
Input: V={...
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Dynamic permutation cycle data
Let $\pi \in S_n$ be a permutation of $\{1, \ldots, n\}$. Does there exist a simple data structure that admits the following operations in polylogarithmic time?
sameCycle($\pi,x,y$): determines ...
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Optimalization of sum $f(x_i, x_j)$ for all $i < j$ pairs through permutation only?
$\DeclareMathOperator*{\argmin}{arg\,min}$Can something be said about the difficulty of minimizing the quantity
$$g(x) = \sum_{i=1}^n\sum_{j=i+1}^n f(x_i, x_j)$$
of some string of symbols $x \in \...
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Characterizing the ANF of Single-Cycle Boolean Permutations
Given a function $F: \{0, 1\}^n \to \{0, 1\}^n$, we say that $F$ is a boolean permutation (also sometimes called a vectorial boolean function or an s-box in the literature) if $F$ is a bijection. We ...
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Reversible polynomial circuit iff polynomial reversible circuit?
My question is about efficiently computable bijective functions. Informally I'm interested in:
If a bijection is computable in polynomial time, can we compute it by a polynomial number of bijective ...
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How to efficiently verify if a semantic symmetry of a CNF formula is valid?
It is easy to verify that a syntactic symmetry of a CNF formula is correct.
Is it also possible to check in polynomial time that a semantic symmetry which is not a syntactic symmetry of a formula is ...
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Finding a largest symmetrical subset of a k-CNF propositional formula
I have a k-CNF propositional formula $F$ which do not admit any global symmetry i.e. there is no permutation $\sigma$ of its variables such that $\sigma(F) = F$ . I'm interested in finding the largest ...
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Can an efficiently computable non-one way permutation be written as the composition of polynomially many easy to compute involutions?
Suppose that $p$ is a polynomial. Then does there exist a polynomial $q$ where if $f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ is a bijection where both $f$ and $f^{-1}$ are computable by circuits with at ...
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Can every efficiently computable permutation be written as the composition of two efficiently computable involutions?
It is well-known that every permutation can be written as the composition of two involutions.
Suppose that $p$ is a polynomial. Then does there exist a polynomial $q$ such that if $f:\{0,1\}^{n}\...
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273
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Distinguishability a set of permutations
Given integers $d<n$, find the largest $k$ such that there exists a set of $k$ permutations $\sigma_1,\cdots,\sigma_k$ on $[n]$, such that any size-$d$ subset $T\subseteq [n]$ is ``distinguishable''...
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Can we compute encodings of binary strings under arbitrary permutation groups?
Given a permutation group $G \leq S_n$, can you construct non-uniformly a circuit computing a function $f : \{0, 1\}^n \rightarrow \{0, 1\}^{ceil(log|\{0, 1\}^n/G_n|)}$ with size $O_n(\frac{|\{0, 1\}^...
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What is the relationship between this notion of symmetry compressibility and Kolmogorov complexity?
If we have some string $x$ and a permutation $g$ which preserves the bits of $x$, we can store the value of $x$ on each of its $g$-orbits as well as $g$ instead of storing $x$ and we can reconstruct $...
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Computing size of permutation group from generators
You're given $k$ permutations $a_1,\dots,a_k$. Consider closure of this set under the composition operation. What are most efficient and simple algorithms to calculate the size of this closure?
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Finding the longest sub-permutation with bounded inversion number
Given a permutation $\sigma\in S_n$ and a positive integer $k$, find the largest integer $0\le m\le n$ such that there exist a subset $I\subseteq [n], |I|=m$, satisfying that the restriction of $\...
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Function which detects rotation of bit string
Consider a function $F: \mathbb{F}_2^d \to \mathbb{Z}^n = (f_1,\ldots,f_n)$ with the property that if $y \in \mathbb{F}_2^d$ is a rotation of $x \in \mathbb{F}_2^d$, i.e. $y$ is $x$ permuted by an ...
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Is this vertex ordering optimization NP-Hard?
Could you help me to prove that the following problem is NP-hard?
Problem. Given an undirected graph $G=(V,E)$, find an ordering of the nodes such that $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
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Complexity of Computing Lexicographically Minimal Element of Orbit
Given strong generators for a group $(G \leq S_n, *)$ acting on bitstrings of length $n$ and an element $s \in \{0, 1\}^n$, how hard is it to compute the lexicographically minimal element of $G.s$, ...
- 1,552
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Candidates for combinatorial one-way permutation
It is known that (worst-case) one-way permutations exist if and only if $P\ne UP \cap coUP$. Almost all candidates that I know are based on hard number theortic problems. I came across a combinatorial ...
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Proving P-Isomorphism between two languges
The famous Isomorphism Conjecture states that all NP-complete problems are isomorphic via polynomial-time computable and invertible bijections (reductions). Padability is the only property that I know ...
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Evaluating symmetric polynomials
Let $f:\mathbb{K}^n \to \mathbb{K}$ be a symmetric polynomial, i.e., a polynomial such that $f(x)=f(\sigma(x))$ for all $x \in \mathbb{K}^n$ and all permutations $\sigma \in S_n$. For convenience, we ...
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Matrix permanent is 0
Valiant's theorem says that computing the permanent of an $n\times n$ matrix is #P-hard. Is the problem of determining if a permanent is 0 any easier? This arises in the context of sequence A006063 in ...
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The complexity of decomposing a bi-stochastic matrix
A bistochastic matrix $A$ is a matrix with positive entries in which
each row/column sums to $1$.
By the Birkhoff von-Neumann theorem $A$ is a convex combination of
permutation matrices.
Further, by ...
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Graph isomorphism problem with invertible adjacency matrices
This question is supplementary to the question asked here.
One of the answers give a class of graphs for which the adjacency matrices are invertible which is defined as follows.
Given a ...
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How hard is recognizing a permutation that is a square for the shift product?
This is a continuation of my attempts to generate simple combinatorial computational problems that turn out to be computationally hard (NP-complete). In this pursuit, I came up with a permutation ...
- 20.6k
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Permutations with forbidden subsequences
Let $[n]$ denote the set $\{1,...,n\}$ and C(n,k) denote the set of all $k$-combinations of elements from $[n]$ without repetition.
Let $p= p_1p_2...p_k$ be a $k$-tuple in $C(n,k)$. We say that a ...
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Recognizing sequences with all permutations of $\{1, \ldots, n\}$ as subsequences
For any $n > 0$, I say that a sequence $s$ of integers in $\{1, \ldots, n\}$ is $n$-complete if, for every permutation $\mathbf{p}$ of $\{1, \ldots, n\}$, written as a sequence of pairwise distinct ...
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Minimize L2 norm by circular permutation
Imagine two lists $L$, $M$ of the same length $n$. How to find $j$ such that
$\sum_{i=1}^n(L[i]-M[i+j])^2$ is minimal, where the index $i+j$ is taken modulo $n$?
Of course one can take all the ...
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Reordering data (set of strings) to optimize for compression?
Are there any algorithms for reordering data to optimize for compression? I understand this is specific to the data and the compression algorithm, but is there a word for this topic? Where can I ...
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Computing parity of a permutation in a streaming-fashion way
I'm looking for a one-pass algorithm which computes parity of a permutation. I assume that an input permutation is given by stream $\pi[1], \pi[2], \cdots, \pi[n]$. The output should be the parity of ...
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Recognition problem of cycle permutation graphs
A cycle permutation graph is a cubic graph composed from two $n$-cycles each labeled $1, 2,..., n$, with additional edges connecting vertex $i$ in the first cycle to vertex $\pi(i)$ in the second, ...
- 20.6k
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Complexity of permutation related problems
Given a group $G$ of permutations on $[n]=\{1, \cdots, n\}$, and two vectors $u,v\in \Gamma^n$ where $\Gamma$ is a finite alphabet which is not quite relevant here, the question is whether there ...
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Probability of generating a desired permutation by random swaps
I'm interested in the following problem. We're given as input a "target permutation" $\sigma\in S_n$, as well as an ordered list of indices $i_1,\ldots,i_m\in [n-1]$. Then, starting with the list $L=...
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Efficient (non-crypto-grade?) pseudorandom permutations with arbitrary domain size
I'm looking for an efficient/simple (even if not necessarily cryptographically strong) algorithm for implementing pseudorandom permutations with domain cardinality other than a power of 2. (FWIW, the ...
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Recent progress in permutation groups algorithms?
I am interested in algorithms for finite groups as implemented in the GAP package. It seems that all known algorithms in this field deal with permutation groups/matrix groups; two fundamental ones are ...
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The rank-polynomial of a graded poset
Let $P$ be a graded poset with rank function $r$. We may then define its rank-polynomial as:
$R_P(q) = \sum_{x \in P} q^{r(x)}$.
This definition can be applied to several interesting posets, for ...
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More questions about permutations
I'm interested in permutations classes defined by one or several excluded patterns of length 4. In particular, the following classes seem interesting but not so-well studied algorithmically.
the skew-...
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Generalizations of the determinant/permanent problem?
A tantalizing open question in computational complexity is to understand the 'behavioral differences' between the determinant and the permanent. While the former is computable in polynomial time with ...
- 1,292
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Some algorithmic questions on permutations
I'm interested in the possibility of fast algorithms for the following two problems on permutations.
1)
Given a permutation $\pi$ and an integer $k$, compute a pair $(\mathcal{C},\mathcal{A})$ where ...
- 1,292
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Hierarchical sorting strategies for pattern-avoiding permutations?
For a class $\mathcal{C}$ of permutations, we cannot expect to sort the permutations of $\mathcal{C}$ with less than $O(\log |\mathcal{C}_n|)$ comparisons, where by convention $\mathcal{C}_n := \...
- 1,292
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Encoding sets of permutations with a generating set and a set of excluded elements
Polynomial-time algorithms are known for finding generating sets of permutation groups, which is interesting since we can then represent those groups succinctly without giving up on polynomial-time ...
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$NP$-completeness of recognizing the difference of two permutations
Shor stated, in his comment to anonymous moose's answer to this question Can you identify the sum of two permutations in polynomial time?, that it is $NP$-complete to identify the difference of two ...
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Permutation optimization problem
Here is the problem as posed by Jerrum:
"The computational complexity of the following problem is investigated: Given a permutation group specified as a set of generators, and a single target ...
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Compatible partial permutations
Please, correct my terminology as I am not a combinatorician
(I am using http://en.wikipedia.org/wiki/Partial_permutation). Please, refer me to the solution if this is a solved problem.
Let $P_k$ be ...
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"Partial" invert a one-way permutation
First of all, to my best understanding, traditionally, if $f$ is a one-way function that maps a length $l$ bit string to another length $l$ bit string (i.e., $f:\{0,1\}^l\rightarrow\{0,1\}^l$), then ...
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Effect of constraints Density on the hardness of Betweness problem
Betweeness problem is well known NP-complete permutation problem. Given a family $M$ of triples $(a_i, a_j, a_k)$, the problem is to decide whether a permutation $\Phi$ of elements $a_1, a_2, ..., a_n$...
- 20.6k
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Algorithm for calculating substitution combination with ordering
I need to calculate the combinations of elements with a substitution element.
For example for elements [A,B] if the substitution is X the results should be ...
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Efficient algorithm for existence of permutation with differences sequence?
This question is motivated by this post, Can you identify the sum of two permutations in polynomial time? , and my interest in computational properties of permutations.
A differences sequence $a_1, ...
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Finding the identity with permutation chains
I have the following problem: I'm given a list of size $K$ of random integer permutations of $[1..n]$, named $P_1$ to $P_K$, and an additional random permutation $Q$.
How hard is to find a sequence $...
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