Theoretical Computer Science

Questions tagged [permutations]

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8
votes
1answer
205 views

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 ...
1
vote
0answers
42 views

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 ...
4
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0answers
73 views

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 ...
4
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0answers
61 views

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 ...
7
votes
0answers
179 views

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}\...
7
votes
1answer
258 views

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''...
2
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0answers
61 views

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\}^...
1
vote
1answer
67 views

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 $...
2
votes
1answer
137 views

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?
3
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0answers
75 views

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 $\...
1
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0answers
71 views

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 ...
9
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2answers
143 views

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 $G.s$, ...
2
votes
0answers
195 views

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 ...
6
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0answers
192 views

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 ...
10
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1answer
245 views

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 ...
5
votes
2answers
1k views

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 ...
4
votes
1answer
210 views

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 ...
0
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1answer
169 views

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 ...
-2
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1answer
104 views

Partitions on Integer Permutations

Let $S_n$ be the set of all permutations of integers from $1$ to $n$. Let $P_1$ and $P_2$ be two partitions defined on $S_n$ as follows. $P_1$ is the set of all those permutations which have even ...
3
votes
1answer
197 views

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 ...
14
votes
1answer
719 views

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 ...
25
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1answer
711 views

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 ...
1
vote
0answers
117 views

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 ...
12
votes
4answers
407 views

Reordering data 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 ...
16
votes
1answer
698 views

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 ...
4
votes
0answers
145 views

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, ...
13
votes
3answers
508 views

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 ...
10
votes
1answer
616 views

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=...
2
votes
1answer
67 views

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 ...
10
votes
1answer
279 views

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 ...
4
votes
0answers
102 views

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 ...
1
vote
1answer
171 views

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-...
11
votes
0answers
280 views

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
vote
0answers
123 views

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 ...
8
votes
1answer
215 views

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 := \...
10
votes
1answer
106 views

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 ...
21
votes
2answers
928 views

$NP$-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 ...
7
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0answers
376 views

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 ...
4
votes
1answer
140 views

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 ...
0
votes
0answers
110 views

“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 ...
2
votes
1answer
109 views

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 $a_1, a_2, ..., a_n$...
-1
votes
1answer
48 views

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 ...
12
votes
1answer
2k views

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, ...
3
votes
2answers
475 views

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 $...
29
votes
2answers
2k views

Can you identify the sum of two permutations in polynomial time?

There were two questions asked recently on cs.se which were either related to or had a special case equivalent to the following question: Suppose you have a sequence $a_1, a_2, \ldots a_n$ of $n$ ...
-3
votes
1answer
262 views

the product of a matrix and a permutation matrix [closed]

Can a permutation matrix (P) be used to change the rank of another matrix (M)? Is there any literature to this effect, or to the contrary? I've tried a few small examples and the resulting matrix (M2)...
3
votes
2answers
171 views

Sort with random deviations

I'm looking for an algorithm that will take a sorted array of numbers and generate a random permutation of this array in such a way that the probability of finding a larger element earlier in a ...
2
votes
1answer
144 views

Computing unique subset intersections

Given a set S = {si : {zj : z ∈ N} }, what is a time-efficient algorithm for computing the unique sets of intersections of all of the subsets of S? As per @JeffE's comment below, there are edge ...
2
votes
0answers
125 views

Complexity of the standardization

Let $(A, \leq)$ be a totally ordered alphabet. The standardization ${\tt std}(u)$ of a word $u \in A^n$ is the unique permutation of $n$ elements having the same inversions as $u$ (recall that an ...
17
votes
1answer
661 views

Asymptotically, how many permutations of $[1..n]$ have at most $k$ inversions?

Consider a permutation $\sigma$ of $[1..n]$. An inversion is defined as a pair $(i, j)$ of indices such that $i < j$ and $\sigma(i) > \sigma(j)$. Define $A_k$ to be the number of permutations ...

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