Variable Neighborhood Search Approach for Solving Roman and Weak Roman Domination Problems on Graphs

In this paper Roman and weak Roman domination problems on graphs are considered. Given that both problems are NP hard, a new heuristic approach, based on a Variable Neighborhood Search (VNS), is presented. The presented algorithm is tested on instances known from the literature, with up to 600 vertices. The VNS approach is justified since it was able to achieve an optimal solution value on the majority of instances where the optimal solution value is known. Also, for the majority of instances where optimization solvers found a solution value but were unable to prove it to be optimal, the VNS algorithm achieves an even better solution value

Theoretical Computer Science and Discrete Mathematics

This book includes 15 articles published in the Special Issue "Theoretical Computer Science and Discrete Mathematics" of Symmetry (ISSN 2073-8994). This Special Issue is devoted to original and significant contributions to theoretical computer science and discrete mathematics. The aim was to bring together research papers linking different areas of discrete mathematics and theoretical computer science, as well as applications of discrete mathematics to other areas of science and technology. The Special Issue covers topics in discrete mathematics including (but not limited to) graph theory, cryptography, numerical semigroups, discrete optimization, algorithms, and complexity

Generalized Domination in Graphs with Applications in Wireless Networks

The objective of this research is to study practical generalization of domination in graphs and explore the theoretical and computational aspects of models arising in the design of wireless networks. For the construction of a virtual backbone of a wireless ad-hoc network, two different models are proposed concerning reliability and robustness. This dissertation also considers wireless sensor placement problems with various additional constraints that reflect different real-life scenarios. In wireless ad-hoc network, a connected dominating set (CDS) can be used to serve as a virtual backbone, facilitating communication among the members in the network. Most literature focuses on creating the smallest virtual backbone without considering the distance that a message has to travel from a source until it reaches its desired destination. However, recent research shows that the chance of loss of a message in transmission increases as the distance that the message has to travel increases. We propose CDS with bounded diameter, called dominating s-club (DsC) for s ≥ 1, to model a reliable virtual backbone. An ideal virtual backbone should retain its structure after the failure of a certain number of vertices. The issue of robustness under vertex failure is considered by studying k-connected m-dominating set. We describe several structural properties that hold form ≥ k, but fail when m < k. Three different formulations based on vertex-cut inequalities are shown depending on the value of k and m. The computational results show that the proposed lazy-constraint approach compares favorably with existing methods for the minimum connected dominating set problem (for k = m = 1). The experimental results for k = m = 2, 3, 4 are presented as well. In the wireless sensor placement problem, the objective is often to place a minimum number of sensors while monitoring all sites of interest, and this can be modeled by dominating set. In some practical situations, however, there could be a location where a sensor cannot be placed because of environmental restrictions. Motivated by these practical scenarios, we introduce varieties of dominating set and the corresponding optimization problems. For these new problems, we consider the computational complexity, mathematical programming formulation, and analytical bounds on the size of structures of interest. We solve these problems using a general commercial solver and compare its performance with that of simulated annealing

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Geometric Inhomogeneous Random Graphs for Algorithm Engineering

The design and analysis of graph algorithms is heavily based on the worst case. In practice, however, many algorithms perform much better than the worst case would suggest. Furthermore, various problems can be tackled more efficiently if one assumes the input to be, in a sense, realistic. The field of network science, which studies the structure and emergence of real-world networks, identifies locality and heterogeneity as two frequently occurring properties. A popular model that captures these properties are geometric inhomogeneous random graphs (GIRGs), which is a generalization of hyperbolic random graphs (HRGs). Aside from their importance to network science, GIRGs can be an immensely valuable tool in algorithm engineering. Since they convincingly mimic real-world networks, guarantees about quality and performance of an algorithm on instances of the model can be transferred to real-world applications. They have model parameters to control the amount of heterogeneity and locality, which allows to evaluate those properties in isolation while keeping the rest fixed. Moreover, they can be efficiently generated which allows for experimental analysis. While realistic instances are often rare, generated instances are readily available. Furthermore, the underlying geometry of GIRGs helps to visualize the network, e.g.,~for debugging or to improve understanding of its structure. The aim of this work is to demonstrate the capabilities of geometric inhomogeneous random graphs in algorithm engineering and establish them as routine tools to replace previous models like the Erd\H{o}s-R{\\u27e}nyi model, where each edge exists with equal probability. We utilize geometric inhomogeneous random graphs to design, evaluate, and optimize efficient algorithms for realistic inputs. In detail, we provide the currently fastest sequential generator for GIRGs and HRGs and describe algorithms for maximum flow, directed spanning arborescence, cluster editing, and hitting set. For all four problems, our implementations beat the state-of-the-art on realistic inputs. On top of providing crucial benchmark instances, GIRGs allow us to obtain valuable insights. Most notably, our efficient generator allows us to experimentally show sublinear running time of our flow algorithm, investigate the solution structure of cluster editing, complement our benchmark set of arborescence instances with a density for which there are no real-world networks available, and generate networks with adjustable locality and heterogeneity to reveal the effects of these properties on our algorithms