This presentation is an exposition of an application of the theory of recurrence relations to enumerating strings over an alphabet with a forbidden factor (consecutive substring). As an illustration we examine the case of binary strings with a forbidden factor of k consecutive symbols 1 for given k, using generating function techniques that deserve to be better known.
For a lattice, finding a nonzero shortest vector is computationally difficult in general. The problem becomes quite complicated even when the dimension of the lattice is five. There are two related notions of reduced bases, say, Minkowski-reduced basis and greedy-reduced basis. When the dimension becomes d = 5, there are greedy-reduced bases without achieving the first minimum while any Minkowski-reduced basis contains the shortest four linearly independent vectors. This suggests that the notion of Minkowski-reduced basis is somewhat strong and the notion of greedy-reduced basis is too weak for a basis to achieve the first minimum of the lattice. In this work, we investigate a more appropriate condition for a basis to achieve the first minimum for d = 5. We present a minimal sufficient condition, APG+, for a five dimensional lattice basis to achieve the first minimum in the sense that any proper subset of the required inequalities is not sufficient to achieve the first minimum.
We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.
Bonus-Malus systems are widely used in the insurance business. For the operation of such systems transition rules and premium scales need to established. Optimization of these systems usually means the calculation of appropriate premium scales while treating transition rules external parameters. In this paper, on the contrary, we show how optimal transition rules can be determined for a given set of premium scales. To this end, integer programming (IP) is used as a basic tool. Numerical examples are also presented to demonstrate the viability of our approach.
In this paper, we modify the iteration process of Agarwal et al. (J. Nonlinear Convex Anal. 8(1) (2007), 61–79) to three multi-valued mappings and prove the strong and △-convergence theorems of this iteration in a hyperbolic space. Our theorems extend and improve some recent results announced in the current literature.
The minimum sum coloring of graphs is a variant of the classical graph coloring problem which is known to be NP-hard. The problem consists on minimizing the sum colorings of different graph vertices. In this paper, we propose a new bi-objective model for the underlying problem. We also propose for the resolution a hybrid schema which combines a bi-objective genetic algorithm with an Iterated Variable Neighborhood Search. The proposed approach relies on the use of different dedicated evolutionary operators mainly crossover and mutation. We also note two important features of the Variable Neighborhood Search: the use of destroy/repair method for shaking step and a multi-neighborhood search. Combined methods led us to preliminary promising results.
Two-line graphs of a given partial Latin rectangle are introduced as vertex-and-edge-coloured bipartite graphs that give rise to new autotopism invariants. They reduce the complexity of any currently known method for computing autotopism groups of partial Latin rectangles.
Let G=(V,E,L) be an edge-labeled graph. Let V be the set of vertices of G, E the set of edges, L the set of labels (colors) such that each edge e∈E has an associated label L(e). The goal of the minimum labeling global cut problem (MLGCP) is to find a subset L′⊆L of labels such that G′=(V,E′,L\L′) is not connected and |L′| is minimized. In this work, we generate random instances for the MLGCP to perform empirical tests. Also propose a set of heuristics using concepts of Genetic Algorithm and metaheuristic VNS, including some of their procedures, like two local search moves, and an auxiliary data structure to speed up the local search. Computational experiments show that the metaheuristic VNS outperforms other methods with respect to solution quality.
We consider first order expressible properties of random perfect graphs. That is, we pick a graph Gn uniformly at random from all (labelled) perfect graphs on n vertices and consider the probability that it satisfies some graph property that can be expressed in the first order language of graphs. We show that there exists such a first order expressible property for which the probability that Gn satisfies it does not converge as n→∞.
The cut polytope CUT(n) is the convex hull of the cut vectors in a complete graph with vertex set {1,…, n}. It is well known in the area of combinatorial optimization and recently has also been studied in a direct relation with admissible correlations of symmetric Bernoulli random variables. That probabilistic interpretation is a starting point of this work in conjunction with a natural binary encoding of the CUT(n). We show that for any n, with appropriate scaling, all encoded vertices of the polytope 1-CUT(n) are approximately on the line y=x−1/2.
Block-modeling is a framework to describe a social network as a small structure. We propose here a Neighborhood Exploration Approach with Multi-start for tackling the Extend Generalized Block-modeling. The Extend Generalized Blockmodeling is the first and most complete model approach: it allows to analyze networks without any a priory knowledge about them. The other models require at least to know the size of the partition (i.e. the number of sub-sets that the partition will contain) and a pre-definition of the ideal models.
Partitions play an important role in numerous combinatorial optimization problems. Here we introduce the number of ordered 3-partitions of a multiset M having equal sums denoted by S(m1,…, mn; α1,…, αn), for which we find the generating function and give a useful integral formula. Some recurrence formulae are then established and new integer sequences are added to OEIS, which are related to the number of solutions for the 3-signum equation.
We study minor related row family inequalities for the set covering polyhedron of circular matrices. We address the issue of generating these inequalities via the Chvátal-Gomory procedure and establish a general upper bound for their Chvátal-rank. Moreover, we provide a construction to obtain facets with arbitrarily large coefficients and examples of facets having Chvátal-rank strictly larger than one.
The paper considers some concepts of trichotomy with different growth rates for evolution operators in Banach spaces. Connections between these concepts and characterizations in terms of Lyapunov-type sequences of norms are given. The main result is an equivalence between the nonuniform trichotomy and a certain type of uniform trichotomy, using sequences of norms.
Let G be a 4-connected graph, and let Ec(G) denote the set of 4-contractible edges of G and V≥5(G) denote the set of vertices of G whose degree is greater than 4. Under this notation, we show that |Ec(G)|≥(1/26)∑u∈V≥5(G)degG(u).
Feature selection, usually adopted as a preprocessing step for data mining, is used to select a subset of predictive features aiming to improve the performance of a predictive model. Despite of the benefits of feature selection for classification task, to the best of our knowledge, there is no work in the literature that addresses feature selection in conjunction with global hierarchical classifiers. Thus, in this paper, we fill this gap proposing a feature selection method based on Variable Neighborhood Search (VNS) metaheuristic for the hierarchical classification context. Computational experiments were carried out on five bioinformatics datasets to evaluate the effect of the proposed algorithm on classification performance when using a global hierarchical classifier. As result, we have obtained a classifier performance improvement for three datasets and a competitive result for a fourth dataset, which indicates the suitability of the proposed method for the hierarchical classification scenario.
•Persistent entropy is a topological variable which is stable and scale in-variant.•It can be used to reduce the dimensionality of data analysis problems to one.•Summary functions for persistent homology based on persistent entropy are stable.•They are more robust to noise than Betti functions and distinguish different features.
The Green Vehicle Routing and Scheduling Problem (GVRSP) aims to minimize carbon Dioxide Emissions (CO2) in logistics systems through better planning of deliveries made by a fleet of vehicles. In this paper, the GVRSP considers heterogeneous vehicles, time-varying traffic congestion, capacity constraints and the delivery can be fractionated. The GVRSP also allows vehicles to stop on arcs, so vehicle is allowed to travel an arc in multiple time periods. We present a mathematical formulation to describe the problem and a hybrid algorithm that combines the metaheuristic Iterated Local Search (ILS), the Random Variable Neighborhood Descent (RVND) procedure and an exact Set Covering model. The results obtained are compared with the state of the art and we show that the proposed approach has a strong performance.
We consider a new variant of the knapsack problem, where the contribution of each item on total profit is determined by its position in the knapsack via a specific function. While in the classic version this function could be considered a constant, we study two non-monotone convex functions motived by several real applications. We propose a binary linear programming (BLP) model and a polynomial time algorithm, called Greedy. Computational experiments are carried out, discussing practical and theoretical aspects of the problem resolution.
In the logistics industry, not only cartons but also polybags are used to pack goods for delivery. We study a single polybag loading problem, which is faced by many e-retailers and courier companies. In the problem, a set of rotatable boxes and a two-dimensional rectangular polybag are given. Because the polybag is flexible, the three-dimensional space inside the polybag can be variable when the boxes are loaded. The problem is to choose a subset of the boxes and pack them orthogonally into the polybag so as to maximize the space utilization. We introduce a mixed integer programming formulation and propose a tree search heuristic for the problem. Methods are evaluated by computational experiments on randomly generated data.
