Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Eduardo G. Pardo, Angelo Sifaleras In this issue we present a peer-reviewed selection of short papers that were accepted for presentation in the 4th International Conference on Variable Neighborhood Search (ICVNS'16) which was held in Málaga, Spain, during October 3–5, 2016. This conference is devoted to the Variable Neighborhood Search metaheuristic which was originally proposed by Nenad Mladenović and Pierre Hansen. This methodology is currently consolidated as a general framework to solve hard optimization problems and it is widely used all around the world.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Samuel Souza Brito, Haroldo Gambini Santos Chvàtal-Gomory cuts are well-known cutting planes for Integer Programming problems. As shown in previous works, the inclusion of these cuts allows to significantly reducing the integrality gap. This work presents a Local Search heuristic approach based on Variable Neighborhood Search to discover violated Chvàtal-Gomory inequalities. Since this problem is known to be NP-hard, this approach was designed to generate violated inequalities in restricted amounts of time. Constraints are grouped in several sets, considering the amount of common variables. These sets are processed in parallel in order to obtain the best multipliers and produce violated cuts. We report some preliminary results obtained for MIPLIB 3.0 and 2003 instance sets, comparing our approach with an integer programming based separation method. Our algorithm was able to separate many violated inequalities, reducing the duality gap. Furthermore, it uses an extended numerical precision implementation, since it is not specifically bound to simplex based solvers.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Thays A. Oliveira, Vitor N. Coelho, Helena Ramalhinho, Marcone J.F. Souza, Bruno N. Coelho, Daniel C. Rezende, Igor M. Coelho This paper focuses on Book Marketing Campaigns, where the benefit of offering each book is calculated based on a bipartite graph (biclique). A quasi Biclique problem is assessed for obtaining the probabilities of success of a given client buy a given book, considering it had received another book as free offer. The remaining optimization decision problem can be solved following the Targeted Offers Problem in Direct Marketing Campaigns. The main objective is to maximize the feedback of customers purchases, offering books to the set of customers with the highest probability of buying others ones from its biclique and, at the same time, minimizing campaign operational costs. Given the combinatorial nature of the problem and the large volume of data, which can involve real cases with up to one million customers, metaheuristics procedures have been used as an efficient way for solving it. Here, a hybrid trajectory search based algorithm, namely GGVNS, which combines the Greedy Randomized Adaptive Search Procedures and General Variable Neighborhood Search, is used. The strategy for generating the quasi Biclique problem is described and a new instance generator for the TOPDMC is introduced. Computational results regarding the GGVNS algorithm shows it is able to find useful and profitable sets of clients.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Olivera Janković, Zorica Stanimirović This paper deals with the uncapacitated r-allocation p-hub maximal covering problem (UrApHMCP) with binary coverage criterion. This problem consists of choosing p hub locations from a set of nodes so as to maximize the total demand covered while satisfying the r-allocation strategy. The applied binary coverage criterion ensures that the distance between any origin–destination pair through located hubs should be shorter than a predetermined distance. An integer linear programming model for the considered problem is introduced. As a solution method to UrApHMCP, a General Variable Neighborhood Search (GVNS) heuristic is proposed. A greedy procedure is used to construct an initial solution to GVNS. Neighborhood structures explored within the GVNS are defined by operators that change a set of chosen hubs and node to hub assignments. Variable Neighborhood Descent with sequential search strategy is used as an improvement procedure. The results of computational experiments on standard p-hub benchmark instances show the efficiency and effectiveness of the proposed GVNS when solving the considered problem.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Rafael A.M. Gomes, Túlio A.M. Toffolo, Haroldo Gambini Santos The Nurse Rostering Problem (NRP) is an optimization problem where nurses with specific skills must be assigned shifts in a schedule. The objective is to obtain a feasible solution while minimizing the number of soft constraint violations. This work presents a Variable Neighborhood Search accelerated Column Generation procedure for the NRP in addition to a Relax-and-fix Heuristic for obtaining feasible solutions. The algorithm improved the best known solutions by at least 10% for all 29 hidden instances from the Second International Nurse Rostering Competition (2014) with 4 weeks. The improved solutions have an optimality gap of at most 8%.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Y. Kochetov, A. Kondakov We study a new variant of the bin packing problem. Given a set of items, each item has a set of colors. Each bin has a color capacity, the total number of colors for a bin is the union of colors for its items and can not exceed the bin capacity. We want to pack all items into the minimal number of bins. For this NP-hard problem we apply the column generation technique based on the VNS matheuristic for the pricing problem. To get optimal or near optimal solutions we apply VNS matheuristic again using optimal solution for the large scale linear programming relaxation. Computational experiments are reported for the randomly generated test instances with large bin capacity and number of items up to 250.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Nikolaos Antoniadis, Angelo Sifaleras In this paper, we study various parallelization schemes for the Variable Neighborhood Search (VNS) metaheuristic on a CPU-GPU system via OpenMP and OpenACC. A hybrid parallel VNS method is applied to recent benchmark problem instances for the multi-product dynamic lot sizing problem with product returns and recovery, which appears in reverse logistics and is known to be NP-hard. We report our findings regarding these parallelization approaches and present promising computational results.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Vívian L. Aguiar Santos, José Elias C. Arroyo In this paper, we study an unrelated parallel machine scheduling problem in which the jobs cause deterioration of the machines. This deterioration decreases the performance of the machines, and therefore, the processing times of the jobs are increased over time. The problem is to find the processing sequence of jobs on each machine in order to reduce the deterioration of the machines and consequently minimize the makespan. This problem is NP-hard when the number of machines is greater or equal than two, and hence we propose a heuristic based on the Iterated Greedy meta-heuristic coupled with a variant of the Variable Neighborhood Descent method that uses a random ordering of neighborhoods in local search phase. The performance of our heuristic, named IG-RVND, is compared with the state-of-the-art meta-heuristic proposed in the literature for the problem under study. The results show that the our heuristic outperform the existing algorithm by a significant margin.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Sana Frifita, Malek Masmoudi, Jalel Euchi Home healthcare centers are facing increasing demands and researchers are attracted by the related routing and scheduling issue that is presented in literature as a VRP with synchronization and time windows constraints. The goal is to optimize the assignment of visits to home caregivers and the sequence of visits execution. In this paper, a General Variable Neighborhood Search is provided. Experiments conducted on benchmark instances from the literature clearly show that our method is fast and outperforms the existing approaches on half of the instances.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Samia Hasni, Said Toumi, Bassem Jarboui, Anis Mjirda We address a multi-product version of the IRP, in which a supplier is responsible for the distribution of several products to a set of geographically dispersed customers using a feet of heterogeneous vehicles. We deal with a deterministic version of demand rates in which the supplier has full knowledge of future demands. To generate good solutions, we have developed an algorithm based on the Variable Neighborhood Search capable of solving this variant. Our comparative tests on a large set of artificial instances have shown that our heuristic can produce high quality solutions.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): V.N. Coelho, I.M. Coelho, B.N. Coelho, M.J.F. Souza, F.G. Guimarães, E.J. da S. Luz, A.C. Barbosa, M.N. Coelho, G.G. Netto, R.C. Costa, A.A. Pinto, A. de P. Figueiredo, M.E.V. Elias, D.C.O.G. Filho, T.A. Oliveira Brain activity can be seen as a time series, in particular, electroencephalogram (EEG) can measure it over a specific time period. In this regard, brain fingerprinting can be subjected to be learned by machine learning techniques. These models have been advocated as EEG-based biometric systems. In this study, we apply a recent Hybrid Focasting Model, which calibrates its if-then fuzzy rules with a hybrid GVNS metaheuristic algorithm, in order to learn those patterns. Due to the stochasticity of the VNS procedure, models with different characteristics can be generated for each individual. Some EEG recordings from 109 volunteers, measured using a 64-channels EEGs, with 160 HZ of sampling rate, are used as cases of study. Different forecasting models are calibrated with the GVNS and used for the classification purpose. New rules for classifying the individuals using forecasting models are introduced. Computational results indicate that the proposed strategy can be improved and embedded in the future biometric systems.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Alba Agustin, Angel Juan, Eduardo G. Pardo In this paper we propose a Variable Neighborhood Search approach for the Crew Pairing Problem. This problem consist in assigning a crew to each of the flights of a flight scheduling, in a predefined time horizon. The main objective of the problem is to minimize the number of cabin crews needed to cover all the flights subject to a set of constraints. These constraints are real-life specifications regulated by airline rules and other operational challenges. In particular we propose a General Variable Neighborhood Search algorithm to tackle the problem and we have tested our approach over a real instance provided by an airline and over an additional set of generated instances. The obtained results have been compared with a previous multi-start approach in the state of the art and with the initial solution provided to the algorithm which, in the case of the real instance, was the solution in use by the airline.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): M. Bruglieri, S. Mancini, F. Pezzella, O. Pisacane, S. Suraci We propose a three-phase matheuristic, combining an exact method with a Variable Neighborhood Search local Branching (VNSB) to route a fleet of Electric Vehicles (EVs). EVs are allowed stopping at the recharging stations along their routes to (also partially) recharge their batteries. We hierarchically minimize the number of EVs used and the total time spent by the EVs, i.e., travel times, charging times and waiting times (due to the customer time windows). The first two phases are based on Mixed Integer Linear Programs to generate feasible solutions, used in a VNSB algorithm. Numerical results on benchmark instances show that the proposed approach finds good quality solutions in reasonable amount of time.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Jerzy Duda The paper presents a genetic algorithm (GA) hybridized with variable neighborhood search (VNS) to solve multi-item capacitated lot-sizing multi-family problem with setup times. The problem has a practical application in production planning e.g., in foundry industry, so test cases for computational experiments were based on the data from the real production process in a foundry. The VNS algorithm is used after a certain number of GA generations for all individuals in the population to improve solutions. The presented method applied for large instances of the problem outperforms both a dedicated genetic algorithm and a CLPEX Solver-based rolling horizon methods known from the literature.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Telmo Pinto, Cláudio Alves, José Valério de Carvalho In this paper, we explore a capacitated vehicle routing problem with loading constraints and mixed linehauls and backhauls. The problem belongs to the subclass of pickup and delivery problems. To solve this problem, we describe a set of variable neighborhood search approaches whose shaking and local search phases rely on different neighborhood structures. Some of these structures were specially developed for this problem. All the strategies were implemented and exhaustively tested. The results of this computational study are discussed at the end of this paper.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Rahma Borchani, Abdelkerim Elloumi, Malek Masmoudi University course timetabling problem refers to schedule a set of lectures, tutorials and practical works to a limited number of teachers, classrooms and time slots over a planning horizon while satisfying a set of hard and soft constraints. In this paper, we investigate Variable Neighborhood Descent approach to tackle a specific course timetabling problem related to the case of the Faculty of Economics and Management Sciences of Sfax in Tunisia. The objective is to minimize the total number of holes and the number of isolated lessons for all student groups. We have developed eleven specific neighborhood structures: six of these are designed for correcting the holes, while the other five are designed for adjusting the isolated lessons. Computations are made on real instances and numerical results show that our approach outperforms the existing solution with the elimination of 52.47 % of holes and isolated sessions on average.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Israel López-Plata, Christopher Expósito-Izquierdo, Belén Melián-Batista, José Marcos Moreno-Vega This work addresses the problem of operational management of the internal delivery vehicles on the yard of a maritime container terminal using a Variable Neighbourhood Search algorithm. An empirical analysis is performed to find the Variable Neighbourhood Search variant that provides the best result for the proposed problem. This analysis tests 144 different variants on a wide range of realistic scenarios.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Ana Anokić, Zorica Stanimirović, Tatjana Davidović, Đorđe Stakić Low price of raw materials in sugar industry and characteristics of production method lead to the specific transport organization problem. A new variant of Vehicle Scheduling Problem (VSP) that arises from transportation of sugar beet is considered. The problem is formulated as a Mixed Integer Quadratically Constraint Programming (MIQCP) model, which reflects the objective and specific constrains from practice. Computational experiments are conducted on real-life instances obtained from a sugar company in Serbia and the set of generated instances of larger dimensions. The proposed MIQCP model is used within the framework of Extended Lingo 15 solver, providing optimal solutions on small-size instances only. In order to find solutions on larger problem instances, a metaheuristic method based on Variable Neighborhood Search (VNS) is designed. Obtained computational results show that the proposed VNS quickly reaches all known optimal solutions on small-size instances. On larger problem instances, for which Lingo 15 solver could not find even a feasible solution, VNS provides best solutions in relatively short running times.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Shima Golmohamadi, Reza Tavakkoli-Moghaddam, Mostafa Hajiaghaei-Keshteli Nowadays finding effective solution for transportation problem is one of the main issues both in industries and academia. According to the real world, in this paper it is assumed that the products are transferred in batches in a fixed charge transportation problem. Furthermore, the fuzzy values are used according to the parameters value in the real world. In addition, six meta-heuristics are utilized in order to solve the presented model. Most of these algorithms are firstly used for a mathematical model in transportation problem literature. Because of the importance of tuning the parameters in solving the given problem, the Taguchi method is used. Finally, computational results with different problem sizes are studied and analyzed.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Antonio Alonso-Ayuso, Laureano F. Escudero, F. Javier Martín-Campo, Nenad Mladenović The aircraft conflict resolution problem needs to provide response to such situation in which two or more aircraft violate the safety distances that must be kept during the flight. Then, given the aircraft trajectories, the aim consists of finding a new configuration such that every conflict situation must be avoided. To deal with conflict avoidance an aircraft may change velocity, heading angle and altitude level. Due to nonlinearities involved in the exact model and the need of an almost real-time response, a Variable Neighborhood Search approach is presented within the multiobjective environment.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Jack Brimberg, Nenad Mladenović, Raca Todosijević, Dragan Urošević In this paper we review a recently proposed variant of variable neighborhood search (VNS) referred to as nested variable neighborhood search (NVNS) and propose a generalization of this approach. In addition, we develop a heuristic stemming from this general framework and apply it on the capacitated clustering problem (CCP). Based on obtained results the proposed heuristic outperforms the current state-of-the-art for the CCP.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Leslie Pérez Cáceres, Thomas Stützle Methods for automatic algorithm configuration integrate some search mechanism for generating candidate algorithm configurations and mechanisms for handling the stochasticity of the algorithm configuration problem. One popular algorithm configurator is ParamILS, which searches the configuration space using an iterated local search algorithm. In our research, we explore variable neighborhood search mechanisms as an alternative for the one-exchange neighborhood that is searched in the local search phase of ParamILS. In this article, we explore a reduced variable neighborhood search for automatic configuration. Our experimental results are promising and indicate directions for extending our work.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Marwa Harzi, Saoussen Krichen This paper is concerned with designing an integrated transportation solver for the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is one of the most tackled transportation problems in real-world situations. It consists in determining the minimum cost distance routes for a number of homogeneous vehicles stationed at a depot, that have a task of delivering goods to a number of customers within a specified time windows. As the VRPTW is known to be NP-hard combinatorial problem, it is hard to be solve in a reasonable computational time. Therefore, numerous metaheuristics-based approaches have been developed for finding near-optimal solution for VRPTW. To cope with the VRPTW, a VND approach is proposed. In order to demonstrate the performance of the approach in term of solution quality, we apply it on benchmark instances. The VND provided better results, in terms of solution quality, than existing approaches results.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): J. Sánchez-Oro, A. Martínez-Gavara, M. Laguna, A. Duarte, R. Martí Graphs are used to represent reality in several areas of knowledge. Drawings of graphs have many applications, from project scheduling to software diagrams. The main quality desired for drawings of graphs is readability, and crossing reduction is a fundamental aesthetic criterion for a good representation of a graph. In this paper we target the edge crossing reduction in the context of incremental graph drawing, in which we want to preserve the layout of a graph over successive drawings. We propose a hybrid method based on the GRASP (Greedy Randomized Adaptive Search Procedure) and VND (Variable Neighborhood Descent) methodologies and compare it with previous methods via simulation.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Souhir Elleuch, Pierre Hansen, Bassem Jarboui, Nenad Mladenović In this paper we propose a new local search method for solving automatic programming problem and we use it within recent Variable Neighborhood Programming (VNP) method. In automatic programming area the solution is a program and the most common way is to present it by a tree T with the specific structure (i.e., it has different types of nodes). New local search tries to adapt well known “elementary tree transformation” in generating neighboring trees (programs) N (T) to the incumbent tree T. Several other neighborhoods N k (T ) are used for the shaking step of the VNP. Method is implemented on some typical problems from the literature. Results are compared with the old VNP method.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Eyder Rios, Luiz Satoru Ochi, Cristina Boeres, Igor M. Coelho, Vitor N. Coelho, Nenad Mladenović Among the methods to deal with optimization tasks, parallel metaheuristics have been used in many real-world and scientific applications to efficiently solve these kind of problems. This paper presents a novel Multi Improvement strategy for dealing with the Minimum Latency Problem (MLP), an extension the classic Traveling Salesman Problem. This strategy is embedded in a Graphics Processing Unit (GPU) local search procedure, in order to take advantage of the highly parallel processors from this architecture. In order to explore multiple neighborhoods simultaneously in a CPU/GPU heterogenous and distributed environment, a variant of the classic Variable Neighborhood Descent (VND) is also proposed, named Distributed VND (DVND). The DVND was inspired by a randomized version of the VND (called RVND) and a comparison was made, achieving competitive results in terms of solution quality. The variant of the DVND using two processes succeeded in achieving superlinear speedups up to 2.85, demonstrating that the DVND scalability and capability to explore both GPUs and CPUs. Finally, experiments demonstrate that the Multi Improvement is capable of finding better quality solutions in shorter computational times, when compared the classic Best Improvement strategy, motivating future applications in other hard optimization problems.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Cézar Augusto N. e Silva, Haroldo Gambini Santos In the Graph Drawing problem a set of symbols must be placed in a plane and their connections routed. To produce clear, easy to read diagrams, this problem is usually solved trying to minimize edges crossing and the area occupied, resulting in a NP-Hard problem. Our research focuses in drawing Entity Relationship (ER) diagrams, a challenging version of the problem where nodes have different sizes. Mathematical Programming models for the two solution phases, node placement and connection routing, are discussed and their exact resolution by an Integer Programming (IP) solver is evaluated. As the first phase proved to be specially hard to be solved exactly, a hybrid Variable Neighborhood Search (VNS) heuristic is proposed. Using IP techniques we obtained provably optimal (or close to optimal) solutions for the two different phases, at the expense of a large computational effort. We also show that our VNS based heuristic approach can produce close to optimal solutions in very short times for the hardest part of the solution process. Using either methods we have produced clearly better drawings than existing solutions.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Rita Macedo, Rachid Benmansour, Abdelhakim Artiba, Nenad Mladenović, Dragan Urošević In this paper, we focus on the scheduling of preventive railway maintenance activities. The objective is to keep the railway infrastructure in good operating conditions at low costs, also taking into account the limited available resources in what concerns crew members. Equipments degrade with usage and age and a good preventive maintenance program can greatly reduce their unreliability in the sense that expectable failures can be anticipated. We propose a mixed integer programming formulation for the problem of scheduling preventive railway maintenance activities and a Variable Neighborhood Search (VNS) algorithm to solve large instances of the problem.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Olga Rusetskaya In this paper a new approach in analyzing city behaviour and its dynamic developments is proposed. Usual attributes are divided into two larger groups: social and economic. This idea makes possible analyzing relative performances of cities from different angles, such as: two-dimensional clustering; city ranking by simple calculating their distance from origin; minimum sum of squares clustering, etc. Such an analysis is helpful to show good and week points in developing city strategy in long term period. Grouping of 77 capital cities of regions in Russia is considered. The averages of both groups, for each city are calculated, covering 12 years. Minimum sum-of-squares criterion is used for clustering and solved by variable neighborhood search. It appeared that there are a few outlier cities, two of them being non dominated or efficient. Detailed comparative analysis of the results before and after crisis (economic sanctions) are also provided.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Marwa Amous, Said Toumi, Bassem Jarboui, Mansour Eddaly This paper studies the capacitated vehicle routing problem (CVRP). Since the problem is NP-hard, a variable neighborhood search (VNS) algorithm is proposed for the CVRP with the objective to minimize the total traveled distance. The proposed algorithm includes a variable neighborhood descent (VND) algorithm based on several different neighborhood structures to intensify the search effort. Various benchmark problems including the number of customers, the capacity of vehicles are tested to evaluate the performance of proposed methodology. The experimental results indicate that the proposed algorithm provides superior solutions for well-known benchmark problems compared to those reported in the literature.

Abstract: Publication date: April 2017 Source:Electronic Notes in Discrete Mathematics, Volume 58 Author(s): Danijela Đorić, Abdessamad Ait El Cadi, Saïd Hanafi, Nenad Mladenović, Abdelhakim Artiba Maintenance optimization of railway infrastructure includes several kinds of aspects, such as safety, economic, operational, organization and regulatory issues. Among them the regulatory issues, that are fixed, increase the maintenance costs significantly. This is especially true in so-called capillary networks (local regional railway networks), where only the freight transport exists. Hence, the question is how to minimize maintenance costs with respect to regulatory issues? To solve this problem, we propose a clustering approach. The idea is to cluster tracks, considering elements of railway infrastructure as attributes. Once railway tracks are clustered in groups with similar attributes, then the maintenance can be organized more efficiently. In this paper, Variable Neighborhood Search metaheuristic is developed to solve minimum sum of squares clustering problem. Based on the results of clustering and available real and simulated data we report 22% savings in maintenance schedule for clusters.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Maksim Alekseev Algebraic manipulation detection codes were introduced in 2008 to protect data against a special type of a modification: an algebraic manipulation. One of the most effective ways of constructing such codes is based on polynomial encoding functions. In this paper a new family of polynomials is proposed, which may lead to higher detection capabilities and lower computational complexity of encoding and decoding procedures.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Tsonka Baicheva, Svetlana Topalova A conflict-avoiding code is used to guarantee that each transmitting user can send at least one packet successfully within a fixed period of time, provided that at most k out of M potential users are active simultaneously in a multiple-access collision channel. The number of codewords in a conflict-avoiding code determines the number M of potential users that can be supported in a system. That is why codes of the maximum cardinality for given parameters (optimal codes) are of interest. In this paper we determine the values of the maximum cardinality and classify up to multiplier equivalence optimal conflict-avoiding codes for 6 and 7 active users and given small lengths.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Daniele Bartoli, Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Upper bounds on the smallest size of a complete cap in PG ( N , q ) under a certain probabilistic conjecture, preprint]. In the projective space PG ( N , q ) over the Galois field of order q, N ≥ 3 , an iterative step-by-step construction of complete caps by adding a new point at every step is considered. It is proved that uncovered points are evenly placed in the space. A natural conjecture on an estimate of the number of new covered points at every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the mentioned conjecture, new upper bounds on the smallest size t 2 ( N , q ) of a complete cap in PG ( N , q ) are obtained. In particular, t 2 ( N , q ) < 1 q − 1 q N + 1 ( N + 1 ) ln q + 1 q − 3 q N + 1 ∼ q N − 1 2 ( N + 1 ) ln q . The effectiveness of the bounds is illustrated by comparison with complete caps sizes obtained by computer searches. The reasonableness of the conjecture is discussed.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Daniele Bartoli, Alexander A. Davydov, Alexey A. Kreshchuk, Stefano Marcugini, Fernanda Pambianco In this work we summarize some recent results to be included in a forthcoming paper [Bartoli, D., A. A. Davydov, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Small complete caps in PG ( 3 , q ) and PG ( 4 , q ) , preprint]. We present and analyze computational results concerning small complete caps in the projective spaces PG ( N , q ) of dimension N = 3 and N = 4 over the finite field of order q. The results have been obtained using randomized greedy algorithms and the algorithm with fixed order of points (FOP). The new complete caps are the smallest known. Based on them, we obtained new upper bounds on the minimum size t 2 ( N , q ) of a complete cap in P G ( N , q ) , N = 3 , 4 . Our investigations and results allow to conjecture that these bounds hold for all q.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Daniele Bartoli, Stefano Marcugini, Alfredo Milani, Fernanda Pambianco We describe an algorithm for testing the completeness of caps in PG ( r , q ) , q even. It allowed us to check that the 95256-cap in PG ( 12 , 4 ) recently found by Fu el al. (see [Fu Q., R. Li, L. Guo and G. Xu, Large caps in projective space PG ( r , 4 ) , Finite Fields Appl. 35 (2015), 231–246]) is complete.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Hannes Bartz, Vladimir Sidorenko Punctured Reed-Solomon (RS) and Gabidulin (G) codes over the field F q m with locators from the subfield F q can be represented as interleaving of m correspondent codes over the subfield F q or can be considered as virtual interleaving of m correspondent codes over the field F q m . Using a probabilistic unique syndrome decoder, m-interleaved or virtually interleaved codes can be decoded up to the same radius m m + 1 ( d − 1 ) , where d is the code distance in Hamming metric for RS codes and in rank metric for G codes. We show that the correspondent decoders over the subfield F q and the field F q m are equivalent and conclude that in practice one should use a decoder over the subfield since it has less complexity.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Nikolay Beluhov, Emil Kolev In this paper we consider a searching game called k-chase. A Princess occupies a vertex of given graph G and a Suitor is trying to find her. On each turn, the Suitor examines k vertices of G looking for the Princess (and, if he finds her, the game ends). Following this, the Princess moves to an adjacent vertex of G and the turn is complete. For k = 1 , we give a complete characterization of graphs for which it is possible for the Suitor to find the Princess. We also find the minimum k for which the Suitor finds the Princess when G is a rectangular grid of size 2 n × 2 n .

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Leandro Bezerra de Lima, Reginaldo Palazzo In [A. Khaleghi, D. Silva, and F.R. Kschischang, Subspace Codes, Lecture Notes in Computer Science, vol. 5921, pp. 1-21, 2009] it is proposed an efficient error-control procedure for use in Network Coding called subspace codes constructed from projective space of order m over a finite field F q , denoted by P ( F q m ) , that is, the set of all subspaces in the vector space F q m , [T. Etzion, and A. Vardy, Error Correcting Codes in Projective Space, Proc. of the 2008 IEEE Int. Symp. on Inf. Theory, 871-875, Toronto, Canada, 2008]. The projective space endowed with the subspace metric is a metric space. Such subspace codes are devised for the one use of the channel. An alternative to improve the rate and the error-correcting capabilities, without increasing the order of the finite field or the vector length, is to make use of the channel n times, this new code is known as the n-shot subspace code, [R. Nóbrega, and B. Uchôa-Filho, Multishot Codes for Network Coding: Bounds and a Multilevel Construction, Proc. of the 2009 IEEE Int. Symp. on Information Theory - ISIT-09, Seoul, South Korea, Jun. 2009]. In this paper we present the concept of geometrically uniform subspace codes and the new n-shot geometrically uniform subspace codes.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Sergey V. Bezzateev, Natalia A. Shekhunova The lower bound of the covering radius of non-binary irreducible Goppa codes is obtained.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Jürgen Bierbrauer, Stefano Marcugini, Fernanda Pambianco A ternary [66, 10, 36]3-code admitting the Mathieu group M12 as a group of automorphisms has recently been constructed by N. Pace, see [N. Pace: New ternary linear codes from projectivity groups, Discrete Mathematics 331 (2014), 22-26]. We give a construction of the Pace code in terms of M12 as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5, 6, 12). We also present a proof that the Pace code does indeed have minimum distance 36.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Iliya Bouyukliev, Dusan Bikov, Stefka Bouyuklieva In this paper we present a construction for S-boxes using quasi-cyclic codes. We obtain S-boxes with good nonlinearity.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Stefka Bouyuklieva, Radka Russeva, Emine Karatash We prove that if an extremal binary self-dual code of length n ≥ 72 has an automorphism of type 3-(c, f) then c ≥ f.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Silvia Boumova We compute the sequence of colengths in the cocharacters of the *-polynomial identities of the *-simple algebra M 2 ( F ) ⊕ M 2 ( F ) o p , char(F) = 0.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Peter Boyvalenkov, Konstantin Delchev Using linear programming techniques we derive bounds for antipodal spherical codes with few possible different distances. The possibilities for attaining our bounds are investigated and Lloyd-type theorems are proved.

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Peter Boyvalenkov, Peter Dragnev, Douglas Hardin, Edward Saff, Maya Stoyanova In this paper we introduce a framework for improvement of universal lower bounds on potential energy using the Delsarte-Yudin linear programming approach for polynomials. As a model example we consider the case of 24 points on S 3 .

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Rumen Daskalov, Mladen Manev An ( n , r ) -arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. The maximum size of an ( n , r ) -arc in PG(2, q) is denoted by m r ( 2 , q ) . It follows from [Kohnert A., Arcs in the projective planes, Online tables, www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/PG_arc_table/index.html] and [Daskalov R., and E. Metodieva, New (n,r)-arcs in PG(2, 17), PG(2, 19), and PG(2, 23), Problemi Peredachi Informatsii, 47, no. 3 (2011), 3–9. English translation: Problems of Information Transmission, 47, no. 3, (2011), 217–223] that m 3 ( 2 , 23 ) ≥ 36 . In this paper we establish that m 3 ( 2 , 23 ) ≥ 37 .

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Rumen Daskalov, Elena Metodieva An ( n , r ) -arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. The maximum size of an ( n , r ) -arc in PG ( 2 , q ) is denoted by m r ( 2 , q ) . In this paper we finish our research on new ( n , r ) -arcs in PG ( 2 , 29 ) , started in [Daskalov R., and E.Metodieva, New good (n, r)-arcs in PG(2, 29), in Proc. of Seventh International Workshop on Optimal Codes and Related Topics, 6-12 September 2013, Albena, Bulgaria, (2013), 79–84], and present six new large ( n , r ) -arcs with parameters (531, 20), (593, 22), (626, 23), (662, 24), (694, 25), and (725, 26). The constructed arcs improve the respective lower bounds on m r ( 2 , 29 ) in [Kohnert A., Arcs in the projective planes, Online tables, www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/PG_arc_table/index.html].

Abstract: Publication date: March 2017 Source:Electronic Notes in Discrete Mathematics, Volume 57 Author(s): Rumen Daskalov, Elena Metodieva An ( n , r ) -arc is a set of n points of a projective plane such that some r, but no r + 1 of them, are collinear. The maximum size of an ( n , r ) -arc in PG ( 2 , q ) is denoted by m r ( 2 , q ) . In this paper a (377, 14)-arc, a (514, 18)-arc, a (535, 19)-arc and a (565, 20)-arc in PG(2,31) are presented. The constructed arcs improve the respective lower bounds on m r ( 2 , 31 ) in [Kohnert A., Arcs in the projective planes, Online tables, www.algorithm.uni-bayreuth.de/en/research/Coding_Theory/PG_arc_table/index.html].