Domination Ratio Research Articles

Domination of triangulated discs and maximal outerplanar graphs

Given an infinite family of graphs, its domination ratio is the smallest real k such that for every large enough graph in the family, the ratio between its domination number and order is at most k. We give bounds for the domination ratios of various families of triangulated discs, disproving two conjectures of Tokunaga. For instance, we show that the domination ratio of triangulated discs of minimum degree 3 is at least 3/10. We also give a natural extension of a theorem on the domination number of maximal outerplane graphs, proven by Campos and Wakabayashi, and independently Tokunaga. Motivated by the Matheson-Tarjan Conjecture, we present observations on dominating triangulations via Jackson-Yu decomposition trees as well as an approach on how to efficiently dominate, given a triangulation, many large subtriangulations thereof. Throughout the paper, results are accompanied by computational experiments.

Perfect Domination Ratios of Archimedean Lattices

An Archimedean lattice is an infinite graph constructed from a vertex-transitive tiling of the plane by regular polygons. A dominating set of vertices is a perfect dominating set if every vertex that is not in the set is dominated exactly once. The perfect domination ratio is the minimum proportion of vertices in a perfect dominating set. Seven of the eleven Archimedean lattices can be efficiently dominated, which easily determines their perfect domination ratios. The perfect domination ratios are determined for the four Archimedean lattices that can not be efficiently dominated.

Domination ratio of a family of integer distance digraphs with arbitrary degree

An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to a finite subset $S\subseteq\mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$, defined as the minimum density of its dominating sets, is related to some number theory problems, such as tiling the integers and finding the maximum density of a set of integers with missing differences. We precisely determine the domination ratio of the integer distance graph $\Gamma(\mathbb{Z},\{1,2,\ldots,d-2,s\})$ for any integers $d$ and $s$ satisfying $d\ge2$ and $s\notin[0,d-2]$. Our result generalizes a previous result on the domination ratio of the graph $\Gamma(\mathbb{Z},\{1,s\})$ with $s\in\mathbb{Z}\setminus\{0,1\}$ and also implies the domination number of certain circulant graphs $\Gamma(\mathbb{Z}_n,S)$, where $\mathbb{Z}_n$ is the finite cyclic group of integers modulo $n$ and $S$ is a subset of $\mathbb{Z}_n$.

Development of new software enabling automatic identification of the optimal anatomical liver resectable region, incorporating preoperative liver function.

Determining the resectable region and volume of the liver prior to anatomical resection is important. The synapse Vincent (SV) system is the current method for surgical liver resection that relies on the surgeon's individual experience and skill. Additionally, in cases involving abnormal liver function, the resectable region is limited due to deteriorating liver function, thus making the determination of the hepatectomy region challenging. The current study outlines a novel 3D Hariyama-Shimoda Soft (HSS) simulation software that can be used to automatically simulate the optimal hepatectomy region under a limited resectable liver volume. The current study recruited patients with hepatic malignant tumors that were scheduled for anatomical resection. The influence of the tumor on each portal vein point was quantified in accordance with the tumor domination ratio (TDR). The resectable region was subsequently determined so that the sum of the TDR was the maximum estimated resectable liver volume (ERLV). The maximum ERLV settings utilized were within Makuuchi's criteria. ERLV was compared with the actual resected liver volume (ARLV) using SV and HSS. A total of 15 patients were included in the present study. The median ERLV was not significantly different between the two groups (P=0.15). However, the correlation between ERLV and ARLV, for SV and HSS, was statistically significant [SV ERLV (ml) = 1.139 × HSS ERLV (ml) + 30.779 (P=0.001)]. In conclusion, HSS may be an effective 3D simulation system. TDR and ERLV were indicated to be novel factors that may be incorporated into simulation software for use in anatomical resection surgery.

Domination ratio of integer distance digraphs

An integer distance digraph is the Cayley graph Γ(Z,S) of the additive group Z of all integers with respect to some finite subset S⊆Z. The domination ratio of Γ(Z,S) is the minimum density of a dominating set in Γ(Z,S). We establish some basic results on the domination ratio of Γ(Z,S) and precisely determine it when S={s,t} with s dividing t.

Average value of solutions of the bipartite quadratic assignment problem and linkages to domination analysis

We study domination analysis of algorithms for the bipartite quadratic assignment problem. A formula for the average objective function value of solutions is presented, whereas computing the median objective function value is shown to be NP-hard. An upper bound on the domination ratio of any polynomial time heuristic is given. Also, we show that heuristics that produce no worse than the average solutions have domination ratio at least 1mn. Heuristics with improved domination ratio are also presented.

A novel metric based on changes in pareto domination ratio for objective reduction of many-objective optimization problems

Although many-objective optimisation can be simplified through reduction of redundant objectives, algorithms that perform this reduction still lack a convenient method of evaluation. In this paper, we address this deficiency by proposing a new method of evaluation, on the basis of changes in the Pareto-domination ratio after a reduction has occurred. Experimental results have shown that the proposed method can perform non- redundant objective set evaluation more accurately than existing evaluation methods, and also does not need the true Pareto front beforehand.

A domination algorithm for {0,1}‐instances of the travelling salesman problem

AbstractWe present an approximation algorithm for ‐instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial‐time algorithm which has domination ratio . In other words, given a ‐edge‐weighting of the complete graph on vertices, our algorithm outputs a Hamilton cycle of with the following property: the proportion of Hamilton cycles of whose weight is smaller than that of is at most . Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial‐time algorithm with domination ratio for arbitrary edge‐weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant such that cannot be replaced by in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016

On maximum independent set of categorical product and ultimate categorical ratios of graphs

We first present polynomial algorithms to compute the independence number of the categorical product for two cographs or two splitgraphs, respectively. Then we prove that computing the maximum independent set of the categorical product of a planar graph of maximum degree three and a K4 is NP-complete. The ultimate categorical independence ratio of a graph G is defined as limk→∞⁡α(Gk)/nk. The ultimate categorical independence ratio can be computed in polynomial time for cographs, splitgraphs, permutation graphs, interval graphs and graphs of bounded treewidth. Also, we present an O⁎(3n/3)-time exact, exponential algorithm for the ultimate categorical independence ratio of general graphs. We further present a PTAS for the ultimate categorical independence ratio of planar graphs. Lastly, we show that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).

Average value of solutions for the bipartite boolean quadratic programs and rounding algorithms

We consider domination analysis of approximation algorithms for the bipartite boolean quadratic programming problem (BBQP) with m+n variables. A closed-form formula is developed to compute the average objective function value A of all solutions in O(mn) time. However, computing the median objective function value of the solutions is shown to be NP-hard. Also, we show that any solution with objective function value no worse than A dominates at least 2m+n−2 solutions and this bound is the best possible. Further, we show that such a solution can be identified in O(mn) time and hence the domination ratio of this algorithm is at least 14. We then show that for any fixed natural numbers a and b such that η=ab>1, no polynomial time approximation algorithm exists for BBQP with domination ratio larger than 1−2(1−η)η(m+n), unless P = NP. It is shown that some powerful local search algorithms can get trapped at a local maximum with objective function value less than A. One of our approximation algorithms has an interesting rounding property which provides a data dependent lower bound on the optimal objective function value. A new integer programming formulation of BBQP is also given and computational results with our rounding algorithms are reported.

Statistical quantification of the effects of viewing distance on texture perception

In general, viewers are more attracted to local features in images at a shorter viewing distance and to global features in images at a longer viewing distance. However, numerical analysis of the effect of viewing distance on human texture perception and how the perception of global and local changes under certain conditions are still undetermined. In this paper, we present statistical prediction of the relationship between the domination ratio of global and local features and the viewing distances under the control of several factors, using the logistic regression model. We synthesized textures by separately controlling global and local textural features using a texture model based on mathematical morphology, namely the primitive, grain, and point configuration texture model. Visual sensory tests were carried out on 80 subjects during two sets of experiments. The collected data were statistically analyzed using logistic regression and Akaike information criteria. Besides the main factor of viewing distance, the factors including gender, changing the order of viewing positions, and prior knowledge were also shown quantitatively to have significant influence on human texture perception. Our results showed that (1) local features of a texture were more attractive to females than males, (2) the first impression might have affected subsequent decisions in texture perception, and (3) subjects who had prior knowledge (supervised) were more sensitive to the changes in global and local dominance. (4) Regarding the interactions of the factors, prior knowledge reduced the effects of individual differences and perception condition differences on human texture perception. This study is dedicated to the construction of numerical relationships between viewing distance and human texture perception as well as to cognitive investigation of biases in global and local perceptions.

Hamilton decompositions of regular expanders: A proof of Kelly’s conjecture for large tournaments

A long-standing conjecture of Kelly states that every regular tournament on n vertices can be decomposed into (n−1)/2 edge-disjoint Hamilton cycles. We prove this conjecture for large n. In fact, we prove a far more general result, based on our recent concept of robust expansion and a new method for decomposing graphs. We show that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. This enables us to obtain numerous further results, e.g. as a special case we confirm a conjecture of Erdős on packing Hamilton cycles in random tournaments. As corollaries to the main result, we also obtain several results on packing Hamilton cycles in undirected graphs, giving e.g. the best known result on a conjecture of Nash-Williams. We also apply our result to solve a problem on the domination ratio of the Asymmetric Travelling Salesman problem, which was raised e.g. by Glover and Punnen as well as Alon, Gutin and Krivelevich.

Species domination in oats phytocenosis and its yield

The paper presents an analysis of oats phytocenosis (the crop and weeds) treated as a competitor group as concerns the species domination developing within it. The study covered oats cultivated on plots after potatoes (in crops rotation system with 25% share of oats) and twice after oats (75% share of oats in the rotation system) during the years 1990-2000. Simpson's species domination ratio and domination structure in the association identifying groups of: dominants, subdominants, influents and accessory species, were computed and studied. Relations between the studied phytocenosis features and weather conditions and oats yield were also determined. It was shown that quantitative relations in the assessed competitor associations were characterized by high variability between years and vegetation periods while no major differences were found between positions in the crop rotation systems. The density of spring phytocenosis during 11 years of studies increased resulting from increasing numbers of weeds appearing; during the late vegetation period they changed the density of associations slightly. In the spring, the groups competitive to the group of dominants consisted usually of the crop and <i>Thlaspi arvense</i> as well as <i>Chenopodium album</i>; with the passage of years their position strengthened. The enrichment of species in phytocenosis was represented mainly by an increase in numbers of accessory species. At the end of vegetation the importance of oats as a dominant increased significantly; during some seasons <i>Chenopodium album</i> co-dominated, however, with the passage of years the group of subdominants grew. Domination ratios for the studied competitor groups decreased with the passage of years, parallel to decrease of oats share in them. The studied features of phytocenosis showed a stronger correlation to the passage of years than the weather development. Invasiveness of weeds increased during consecutive seasons of cultivation while it showed less correlation with the density of oats. Oats yields showed a slight decreasing trend from year to year and was highly determined by the density of the entire weeds assemblage and in the spring by density and share of dominating species.

On total domination and support vertices of a tree

The total domination number γt(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is adjacent to some vertex in D. In this paper we prove two new upper bounds on the total domination number of a tree related to particular support vertices (vertices adjacent to leaves) of the tree. One of these bounds improves a 2004 result of Chellali and Haynes [1]. In addition, we prove some bounds on the total domination ratio of trees.

Automatic sound recognition for the hearing impaired

We present a wearable sound recognition system to assist the hearing impaired. Traditionally, hearing aid dogs are specially trained to facilitate the daily life of the hearing impaired. However, since training hearing aid dogs is costly and time-consuming, it would be desirable to substitute them with an automatic sound recognition system using speech recognition technologies. As the sound recognition system will be used in home environments where background noises and reverberations are high, conventional speech recognition techniques are not directly applicable, since their performance drops off rapidly in these environments. In this paper, we introduce a new sound recognition algorithm which is optimized for mechanical sounds such as doorbells. The new algorithm uses a new distance measure called the normalized peak domination ratio (NPDR) that is based on the characteristic spectral peaks of these sounds. The proposed algorithm showed a sound recognition accuracy of 99.7%, and noise rejection accuracy of 99.7%.

Dominance guarantees for above-average solutions

Gutin et al. [G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1–2) (2002) 107–116] proved that, in the ATSP problem, a tour of weight not exceeding the weight of an average tour is of dominance ratio at least 1 / ( n − 1 ) for all n ≠ 6 . (Tours with this property can be easily obtained.) In [N. Alon, G. Gutin, M. Krivelevich, Algorithms with large domination ratio, Journal on Algorithms 50 (2004) 118–131; G. Gutin, A. Vainshtein, A. Yeo, Domination analysis of combinatorial optimization problems, Discrete Applied Mathematics 129 (2–3) (2003) 513–520; G. Gutin, A. Yeo, Polynomial approximation algorithms for the TSP and the QAP with a factorial domination number, Discrete Applied Mathematics 119 (1–2) (2002) 107–116], algorithms with large dominance ratio were provided for Max Cut, Max r - Sat, ATSP, and other problems. All these algorithms share a common property — they provide solutions of quality guaranteed to be not worse than the average solution value. In this paper we show that, in general, this property by itself does not necessarily ensure a good performance in terms of dominance. Specifically, we show that for the MaxSat problem, algorithms with this property might perform poorly in terms of dominance.

Domination analysis for minimum multiprocessor scheduling

Let P be a combinatorial optimization problem, and let A be an approximation algorithm for P . The domination ratio domr ( A , s ) is the maximal real q such that the solution x ( I ) obtained by A for any instance I of P of size s is not worse than at least the fraction q of the feasible solutions of I . We say that P admits an asymptotic domination ratio one (ADRO) algorithm if there is a polynomial time approximation algorithm A for P such that lim s → ∞ domr ( A , s ) = 1 . Alon, Gutin and Krivelevich [Algorithms with large domination ratio, J. Algorithms 50 (2004) 118–131] proved that the partition problem admits an ADRO algorithm. We extend their result to the minimum multiprocessor scheduling problem.

Algorithms with large domination ratio

Let P be an optimization problem, and let A be an approximation algorithm for P. The domination ratio domr( A, n) is the maximum real q such that the solution x( I) obtained by A for any instance I of P of size n is not worse than at least a fraction q of the feasible solutions of I. We describe a deterministic, polynomial-time algorithm with domination ratio 1− o(1) for the partition problem, and a deterministic, polynomial-time algorithm with domination ratio Ω(1) for the MaxCut problem and for some far-reaching extensions of it, including Max- r-Sat, for each fixed r. The techniques combine combinatorial and probabilistic methods with tools from harmonic analysis.

Domination analysis of combinatorial optimization problems

We use the notion of domination ratio introduced by Glover and Punnen in 1997 to present a new classification of combinatorial optimization (CO) problems: DOM -easy and DOM -hard problems. It follows from results already proved in the 1970s that min TSP (both symmetric and asymmetric versions) is DOM -easy. We prove that several CO problems are DOM -easy including weighted max k- SAT and max cut. We show that some other problems, such as max clique and min vertex cover, are DOM -hard unless P=NP.

Some indicators of multinational domination of national economies: analysis for the UK and other developed countries

The paper starts by giving an overview of the foreign direct investment position, relative to the size of the domestic economy, for major developed western economies. It then develops the view that, in assessing the full impact of foreign direct investment on national economies, a global approach should be considered: such approach stresses the relevance of multinational spread of activities and the need to consider the cumulative effect of both outward and inward direct investment. This leads to the introduction of a ‘multinational domination ratio’ for national economies and to estimates of it for some main industrialized western economies over a 20-year period. A first attempt to use this global approach in order to assess the possible impact of UK foreign direct investment (both inward and outward) on trade leads to the computation of elasticities of trade with respect to foreign direct investment.