The number of oriented rational links with a given deficiency number

Let [Formula: see text] be the set of un-oriented and rational links with crossing number [Formula: see text], a precise formula for [Formula: see text] was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] and let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] ([Formula: see text]) and deficiency [Formula: see text]. In this paper, we derive precise formulas for [Formula: see text] and [Formula: see text] for any given [Formula: see text] and [Formula: see text] and show that [Formula: see text] where [Formula: see text] is the [Formula: see text]th convolution of the convolved Fibonacci sequences.

Recognizing Lexicographically Smallest Words and Computing Successors in Regular Languages

For a formal language [Formula: see text], the problem of language enumeration asks to compute the length-lexicographically smallest word in [Formula: see text] larger than a given input [Formula: see text] (henceforth called the [Formula: see text]-successor of [Formula: see text]). We investigate this problem for regular languages from a computational complexity and state complexity perspective. We first show that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are (in general) necessary and sufficient for an unambiguous finite-state transducer to compute [Formula: see text]-successors. As a byproduct, we obtain that if [Formula: see text] is recognized by a DFA with [Formula: see text] states, then [Formula: see text] states are sufficient for a DFA to recognize the subset [Formula: see text] of [Formula: see text] composed of its lexicographically smallest words. We give a matching lower bound that holds even if [Formula: see text] is represented as an NFA. It has been known that [Formula: see text]-successors can be computed in polynomial time, even if the regular language is given as part of the input (assuming a suitable representation of the language, such as a DFA). In this paper, we refine this result in multiple directions. We show that if the regular language is given as part of the input and encoded as a DFA, the problem is in [Formula: see text]. If the regular language [Formula: see text] is fixed, we prove that the enumeration problem of the language is reducible to deciding membership to the Myhill-Nerode equivalence classes of [Formula: see text] under [Formula: see text]-uniform [Formula: see text] reductions. In particular, this implies that fixed star-free languages can be enumerated in [Formula: see text], arbitrary fixed regular languages can be enumerated in [Formula: see text] and that there exist regular languages for which the problem is [Formula: see text]-complete.

A skeleton model to enumerate standard puzzle sequences

<abstract><p>Guo-Niu Han [Sémin. Lothar. Comb. 85 (2021) B85c (electronic)] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the enumeration problem of over fifty thousand standard puzzle sequences. Most of them can be represented by classical numbers, such as Catalan numbers, double factorials, secant numbers and so on. Also, we prove several identities for standard puzzle sequences.</p></abstract>

An iterative vertex enumeration method for objective space based vector optimization algorithms

An application area of vertex enumeration problem (VEP) is the usage within objective space based linear/convex vector optimization algorithms whose aim is to generate (an approximation of) the Pareto frontier. In such algorithms, VEP, which is defined in the objective space, is solved in each iteration and it has a special structure. Namely, the recession cone of the polyhedron to be generated is the ordering cone. We consider and give a detailed description of a vertex enumeration procedure, which iterates by calling a modified `double description (DD) method' that works for such unbounded polyhedrons. We employ this procedure as a function of an existing objective space based vector optimization algorithm (Algorithm 1); and test the performance of it for randomly generated linear multiobjective optimization problems. We compare the efficiency of this procedure with another existing DD method as well as with the current vertex enumeration subroutine of Algorithm 1. We observe that the modified procedure excels the others especially as the dimension of the vertex enumeration problem (the number of objectives of the corresponding multiobjective problem) increases.

A Comprehensive Study of the Key Enumeration Problem

In this paper, we will study the key enumeration problem, which is connected to the key recovery problem posed in the cold boot attack setting. In this setting, an attacker with physical access to a computer may obtain noisy data of a cryptographic secret key of a cryptographic scheme from main memory via this data remanence attack. Therefore, the attacker would need a key-recovery algorithm to reconstruct the secret key from its noisy version. We will first describe this attack setting and then pose the problem of key recovery in a general way and establish a connection between the key recovery problem and the key enumeration problem. The latter problem has already been studied in the side-channel attack literature, where, for example, the attacker might procure scoring information for each byte of an Advanced Encryption Standard (AES) key from a side-channel attack and then want to efficiently enumerate and test a large number of complete 16-byte candidates until the correct key is found. After establishing such a connection between the key recovery problem and the key enumeration problem, we will present a comprehensive review of the most outstanding key enumeration algorithms to tackle the latter problem, for example, an optimal key enumeration algorithm (OKEA) and several nonoptimal key enumeration algorithms. Also, we will propose variants to some of them and make a comparison of them, highlighting their strengths and weaknesses.

Predictive models and abstract argumentation: the case of high-complexity semantics

In this paper, we describe how predictive models can be positively exploited in abstract argumentation. In particular, we present two main sets of results. On one side, we show that predictive models are effective for performing algorithm selection in order to determine which approach is better to enumerate the preferred extensions of a given argumentation framework. On the other side, we show that predictive models predict significant aspects of the solution to the preferred extensions enumeration problem. By exploiting an extensive set of argumentation framework features—that is, values that summarize a potentially important property of a framework—the proposed approach is able to provide an accurate prediction about which algorithm would be faster on a given problem instance, as well as of the structure of the solution, where the complete knowledge of such structure would require a computationally hard problem to be solved. Improving the ability of existing argumentation-based systems to support human sense-making and decision processes is just one of the possible exploitations of such knowledge obtained in an inexpensive way.

Enumeration of maps

This article discusses the relationship between random matrices and maps, i.e. graphs drawn on surfaces, with particular emphasis on the one-matrix model and how it can be used to solve a map enumeration problem. It first provides an overview of maps and related objects, recalling the basic definitions related to graphs and defining maps as graphs embedded into surfaces before considering a coding of maps by pairs of permutations. It then examines the connection between matrix integrals and maps, focusing on the Hermitian one-matrix model with a polynomial potential and how the formal expansion of its free energy around a Gaussian point (quadratic potential) can be represented by diagrams identifiable with maps. The article also illustrates how the solution of the map enumeration problem can be deduced by means of random matrix theory (RMT). Finally, it explains how the matrix model result can be translated into a bijective proof.

Formal Development of Distributed Enumeration Algorithms By Refinement-Based Techniques

The enumeration problem addresses a collection of important algorithmic issues related to distributed computations. Among existing solutions, we are interested on the seminal algorithm of Mazurkiewicz, based on local computations. Our paper contributes to the design of a correct-by-construction enumeration algorithm. The main idea relies upon the development of the problem following a top/down approachthat can be supported by an incremental process controlled by the refinement of models. Event-B modelling language is supporting our methodological. Our main objective is to provide a verified component for distributed enumeration inorder to be used and extended for solving other problems of distributed algorithms.