Exact and heuristic algorithms for the weighted total domination problem

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

We consider the Laplacian matrix of a weighted graph, and how the algebraic connectivity, α, behaves when considered as a function of a single edge weight. Under suitable differentiability conditions, we bound the first derivative of α from above, show that α is necessarily concave down, and produce a lower bound on the second derivative of α. When α is simple, we discuss the effect of increasing an edge weight on the corresponding Fiedler vector. We also compute the limiting value of α as the edge weight increases to infinity.

We study a large-scale resource allocation problem with a convex, separable, not necessarily differentiable objective function that includes uncertain parameters falling under an interval uncertainty set, considering a set of deterministic constraints. We devise an exact algorithm to solve the minimax regret formulation of this problem, which is NP-hard, and we show that the proposed Benders-type decomposition algorithm converges to an [Formula: see text]-optimal solution in finite time. We evaluate the performance of the proposed algorithm via an extensive computational study, and our results show that the proposed algorithm provides efficient solutions to large-scale problems, especially when the objective function is differentiable. Although the computation time takes longer for problems with nondifferentiable objective functions as expected, we show that good quality, near-optimal solutions can be achieved in shorter runtimes by using our exact approach. We also develop two heuristic approaches, which are partially based on our exact algorithm, and show that the merit of the proposed exact approach lies in both providing an [Formula: see text]-optimal solution and providing good quality near-optimal solutions by laying the foundation for efficient heuristic approaches.

We present an algorithm which calculates a minimum cut and its weight in an undirected graph with nonnegative real edge weights, n vertices and m edges, in time O(max(log n, min(m/n,δG/e)) n2), where e is the minimal edge weight, and δG is the minimal weighted degree. For integer edge weights this time is further improved to O(δG n2) and O(λG n2). In both cases these bounds are improvements of the previously known best bounds of deterministic algorithms. These were O(nm + n2 log n) for real edge weights and O(nM + n2) and O(M + λG n2) for integer weights, where M is the sum of all edge weights.

In the Orienteering Problem (OP), we are given an undirected graph with edge weights and node prizes. The problem calls for a simple cycle whose total edge weight does not exceed a given threshold, while visiting a subset of nodes with maximum total prize. This NP-hard problem arises in routing and scheduling applications. We describe a branch-and-cut algorithm for finding an optimal OP solution. The algorithm is based on several families of valid inequalities. We also introduce a family of cuts, called conditional cuts, which can cut off the optimal OP solution, and propose an effective way to use them within the overall branch-and-cut framework. Exact and heuristic separation algorithms are described, as well as heuristic procedures to produce near-optimal OP solutions. An extensive computational analysis on several classes of both real-world and random instances is reported. The algorithm proved to be able to solve to optimality large-scale instances involving up to 500 nodes, within acceptable computing time. This compares favorably with previous published methods.

The weighted total domination problem (WTDP) belongs to the family of dominating set problems. Given an edge- and vertex- weighted graph, the WTDP consists in selecting a total dominating set D, such that the sum of vertices and edges weights of the subgraph induced by D plus, for each vertex not in D, the minimum weight of its edge to a vertex in D is minimized. A total dominating set D is a subset of the graph’s vertices, such that every vertex, including those in D, is at least adjacent to one vertex in D. This problem arises in many real-life applications closely related to covering and independent set problems; however, it remains computationally challenging due to its $$\mathcal{N}\mathcal{P}$$ N P -hardness. This work presents a variable neighborhood search (VNS) procedure to tackle the WTDP, and investigates the advantages and disadvantages of a multi-start strategy within VNS methodology. In addition, we develop a biased greedy randomized adaptive search procedure (Biased GRASP) that keeps adding elements once a feasible solution is found to produce high-quality initial solutions. We perform extensive numerical analysis to look into the influences of the algorithmic components and to disclose the contribution of the elements and strategies of our method. Finally, the empirical analysis shows that our proposal outperforms the state-of-art results, and the statistical analysis confirms the superiority of our proposal to find the best total dominating sets.

For a graph G = (V; E), a set S V (G) is a total dominating set if it is dominating and both hSi has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S V (G) is a total restrained dominating set if it is total dominating and hV (G) Si has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.

The game total domination problem is log-complete in PSPACE

A set S ofvertices in a graphG is a total dominating set of G ifevery vertex ofG is adjacentto some vertex in S. We characterize the set ofvertices ofa tree that are contained in all, or inno, minimum total dominating sets ofthe tree. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Total domination; Total domination number; Tree 1. IntroductionLet G=(V;E) be a simple undirected graph with | V | =n. The open neighborhoodof v ∈ V is N(v)= { u ∈ V | uv ∈ E } . A set S ⊆ V is a dominating set of G ifeveryvertex in V − S is adjacent to a vertex of S. The domination number (G)istheminimum cardinality ofa dominating set ofG. A set S ⊆ V is a total dominating set,abbreviated TDS,ofG ifevery vertex inV is adjacent to a vertex in S. Every graphwithout isolated vertices has a TDS, since S =V is such a set. The total dominationnumber of G, denoted by t (G), is the minimum cardinality ofa TDS. A TDS ofcardinality t (G) is called a t (G)-set. ∗ Corresponding author.E-mail address: henning@nu.ac.za (M.A. Henning).

As a promising computing architecture, fog computing has attracted lots of attention from industry and research communities in recent years. With the huge number of heterogeneous and distributed devices performing computational and storage tasks between the cloud and users, fog computing can be an answer to the surging challenges in today's networks. However, due to the decentralized and heterogeneous nature of fog networks, planning a fog network can be a complicated and challenging task. To our best knowledge, little work has been done on the planning and designing of fog computing networks. To deal with this problem, we first propose a multi-objective mathematical model that simultaneously deals with the fog node placement, fog node dimensioning and demand routing. The model optimizes the tradeo front (Pareto front) between capital expenditure and network delay in dual objective functions. Then, we analyze the performance of an exact algorithm (branch and bound) and two evolutionary algorithms (genetic algorithm and particle swarm algorithm) on this problem, showing that the evolutionary algorithms oer a good balance between the Pareto optimality and computation time e ciency. Inspired by the existing evolutionary algorithms, we proposed a new evolutionary algorithm, named PSONSGA, which combines the convergence e ciency from NSGA-II and the searching e ciency from SMPSO. The results demonstrate that the evolutionary algorithms are highly e cient compared to the exact algorithm. Among the three evolutionary algorithms, the algorithm we proposed (PSONSGA) gives the best Pareto front solutions which shows the good convergence to the true optimal front and the evenly distribution character. The proposed algorithm can be a valuable planning tool for real-world fog network planning.

The vehicle routing problem for several agents among the objects of two types is considered. The comparative analysis of the exact algorithm for solving this problem and heuristic algorithms is proposed. The exact algorithm is based on the branch and bound method. The ant colony algorithm and the genetic algorithm were chosen as approximate. The algorithms run time is analyzed and the results of their work are compared in terms of proximity to the optimal solution.

A set S of vertices of a graph G is a total dominating set, if every vertex of V(G) is adjacent to some vertex in S. The total domination number of G, denoted by t(G), is the minimum cardinality of...

As we have seen in Chapter 3, the decision problems associated with computing the domination, total domination, and independent domination numbers of arbitrary graphs are all NP-complete. Given the difficulty of determining the exact values of these domination numbers, much of the research involves the determination of tight lower and upper bounds for these numbers. In this chapter, we present some of the more basic bounds on the domination, total domination, and independent domination numbers of graphs. Additional bounds on these parameters will be given throughout the text, particularly in Chapter 6 in terms of minimum degree and Chapter 8 in terms of size.

The prize-collecting dominating set (PCDS) problem is the generalization of the minimum dominating set (MDS) problem. The MDS problem requires a dominating set in a given graph, namely a subset of vertexes. Any vertex in the graph belongs to the closed neighborhood of the subset of vertexes, where the subset of vertexes with the smallest cardinality is the MDS. In the PCDS problem: given an undirected graph, its vertex setV has nonnegative weighting functionω and nonnegative penalty functionπ . Finding a vertex subset D⊂V, for vertexes do not belong to the closed neighborhood of D , we need to pay penalties for them. The objective of this issue is to minimize the sum of weights and penalties. As we all know that the MDS problem is NP-hard in general graph, obviously, PCDS problem is NP-hard, too., In certain cases, the PCDS problem can achieve better results than the MDS problem, which has a certain value in practical applications such as facility location and logistics management. Whether the problem is limited to a special graph will get useful structural characterization and corresponding algorithm results is considered, designed polynomial time exact algorithm on the star and according to the r -LMP approximation algorithm, designed the 2-LMP approximation algorithm on the path and cycle.

Genetic algorithms have demonstrated considerable success in providing good solutions to many NP-hard optimization problems. For such problems, exact algorithms that always find an optimal solution are only useful for small toy problems, so heuristic algorithms such as the genetic algorithm must be used in practice. In this paper, we apply the genetic algorithm to the NP-hard problem of multiple fault diagnosis (MFD). We compare a pure genetic algorithm with several variants that include local improvement operators. These operators, which are often domain-specific, are used to accelerate the genetic algorithm in converging on optimal solutions. Our empirical results indicate that by using the appropriate local improvement operator, the genetic algorithm is able to find an optimal solution in all but a tiny fraction of the cases and at a speed orders of magnitude faster than exact algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>