Discrete Mathematics, Algorithms and Applications
About: Discrete Mathematics, Algorithms and Applications is an academic journal published by World Scientific. The journal publishes majorly in the area(s): Mathematics & Vertex (graph theory). It has an ISSN identifier of 1793-8309. Over the lifetime, 1048 publications have been published receiving 4086 citations.
TL;DR: Here it is proved the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles, and an approximation algorithm and an exact algorithm are presented for the Problem ofFinding a maximum weight exchange involving cycles of bounded length.
Abstract: Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration.
TL;DR: The ratio to compare the size of an MIS with a minimum CDS becomes a theoretical upper bound for approximation algorithms to compute CDS and this upper bound is improved with the help of Voronoi diagram and Euler's formula.
Abstract: Connected Dominating Set is widely used as virtual backbone in wireless networks to improve network performance and optimize routing protocols. Based on special characteristics of ad-hoc and sensor networks, we usually use unit disk graph to represent the corresponding geometrical structures, where each node has a unit transmission range and two nodes are said to be adjacent if the distance between them is less than 1. Since every Maximal Independent Set (MIS) is a dominating set and it is easy to construct, we can firstly find an MIS and then connect it into a Connected Dominating Set (CDS). Therefore, the ratio to compare the size of an MIS with a minimum CDS becomes a theoretical upper bound for approximation algorithms to compute CDS. In our paper, with the help of Voronoi diagram and Euler's formula, we improved this upper bound, so that improved the approximations based on this relation.
TL;DR: The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks.
Abstract: Circulant graphs have been extensively investigated over the past 30 years because of their broad application to different fields of theory and practice. Two known surveys on circulant networks including a survey on undirected circulants have been published: by Bermond et al. [Distributed loop computer networks: A survey, J. Parallel Distributed Comput.24 (1995) 2–10] and by Hwang [A survey on multi-loop networks, Theoret. Comput. Sci.299 (2003) 107–121]. The present paper includes the results which have not been presented there, in particular the works of Russian researchers, and also a lot of new results obtained in the area of research of circulant networks. We focus on the survey connected with study of structural and communicative properties of circulant networks.
TL;DR: This work explores group testing strategies that use a nearly optimal number of pools and a few stages although d is not known beforehand, and provides a classification of types of randomized search strategies in general.
Abstract: Suppose that we are given a set of n elements d of which have a property called defective. A group test can check for any subset, called a pool, whether it contains a defective. It is known that a nearly optimal number of O(d log (n/d)) pools in 2 stages (where tests within a stage are done in parallel) are sufficient, but then the searcher must know d in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known beforehand. We prove a lower bound of O(log d log log d) stages and a more general pools vs. stages tradeoff. This is almost tight, since O(log d) stages are sufficient for a strategy with O(d log n) pools. As opposed to this negative result, we devise a randomized strategy using O(d log (n/d)) pools in 3 stages, with any desired success probability 1-epsilon. With some additional measures even 2 stages are enough. Open questions concern the optimal constant factors and practical implications. A related problem motivated by, e.g., biological network analysis is to learn hidden vertex covers of a small size k in unknown graphs by edge group tests. Does a given subset of vertices contain an edge?) We give a 1-stage strategy using O(k^3 log n) pools, with any parameterized algorithm for vertex cover enumeration as a decoder. During the course of this work we also provide a classification of types of randomized search strategies in general.
TL;DR: Here, the behavior of this index under several graph operations is studied and the results are applied to find the F-index of different chemically interesting molecular graphs and nano-structures.
Abstract: The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total π-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.