Questions tagged [permutations]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
9 votes
1 answer
372 views

Is it NP-hard to find an order on a set of strings so that the concatenation is a given string?

Consider the following decision problem over a fixed alphabet $\Sigma$: Input: strings $s_1, \ldots, s_n$ of $\Sigma^*$ and a target string $t \in \Sigma^*$ Output: does there exist a permutation $\...
  • 8,223
2 votes
0 answers
42 views

Is there a name and context where function for permutation defined as f(k) = #σ({1,2,…,k})∩{1,2,…,k} appeared ? (Measure of locality)

Context: Thinking on some machine learning questions on comparing the feature importances produced by different methods - the following measure of concordance between two ordered lists seems to be ...
2 votes
0 answers
52 views

hardness of partition of permutation into a minimum number of monotone subsequences

Given a permutation P, a monotone subsequence is a subsequence (i.e. the elements do not have to be consecutive in P) that increases or decreases. This leads naturally to the following optimization ...
6 votes
1 answer
469 views

Number of permutations that satisfy a given set of comparisons

We are given a set of comparisons of the form z[i] < z[j] for various i and j and an ...
  • 419
1 vote
1 answer
167 views

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 ...
  • 137
4 votes
0 answers
204 views

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 ...
  • 12.6k
4 votes
0 answers
92 views

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={...
  • 61
2 votes
0 answers
52 views

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 ...
  • 121
0 votes
1 answer
132 views

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 \...
  • 690
4 votes
0 answers
142 views

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 ...
11 votes
1 answer
313 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
0 answers
81 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 ...
  • 301
4 votes
0 answers
82 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 ...
  • 301
4 votes
0 answers
76 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
0 answers
248 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
1 answer
278 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''...
  • 437
2 votes
0 answers
70 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
1 answer
95 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
1 answer
164 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 votes
0 answers
118 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 $\...
  • 437
2 votes
1 answer
124 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 ...
8 votes
3 answers
922 views

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 $\sum\limits_{v\in V}|succ(v)|\times|pred(v)|$ ...
  • 563
9 votes
2 answers
171 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
0 answers
206 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 votes
1 answer
321 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 ...
11 votes
1 answer
378 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 ...
  • 11.1k
5 votes
2 answers
2k 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 ...
  • 1,697
4 votes
1 answer
336 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 ...
  • 1,224
0 votes
1 answer
218 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 ...
3 votes
1 answer
218 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 ...
16 votes
1 answer
801 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 votes
1 answer
789 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 ...
  • 8,223
1 vote
0 answers
127 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
4 answers
843 views

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 ...
  • 252
17 votes
1 answer
990 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
0 answers
160 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
2 answers
966 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 ...
  • 133
10 votes
1 answer
750 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
1 answer
69 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 ...
  • 531
10 votes
1 answer
336 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
0 answers
121 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 ...
  • 1,292
1 vote
1 answer
178 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-...
  • 1,292
11 votes
0 answers
301 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 ...
  • 1,292
1 vote
0 answers
125 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 ...
  • 1,292
8 votes
1 answer
246 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 := \...
  • 1,292
11 votes
1 answer
133 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
2 answers
1k 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 ...
8 votes
0 answers
402 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
1 answer
166 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 ...
  • 223
0 votes
0 answers
111 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 ...
  • 1