Permutation Domain Research Articles

New characterizations of primitive permutation groups with applications to synchronizing automata

For a finite permutation group on n elements we show the following (and variants thereof) equivalences: (1) the permutation group is primitive, (2) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every non-empty subset, an element mapping the whole permutation domain onto this subset, (3) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every two distinct subsets, an element mapping precisely one to a singleton set. We also investigate further properties related to the reachability of subsets. Lastly, we apply our results to automata and show that automata whose transformation monoids contain a primitive permutation group and a mapping that excludes precisely one state from its image are completely reachable and have the property that a minimal automaton for the set of synchronizing words has the maximal possible number of states.

Three-Dimensional Mesh Steganography and Steganalysis: A Review.

Three-dimensional (3-D) meshes are commonly used to represent virtual surfaces and volumes. Over the past decade, 3-D meshes have emerged in industrial, medical, and entertainment applications, being of large practical significance for 3-D mesh steganography and steganalysis. In this article, we provide a systematic survey of the literature on 3-D mesh steganography and steganalysis. Compared with an earlier survey (Girdhar et al., 2017), we propose a new taxonomy of steganographic algorithms with four categories: 1) two-state domain, 2) LSB domain, 3) permutation domain, and 4) transform domain. Regarding steganalysis algorithms, we divide them into two categories: 1) universal steganalysis and 2) specific steganalysis. For each category, the history of technical developments and the current technological level are introduced and discussed. Finally, we highlight some promising future research directions and challenges in improving the performance of 3-D mesh steganography and steganalysis.

A factorial based particle swarm optimization with a population adaptation mechanism for the no-wait flow shop scheduling problem with the makespan objective

The no-wait flow shop scheduling problem (NWFSP) performs an essential role in the manufacturing industry. In this paper, a factorial based particle swarm optimization with a population adaptation mechanism (FPAPSO) is implemented for solving the NWFSP with the makespan criterion. The nearest neighbor mechanism and NEH method are employed to generate a potential initial population. The factorial representation, which uniquely represents each number as a string of factorial digits, is designed to transfer the permutation domain to the integer domain. A variable neighbor search strategy based on the insert and swap neighborhood structure is introduced to perform a local search around the current best solution. A population adaptation (PA) mechanism is designed to control the diversity of the population and to avoid the particles being trapped into local optima. Furthermore, a runtime analysis of FPAPSO is performed with the level-based theorem. The computational results and comparisons with other state-of-the-art algorithms based on the Reeve's and Taillard's instances demonstrate the efficiency and performance of FPAPSO for solving the NWFSP.

A Fitness Landscape Analysis for the No-Wait Flow Shop Scheduling Problem With Factorial Representation

The no-wait flow shop scheduling problem (NWFSP) is one of the essential models in the manufacturing systems. In this paper, the fitness landscape of the factorial representation for NWFSP with the makespan criterion is studied. The encoding and decoding schemes based on the factorial representation are constructed to transfer the permutation domain to the integer domain. The position-type distributions and fitness distance correlation are implemented to analyze the fitness landscape of the classic benchmarks. The multiple big valleys' structure in the fitness landscape is confirmed through the observation of fitness distance plot and the analysis of factorial coding theory. The various local optima and high ruggedness of the fitness landscape are visualized through the statistical results of position-type distributions. The results of fitness landscape analysis show the suitability of the landscape for the searchability with evolutionary algorithms and local search methods for solving NWFSP.

Chaotic distribution of QAM symbols for secure OFDM signal transmission

Abstract This paper proposes and demonstrates a novel dynamic quadrature amplitude modulation (QAM) mapping scheme to provide a secure physical-layer transmission in orthogonal frequency division multiplexing passive optical network (OFDM-PON). In the proposed scheme, permutated QAM symbols are independently and randomly mapped onto chaotically selected new locations on the complex plane. With the proposed encryption scheme these new locations are within the dimensions of a conventional QAM. A hyper digital chaos is applied to perform a multi-fold encryption in the proposed scheme which involves dynamic QAM mapping as well as time domain and frequency domain permutations on QAM symbols. As a result, a noise-like constellation with a key space of ∼10310 is achieved to enhance the high-level security in physical-layer transmission. A successful transmission experiment of 9.4-Gb/s encrypted 16-QAM OFDM signals is demonstrated over 25-km standard single mode fiber (SSMF), where the robust security and transmission performances are validated.

Multi-Objectivising Combinatorial Optimisation Problems by Means of Elementary Landscape Decompositions.

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this article, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation problems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their performance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.

Zermelo's Analysis of ‘General Proposition’

On Zermelo's view, any mathematical theory presupposes a non-empty domain, the elements of which enjoy equal status; furthermore, mathematical axioms must be chosen from among those propositions that reflect the equal status of domain elements. As for which propositions manage to do this, Zermelo's answer is, those that are ‘symmetric’, meaning ‘invariant under domain permutations’. We argue that symmetry constitutes Zermelo's conceptual analysis of ‘general proposition’. Further, although others are commonly associated with the extension of Klein's Erlanger Programme to logic, Zermelo's name has a place in that story.

Symmetric Propositions and Logical Quantifiers

Symmetric propositions over domain \(\mathfrak{D}\) and signature \(\Sigma = \langle R^{n_1}_1, \ldots, R^{n_p}_p \rangle\) are characterized following Zermelo, and a correlation of such propositions with logical type-\(\langle \vec{n} \rangle\) quantifiers over \(\mathfrak{D}\) is described. Boolean algebras of symmetric propositions over \(\mathfrak{D}\) and Σ are shown to be isomorphic to algebras of logical type-\(\langle \vec{n} \rangle\) quantifiers over \(\mathfrak{D}\). This last result may provide empirical support for Tarski’s claim that logical terms over fixed domain are all and only those invariant under domain permutations.

Domain permutation reduction for constraint satisfaction problems

This paper is concerned with the Constraint Satisfaction Problem (CSP). It is well-known that the general CSP is NP-hard. However, there has been significant success in discovering subproblems which are tractable (polynomial time solvable). One of the most effective ways to obtain a tractable subproblem has been to force all of the constraint relations to lie in some tractable language. In this paper we define a new way of identifying tractable subproblems of the CSP. Let P be an arbitrary CSP instance and Γ be any tractable language. Suppose there exists, for each variable of P, a permutation of the domain such the resultant permuted constraint relations of P all lie in Γ. The domain permuted instance is then an instance of a tractable class and can be solved by the polynomial time algorithm for Γ. Solutions to P can be obtained by inverting the domain permutations. The question, for a given class of instances and language Γ, whether such a set of domain permutations can be found efficiently is the key to this method's tractability. One of the important contributions made in this paper is the notion of a “lifted constraint instance” which is a powerful tool to study this question. • We consider the open problem of discovering domain permutations which make instances max-closed. We show that, for bounded arity instances over a Boolean domain this problem is tractable, while for domain size three it is intractable even for binary instances. • We give a simple proof verifying the tractability of discovering domain permutations which make instances row convex. We refute a published result by giving a simple proof of the intractability of discovering domain permutations which make instances, even with domain size four, connected row convex. • We demonstrate that triangulated and stable marriage instances are reducible, via domain permutations, to max-closed instances. This provides a simple explanation for arc consistency deciding these instances. • We verify with a simple direct proof the tractability of identification of renamable Horn instances, and the intractability of finding the largest renamable Horn subset of clauses of an instance of SAT. • We describe natural tractable classes which properly extend the maximal relational classes arising from tractable constraint languages. We believe that domain permutation reductions have a significant chance of being useful in practical applications.

Nash implementable domains for the Borda count

We characterize the preference domains on which the Borda count satisfies Maskin monotonicity. The basic concept is the notion of a “cyclic permutation domain” which arises by fixing one particular ordering of alternatives and including all its cyclic permutations. The cyclic permutation domains are exactly the maximal domains on which the Borda count is strategy-proof when combined with every possible tie breaking rule. It turns out that the Borda count is monotonic on a larger class of domains. We show that the maximal domains on which the Borda count satisfies Maskin monotonicity are the “cyclically nested permutation domains” which are obtained from the cyclic permutation domains in an appropriately specified recursive way.

Enumerating big orbits and an application: B acting on the cosets of [formula omitted

We describe a novel technique to handle big permutation domains for large groups. It is applied to the multiplicity-free action of the sporadic simple Baby Monster group on the cosets of its maximal subgroup Fi 23 , to determine the character table of the associated endomorphism ring.

The minimum length of a base for the symmetric group acting on partitions

A base for a permutation group, G , is a sequence of elements of its permutation domain whose stabiliser in G is trivial. Using purely elementary and constructive methods, we obtain bounds on the minimum length of a base for the action of the symmetric group on partitions of a set into blocks of equal size. This upper bound is a constant when the size of each block is at most equal to the number of blocks and logarithmic in the size of a block otherwise. These bounds are asymptotically best possible.

The probability of generating the symmetric group when one of the generators is random

A classical result of John Dixon (1969) asserts that a pair of random permutations of a set of n elements almost surely generates either the symmetric or the alternating group of degree n. We answer the question, For what permutation groups G ≤ Sn do G and a random permutation σ ∈ Sn almost surely generate the symmetric or the alter- nating group? Extending Dixon's result, we prove that this is the case if and only if G fixes o(n) elements of the permutation domain. The question arose in connection with the study of the diameter of Cayley graphs of the symmetric group. Our proof is based on ar esult byLuczak and Pyber on the structure of random permutations.

Non-manipulable domains for the Borda count

We characterize the preference domains on which the Borda count satisfies Arrow’s “independence of irrelevant alternatives” condition. Under a weak richness condition, these domains are obtained by fixing one preference ordering and including all its cyclic permutations (“Condorcet cycles”). We then ask on which domains the Borda count is non-manipulable. It turns out that it is non-manipulable on a broader class of domains when combined with appropriately chosen tie-breaking rules. On the other hand, we also prove that the rich domains on which the Borda count is non-manipulable for all possible tie-breaking rules are again the cyclic permutation domains.

Nash Implementable Domains for the Borda Count

We characterize the preference domains on which the Borda count satisfies Maskin monotonicity. The basic concept is the notion of a cyclic permutation domain which arises by fixing one particular ordering of alternatives and including all its cyclic permutations. The cyclic permutation domains are exactly the maximal domains on which the Borda count is strategy-proof (when combined with every tie breaking rule). It turns out that the Borda count is monotonic on a larger class of domains. We show that the maximal domains on which the Borda count satisfies Maskin monotonicity are the cyclically nested permutation domains. These are the preference domains which can be obtained from the cyclic permutation domains in an appropriate recursive way.

Probabilistic Model Building Based GAs in Permutation Domains Using Edge Histograms.

Recently, there has been a growing interest in developing evolutionary algorithms based on probabilistic modeling. They are called probabilistic model-building genetic algorithms (PMBGAs) or estimation of distribution algorithms (EDAs). In this scheme, the offspring population is generated according to the estimated probability density model of the parent instead of using recombination and mutation operators. In this paper, we have proposed PMBGAs in permutation domains using edge histogram based sampling algorithms (EHBSAs). Two types of sampling algorithms, without template (EHBSA/WO) and with template (EHBSA/WT), are presented. The results were tested in the TSP and showed EHBSA/WT worked fairly well with a small population size in the test problems used. It also worked better than well-known traditional two-parent recombination operators.

Finding Symmetries of Algebraic System Nets

The problem of finding symmetry information from algebraic system nets prior to the reachability graph generation is studied. The approach presented is based on well-formedness of transition descriptions, meaning that some data types in a net may be used in a symmetric way. Permutations on the domains of such data types produce symmetries on the state space level of the net, which in turn can be exploited during the reachability analysis. To ensure that the transitions behave symmetrically with respect to the chosen data domain permutations, a sufficient compatibility condition between data domain permutations and the algebraic terms used as transition guards and arc annotations is proposed. The solution is a general and flexible one as it does not fix the set of applicable operations, enabling the design of customized net classes. To help the process of deciding whether a term is compatible with a data domain permutation, an approximation rule for the compatibility condition is given.

Bases for Primitive Permutation Groups and a Conjecture of Babai

Abaseof a permutation groupGis a sequenceBof points from the permutation domain such that only the identity ofGfixesBpointwise. We show that primitive permutation groups with no alternating composition factors of degree greater thandand no classical composition factors of rank greater thandhave a base of size bounded above by a function ofd. This confirms a conjecture of Babai.

Primitive Groups with no Regular Orbits on the Set of Subsets

We determine all primitive groups which do not have a regular orbit on the power set of the permutation domain. As a corollary, we also determine all families of orbit equivalent primitive permutation groups.

Fast Management of Permutation Groups I

We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.