Forbidden Set Research Articles

String editing under pattern constraints

We introduce the novel Nearest Pattern Constrained String (NPCS) problem of finding a minimum set Q of character mutation, insertion, and deletion edit operations sufficient to modify a string x to contain all contiguous substrings in a pattern set P and no contiguous substrings in a forbidden pattern set F. Letting Σ be the alphabet of allowed characters, and letting η and ϒ be the longest string length and sum of all string lengths in P∪F, respectively, we show that NPCS is fixed-parameter tractable in |P| with time complexity O(2|P|⋅ϒ⋅|Σ|⋅(|P|+η)(|x|+1)). Additionally, we consider a generalization of the NPCS problem in which we allow for constraints based on the membership of substrings in regular languages. In particular, we introduce a problem we denote String Editing under Substring in Language Constraints (StrEdit-SILC), where provided a wildcard-free string x∈Σ⁎, a finite set of regular languages R={L1,L2,…}, and a regular language LF, the objective is to find a minimum cost set of mutation, insertion, and deletion edit operations Q that suffice to convert the input string x into a string x′∈Σ⁎, where no substring has membership in LF, and ∀Li∈R, there exists a substring in Li. Here, letting Ψ and ϖ be the sum of all regular expression lengths and longest regular expression length for languages in R∪{LF}, respectively, and letting Cmid∈N be the maximum cost of an edit operation, we show that StrEdit-SILC is fixed-parameter tractable with respect to Ψ, having time complexity O(2Ψ⋅|x|⋅(ϖ⋅|Σ|+Cmid)). However, we also show that StrEdit-SILC is MAX-SNP-hard and otherwise difficult to approximate under stringent constraints.

On the qualitative and quantitative analysis for two fourth–order difference equations

Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations xn+1=xn-2xn-3axn+bxn-3,n≥0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{n+1}=\\frac{x_{n-2}x_{n-3}}{ax_{n}+bx_{n-3}}, \\ n\\ge 0, \\end{aligned}$$\\end{document}and xn+1=xn-2xn-3-axn+bxn-3,n≥0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{n+1}=\\frac{x_{n-2}x_{n-3}}{-ax_{n}+bx_{n-3}}, \\ n\\ge 0, \\end{aligned}$$\\end{document}with positive arbitrary real parameters a, b and arbitrary real initial conditions, as well as study the qualitative behaviors for each. For the first equation, we show that every admissible solution converges to a period-3 solution when a+b=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a+b=1$$\\end{document}. For the second equation, we show that every admissible solution converges to zero if b>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b>2$$\\end{document} when b2≥4a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^2\\ge 4a$$\\end{document}. When b2<4a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^2<4a$$\\end{document}, we show the existence of periodic solutions under certain conditions. We introduce the forbidden sets as well as provide some illustrative examples for the above-mentioned equations.

Extension of some edge graph problems: Standard, parameterized and approximation complexity

We consider extension variants of some edge optimization problems in graphs containing the classical Edge Cover, Matching, and Edge Dominating Set problems and generalizations thereof. Given a graph G=(V,E) and an edge set U⊆E, it is asked whether there exists an inclusion-wise minimal (or maximal, respectively) feasible solution E′ which satisfies a given property, for instance, being an edge dominating set (or a matching, respectively) and containing the forced edge set U (or avoiding any edges from the forbidden edge set E∖U, respectively). We present hardness results for these problems, for restricted instances such as bipartite or planar graphs. We counter-balance these negative results with parameterized complexity results. We also consider the price of extension, a natural optimization problem variant of extension problems, leading to some approximation and inapproximability results.

Deletion to scattered graph classes I - Case of finite number of graph classes

Graph-deletion problems involve deleting a small number of vertices so that the resulting graph belong to a given hereditary graph class. We initiate a study of a natural variation of the problem of deletion to scattered graph classes. We want to delete at most k vertices so that each connected component of the resulting graph belongs to one of the constant number of graph classes. As our main result, we show that this problem is non-uniformly fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the constant number of graph classes is known to be FPT and the properties that a graph belongs to these classes are expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set.

The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations

&lt;abstract&gt;&lt;p&gt;In this paper, the circular system of Riccati type complex difference equations of the form&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ u_{n+1}^{(j)} = \frac{a_ju_n^{(j-1)}+b_j}{c_ju_n^{(j-1)}+d_j}, \; n = 0, 1, 2, \cdots, \; j = 1, 2, \cdots, k, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ u_n^{(0)}: = u_n^{(k)} $ for all $ n $, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and the expression of the solutions, given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ R_j: = \frac{a_jd_j-b_jc_j}{[a_j+d_j]^2}, \; j\in\overline{1, k}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.&lt;/p&gt;&lt;/abstract&gt;

Global behaviour of quaternion Riccati rational difference equation

Since non-commutative algebraic structure of quaternions, it is difficult to study the forbidden sets and asymptotic stability of quaternion rational difference equations. In this paper, by generalized Niven's algorithm and its related properties, the representations of general solutions for the third order linear quaternion difference equation are given through the roots of the corresponding quaternion characteristic equation, based on which, the corresponding forbidden sets of the Riccati quaternion rational difference equation are computed. Moreover, we obtain the quaternion curves of initial points located in two-dimensional quaternion space outside which we determine the convergence, divergence and global asymptotic stability of the solutions for the second order Riccati quaternion rational difference equations, some sufficient and necessary conditions for global asymptotic stability are established. Finally, several examples are illustrated to show the feasibility of our obtained results.

Characterizing Forbidden Pairs for the Edge-Connectivity of a Connected Graph to Be Its Minimum Degree

Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for any H∈H. In this article, we characterize all connected subgraph pairs {R,S} guranteeing the edge-connectivity of a connected {R,S}-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation.

Towards cheat-preventing in block-based progressive visual cryptography for general access structures

Existing block-based progressive visual cryptography schemes (BPVCS) that prevent cheating are limited to (2,n) and (k,n) thresholds. They are unsuitable for implementing complex sharing techniques in the actual world. A cheating immune BPVCS for general access structure (GAS) is proposed in this paper. To find the parameters that make up the proposed scheme, a simplified multiple assignment procedure based on integer linear programming is introduced. The secret parts can be decrypted by the members of a qualified set. Participants in a forbidden set, on the other hand, are not allowed to know about the secret. The proposed scheme includes cheat-prevention and progressive recovery capabilities. In addition, to generate meaningful shadows for BPVCS, a modified extended VCS (EVCS) is proposed. Our modified EVCS can solve the cover-interference problem in conventional BPVCS. The effectiveness and benefits of the proposed BPVCS are supported through the experimental findings and comparisons.

Safety of stochastic systems: An analytic and computational approach

We refine the concept of stochastic reach avoidance for a general class of Markov processes introducing a threshold of p for the reaching probability. This new problem is called p-safety, and it aims to ensure that the given process reaches a forbidden set before leaving its ‘working’ state space with a probability of less than p. In the situation when an initial probability measure characterizes the initial states, a variant of p-safety is put forward. We call this form of safety weak p-safety. In this work, we characterize both p-safety and weak p-safety and show how to compute them. We employ semi-definite programming to compute p-safety and linear programming to compute weak p-safety. To get to this point, we use certificates of positivity of polynomials translated into the sum of squares and the Bernstein forms.

Solvability of a $\left( k+l\right)$-order nonlinear difference equation

It is shown that the following $\left( k+l\right) $-order nonlinear difference equation $$x_{n}=\frac{x_{n-k}x_{n-k-l}}{x_{n-l}\left( a_{n}+b_{n}x_{n-k}x_{n-k-l}\right)}, \ n\in \mathbb{N}_{0},$$ where $k,l\in \mathbb{N}$, $\left(a_{n} \right)_{n\in \mathbb{N}_{0}}$, $\left(b_{n} \right)_{n\in \mathbb{N}_{0}}$ and the initial values $x_{-i}$, $i=\overline {1,k+l}$, are real numbers, can be solved and extended some results in literature. Also, by using obtained formulas, we give the forbidden set of the initial values for aforementioned equation and study the asymptotic behavior of well-defined solutions of above difference equation for the case $k=3$, $l=k$.

Preferred Explanations for Ontology-Mediated Queries under Existential Rules

Recently, explanations for query answers under existential rules have been investigated, where an explanation is an inclusion-minimal subset of a given database that, together with the ontology, entails the query. In this paper, we take a step further and study explanations under different minimality criteria. In particular, we first study cardinality-minimal explanations and hence focus on deriving explanations of minimum size. We then study a more general preference order induced by a weight distribution. We assume that every database fact is annotated with a (penalization) weight, and we are interested in explanations with minimum overall weight. For both preference orders, we study a variety of explanation problems, such as recognizing a preferred explanation, all preferred explanations, a relevant or necessary fact, and the existence of a preferred explanation not containing forbidden sets of facts. We provide a detailed complexity analysis for all the aforementioned problems, thereby providing a more complete picture for explaining query answers under existential rules.

ON A SOLVABLE SYSTEM OF NON-LINEAR DIFFERENCE EQUATIONS WITH VARIABLE COEFFICIENTS

In this paper, we show that the system of difference equations can be solved in the closed form. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, we obtain periodic solutions of aforementioned system.

Strong Algorithms for the Ordinal Matroid Secretary Problem

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s [Formula: see text] utility-competitive algorithm for uniform matroids of rank [Formula: see text] in order to obtain a [Formula: see text] probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.

On [formula omitted]-good-neighbor conditional connectivity and diagnosability of hierarchical star networks

In this paper, we study the connectivity and diagonosability of n-dimensional hierarchical star network HSn based on the concept of forbidden faulty sets. In a forbidden faulty set, certain nodes cannot be faulty at the same time and this model can better reflect fault patterns in a real system than the existing ones. Under the condition every fault-free node in a network contains at least g fault-free neighbors, the g-good neighbor conditional connectivity is defined as the minimum number of faulty processors whose deletion makes the network disconnected, the g-good neighbor conditional diagnosability tg(G) is defined as the maximum number of faulty processors that the network can guarantee to identify solely by performing mutual tests among the processors. We investigate the g-good neighbor conditional connectivity and the g-good-neighbor conditional diagnosability of n-dimensional hierarchical star network HSn. Our results show that the g-good-neighbor conditional connectivity of HSn is (n−g)(g+1)!−1 and the g-good-neighbor conditional diagnosability of HSn under the PMC and MM* models is (n−g+1)(g+1)!−1 when 0≤g≤n−3,g≠2 and n≥4. In addition, we show the 2-good-neighbor conditional connectivity of HSn is 4n−8 and the 2-good-neighbor conditional diagnosability of HSn under the PMC and MM* models is 4n−5 when n≥4.

Hamiltonian Cycle Embeddings in Faulty Hypercubes Under the Forbidden Faulty Set Model

In this paper, we study the fault-tolerant capability of hypercubes with respect to the hamiltonian property based on the concept of forbidden faulty sets. We show, with the assumption that each vertex is incident with at least three fault-free edges, that an [Formula: see text]-dimensional hypercube contains a fault-free hamiltonian cycle, even if there are up to [Formula: see text] edge faults. Moreover, we give an example to show that the result is optimal with respect to the number of edge faults tolerated.

A Novel ( t , s , k , n )-Threshold Visual Secret Sharing Scheme Based on Access Structure Partition

Visual secret sharing (VSS) is a new technique for sharing a binary image into multiple shadows. For VSS, the original image can be reconstructed from the shadows in any qualified set, but cannot be reconstructed from those in any forbidden set. In most traditional VSS schemes, the shadows held by participants have the same importance. However, in practice, a certain number of shadows are given a higher importance due to the privileges of their owners. In this article, a novel ( t , s , k , n )-threshold VSS scheme is proposed based on access structure partition. First, we construct the basis matrix of the proposed ( t , s , k , n )-threshold VSS scheme by utilizing a new access structure partition method and sub-access structure merging method. Then, the secret image is shared by the basis matrix as n shadows, which are divided into s essential shadows and n - s non-essential shadows. To reconstruct the secret image, k or more shadows should be collected, which include at least t essential shadows; otherwise, no information about the secret image can be obtained. Compared with related schemes, our scheme achieves a smaller shadow size and a higher visual quality of the reconstructed image. Theoretical analysis and experiments indicate the effectiveness of the proposed scheme.

Solvability of a system of higher order nonlinear difference equations

In this paper we show that the system of difference equations\[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

Exact Separation of Forbidden-Set Cuts Associated With Redundant Parity Checks of Binary Linear Codes

In recent years, several integer programming (IP) approaches were developed for maximum-likelihood decoding and minimum distance computation for binary linear codes. Two aspects in particular have been demonstrated to improve the performance of IP solvers as well as adaptive linear programming decoders: the dynamic generation of forbidden-set (FS) inequalities, a family of valid cutting planes, and the utilization of so-called redundant parity-checks (RPCs). However, to date, it had remained unclear how to determine whether or not there exists any violated FS inequality w.r.t. any known or unknown parity-check. In this note, we prove NP -hardness of this RPC separation problem. Moreover, we formulate an IP model that combines the search for most violated FS cuts with the generation of RPCs, and report on computational experiments. Empirically, for various instances of the minimum distance problem, it turns out that while utilizing the exact separation IP does not appear to provide a computational advantage, it can apparently be avoided altogether by combining heuristics to generate RPC-based cuts.

On the Chromatic Numbers Corresponding to Exponentially Ramsey Sets

In this paper, nontrivial upper bounds on the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n=\left({\mathrm{\mathbb{R}}}^n, lp\right) $$ with forbidden monochromatic sets are proved. In the case of a forbidden rectangular parallelepiped or a regular simplex, explicit exponential lower bounds on the chromatic numbers are obtained. Exact values of the chromatic numbers of the spaces $$ {\mathrm{\mathbb{R}}}_p^n $$ with a forbidden regular simplex in the case p = ∞ are found. Bibliography: 39 titles.

An explicit formula and forbidden set for a higher order difference equation

In this paper, we determine the forbidden set, introduce an explicit formula for the solutions and discuss the global behavior of solutions of the difference equation $$\begin{aligned} x_{n+1}=\frac{ax_{n}x_{n-k+1}}{ bx_{n-k+1}+ cx_{n-k}},\quad n=0,1,\ldots , \end{aligned}$$where a, b, c are positive real numbers, the initial conditions $$ x_{-k}, x_{-k+1},\ldots ,x_{-1},x_0$$ are real numbers and k is a positive integer.