Quantum algorithm for minimum dominating set problem with circuit design

Using quantum algorithms to solve various problems has attracted widespread attention with the development of quantum computing.Researchers are particularly interested in using the acceleration properties of quantum algorithms to solve NP-complete problems.This paper focuses on the well-known NP-complete problem of finding the minimum dominating set in undirected graphs.To expedite the search process,a quantum algorithm employing Grover’s search is proposed.However,a challenge arises from the unknown number of solutions for the minimum dominating set,rendering direct usage of original Grover’s search impossible.Thus,a swap test method is introduced to ascertain the number of iterations required.The oracle,diffusion operators,and swap test are designed with achievable quantum gates.The query complexity is O(1.414^(n))and the space complexity is O(n).To validate the proposed approach,qiskit software package is employed to simulate the quantum circuit,yielding the anticipated results.

A Game Theoretic Approach for a Minimal Secure Dominating Set

The secure dominating set(SDS),a variant of the dominating set,is an important combinatorial structure used in wireless networks.In this paper,we apply algorithmic game theory to study the minimum secure dominating set(Min SDS) problem in a multi-agent system.We design a game framework for SDS and show that every Nash equilibrium(NE) is a minimal SDS,which is also a Pareto-optimal solution.We prove that the proposed game is an exact potential game,and thus NE exists,and design a polynomial-time distributed local algorithm which converges to an NE in O(n) rounds of interactions.Extensive experiments are done to test the performance of our algorithm,and some interesting phenomena are witnessed.

On the Maximum Number of Dominating Classes in Graph Coloring

We investigate the dominating-c-color number,, of a graph G. That is the maximum number of color classes that are also dominating when G is colored using colors. We show that where is the join of G and . This result allows us to construct classes of graphs such that and thus provide some information regarding two questions raised in [1] and [2].

Twin domination in generalized Kautz digraphs

Let G = （V,A） be a digraph.A set T of vertices of G is a twin dominating set of G if for every vertex v ∈ V ／ T.There exist u,w ∈ T （possibly u = w） such that （u,v）,（v,w） ∈ A.The twin domination number γ＊（G） of G is the cardinality of a minimum twin dominating set of G.In this paper we consider the twin domination number in generalized Kautz digraphs GK（n,d）.In these digraphs,we establish bounds on the twin domination number and give a sufficient condition for the twin domination number attaining the lower bound.We give the exact values of the twin domination numbers by constructing minimum twin dominating sets for some special generalized Kautz digraphs.

Backbone formulation algorithm in wireless sensor network based on cross-entropy method

In wireless sensor network,virtual backbone is a cost effective broadcasting method.Connected dominating set formation is proposed to construct a virtual backbone.However,it is NP-Hard to find a minimum connected dominating set in an arbitrary graph.In this paper,based on cross-entropy method,we present a novel backbone formulation algorithm(BFA-CE)in wireless sensor network.In BFA-CE,a maximal independent set is got at first and nodes in the independent set are required to get their action sets.Based on those action sets,a backbone is generated with the cross-entropy method.Simulation results show that our algorithm can effectively reduce the size of backbone network within a reasonable message overhead,and it has lower average node degree.This approach can be potentially used in designing efficient broadcasting strategy or working as a backup routing of wireless sensor network.

An improved master-apprentice evolutionary algorithm for minimum independent dominating set problem

The minimum independent dominance set(MIDS)problem is an important version of the dominating set with some other applications.In this work,we present an improved master-apprentice evolutionary algorithm for solving the MIDS problem based on a path-breaking strategy called MAE-PB.The proposed MAE-PB algorithm combines a construction function for the initial solution generation and candidate solution restarting.It is a multiple neighborhood-based local search algorithm that improves the quality of the solution using a path-breaking strategy for solution recombination based on master and apprentice solutions and a perturbation strategy for disturbing the solution when the algorithm cannot improve the solution quality within a certain number of steps.We show the competitiveness of the MAE-PB algorithm by presenting the computational results on classical benchmarks from the literature and a suite of massive graphs from real-world applications.The results show that the MAE-PB algorithm achieves high performance.In particular,for the classical benchmarks,the MAE-PB algorithm obtains the best-known results for seven instances,whereas for several massive graphs,it improves the best-known results for 62 instances.We investigate the proposed key ingredients to determine their impact on the performance of the proposed algorithm.

Paired, Total, and Connected Domination on the Queen’s Graph Revisited

The question associated with total domination on the queen’s graph has a long and rich history, first having been posed by Ahrens in 1910 [1]. The question is this: What is the minimum number of queens needed so that every square of an n × n board is attacked? Beginning in 2005 with Amirabadi, Burchett, and Hedetniemi [2] [3], work on this problem, and two other related problems, has seen progress. Bounds have been given for the values of all three domination parameters on the queen’s graph. In this paper, formations of queens are given that provide new bounds for the values of total, paired, and connected domination on the queen’s graph, denoted , , and respectively. For any n × n board size, the new bound of is arrived at, along with the separate bounds of , for with , and , for with .

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS （PART I：EQUAL TIMES AND COSTS）

Abstract Most papers in scheduling research have treated individual job processing times as fixed parameters. However, in many practical situations, a manager may control processing time by reallocating resources. In this paper, authors consider a machine scheduling problem with controllable processing times. In the first part of this paper, a special case where the processing times and compression costs are uniform among jobs is discussed. Theoretical results are derived that aid in developing an O(n 2) algorithm to slove the problem optimally. In the second part of this paper, authors generalize the discussion to general case. An effective heuristic to the general problem will be presented.

SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND COMPRESSION COSTS ： PROOF OF THEOREMS

Abstract This report is virtually the appendix part of the author's previous paper which includes the proofs for the theorems and lemmas.

SOME RESULTS ON DOMINATION NUMBER OF PRODUCTS OF GRAPHS

Let G=(V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex x∈V-D,x is adjacent to at least one vertex of D . Let γ(G) and γ c(G) denote the domination and connected domination number of G , respectively. In 1965,Vizing conjectured that if G×H is the Cartesian product of G and H , thenγ(G×H)≥γ(G)·γ(H).In this paper, it is showed that the conjecture holds if γ(H) ≠ γ c(H) .And for paths P m and P n , a lower bound and an upper bound for γ(P m×P n) are obtained.

A Distributed Design for Minimum 2-Connected m-Dominating Set in Bidirectional Wireless Ad-Hoc Networks

Wireless ad-hoc network is widely used in many fields for its convenience and outstanding suitability. Because of the inherent lack of infrastructure and the nature of wireless channels, people select the k-Connected m-Dominating Set （（k,m）-CDS） in a network as a fault-tolerant virtual backbone to help the routing process, which will save the energy of non-dominators and improve the network performance significantly. Considering the economic cost and efficiency, we choose （2,m）-CDS as the object of this paper, which is helpful enough in practical applications and has a smaller size. We firstly study the existing algorithms for （k,m）- CDS and figure out the problems of these designs. Then we propose a new distributed algorithm named Dominating Set Based AIgorithm （DSBA） with three sub-routines： Dominating Set AIgorithm （DSA）, Connection Algorithm （CA）, and Connectivity Expansion Algorithm （CEA）. Instead of commonly used Maximal Independent Set （MIS）, we pick dominating set directly from the given graph, and then connect them by a two-step ring based connecting strategy to satisfy the 2-connectivity. We also provide the correctness and complexity analysis of DSBA. At last, we compare DSBA with the last construction Distributed Deterministic Algorithm （DDA） by several numerical experiments. The simulation results show that DSBA improves over 30 percent of the performance of DDA, proving that DSBA is more practical for real-world applications.

A Binary Particle Swarm Optimization for the Minimum Weight Dominating Set Problem

The minimum weight dominating set problem （MWDSP） is an NP-hard problem with a lot of real-world applications. Several heuristic algorithms have been presented to produce good quality solutions. However, the solution time of them grows very quickly as the size of the instance increases. In this paper, we propose a binary particle swarm optimization （FBPSO） for solving the MWDSP approximately. Based on the characteristic of MWDSP, this approach designs a new position updating rule to guide the search to a promising area. An iterated greedy tabu search is used to enhance the solution quality quickly. In addition, several stochastic strategies are employed to diversify the search and prevent premature convergence. These methods maintain a good balance between the exploration and the exploitation. Experimental studies on 106 groups of 1 060 instances show that FBPSO is able to identify near optimal solutions in a short running time. The average deviation between the solutions obtained by FBPSO and the best known solutions is 0.441%. Moreover, the average solution time of FBPSO is much less than that of other existing algorithms. In particular, with the increasing of instance size, the solution time of FBPSO grows much more slowly than that of other existing algorithms.

Directed Dominating Set Problem Studied by Cavity Method:Warning Propagation and Population Dynamics

The minimal dominating set for a digraph(directed graph) is a prototypical hard combinatorial optimization problem. In a previous paper, we studied this problem using the cavity method. Although we found a solution for a given graph that gives very good estimate of the minimal dominating size, we further developed the one step replica symmetry breaking theory to determine the ground state energy of the undirected minimal dominating set problem. The solution space for the undirected minimal dominating set problem exhibits both condensation transition and cluster transition on regular random graphs. We also developed the zero temperature survey propagation algorithm on undirected Erds-Rnyi graphs to find the ground state energy. In this paper we continue to develope the one step replica symmetry breaking theory to find the ground state energy for the directed minimal dominating set problem. We find the following.(i) The warning propagation equation can not converge when the connectivity is greater than the core percolation threshold value of 3.704. Positive edges have two types warning, but the negative edges have one.(ii) We determine the ground state energy and the transition point of the Erd?os-R′enyi random graph.(iii) The survey propagation decimation algorithm has good results comparable with the belief propagation decimation algorithm.

Statistical mechanics of the directed 2-distance minimal dominating set problem

The directed L-distance minimal dominating set(MDS) problem has wide practical applications in the fields of computer science and communication networks. Here, we study this problem from the perspective of purely theoretical interest. We only give results for an Erdós Rényi(ER)random graph and regular random(RR) graph, but this work can be extended to any type of network. We develop spin glass theory to study the directed 2-distance MDS problem. First, we find that the belief propagation(BP) algorithm does not converge when the inverse temperatureβ exceeds a threshold on either an ER random network or RR network. Second, the entropy density of replica symmetric theory has a transition point at a finite β on a regular random graph when the arc density exceeds 2 and on an ER random graph when the arc density exceeds3.3;there is no entropy transition point(or β = ■) in other circumstances. Third, the results of the replica symmetry(RS) theory are in agreement with those of BP algorithm while the results of the BP decimation algorithm are better than those of the greedy heuristic algorithm.

Approximation Algorithms for Steiner Connected Dominating Set

Steiner connected dominating set （SCDS） is a generalization of the famous connected dominating set problem, where only a specified set of required vertices has to be dominated by a connected dominating set, and known to be NP- hard. This paper firstly modifies the SCDS algorithm of Guha and Khuller and achieves a worst case approximation ratio of （2 ＋ 1/（m - 1））H（min（△, k）） ＋O（1）, which outperforms the previous best result （c ＋ 1）H（min（△, k）） ＋ O（1） in the case of m ≥ 1 ＋1/（c - 1）, where c is the best approximation ratio for Steiner tree, A is the maximum degree of the graph, k is the cardinality of the set of required vertices, m is an optional integer satisfying 0 ≤ m ≤ min（△, k） and H is the harmonic function. This paper also proposes another approximation algorithm which is based on a greedy approach. The second algorithm can establish a worst case approximation ratio of 2 ln（min（△, k）） ＋ O（1）, which can also be improved to 2 lnk if the optimal solution is greater than c·e^2c＋1/2（c＋1）.

Integrated dominating and hit set-inspired unequal clusteringbased data aggregation in wireless sensor networks

Purpose-In wireless sensor networks,improving the network lifetime is considered as the prime objective that needs to be significantly addressed during data aggregation.Among the traditional data aggregation techniques,cluster-based dominating set algorithms are identified as more effective in aggregating data through cluster heads.But,the existing cluster-based dominating set algorithms suffer from a major drawback of energy deficiency when a large number of communicating nodes need to collaborate for transferring the aggregated data.Further,due to this reason,the energy of each communicating node is gradually decreased and the network lifetime is also decreased.To increase the lifetime of the network,the proposed algorithm uses two sets:Dominating set and hit set.Design/methodology/approach-The proposed algorithm uses two sets:Dominating set and hit set.The dominating set constructs an unequal clustering,and the hit set minimizes the number of communicating nodes by selecting the optimized cluster head for transferring the aggregated data to the base station.The simulation results also infer that the proposed optimized unequal clustering algorithm(OUCA)is greater in improving the network lifetime to a maximum amount of 22%than the existing cluster head selection approach considered for examination.Findings-In this paper,lifetime of the network is prolonged by constructing an unequal cluster using the dominating set and electing an optimized cluster head using hit set.The dominator set chooses the dominator based on the remaining energy and its node degree of each node.The optimized cluster head is chosen by the hit set to minimize the number of communicating nodes in the network.The proposed algorithm effectively constructs the clusters with a minimum number of communicating nodes using the dominating and hit set.The simulation result confirms that the proposed algorithm prolonging the lifetime of the network efficiently when compared with the existing algorithms.Originality/value-The proposed algorithm effectively constructs the clusters with a minimum number of communicating nodes using the dominating and hit sets.The simulation result confirms that the proposed algorithm is prolonging the lifetime of the network efficiently when compared with the existing algorithms.

Analytical solution for the size of the minimum dominating set in complex networks

Domination is the fastest-growing field within graph theory with a profound diversity and impact in real-world applications,such as the recent breakthrough approach that identifies optimized subsets of proteins enriched with cancer-related genes.Despite its conceptual simplicity,domination is a classical NP-complete decision problem which makes analytical solutions elusive and poses difficulties to design optimization algorithms for finding a dominating set of minimum cardinality in a large network.Here,we derive for the first time an approximate analytical solution for the density of the minimum dominating set(MDS)by using a combination of cavity method and ultra-discretization(UD)procedure.The derived equation allows us to compute the size of MDS by only using as an input the information of the degree distribution of a given network.

Partitioning a Graph into Defensive k-Alliances

A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating Set. In this paper we study the problem of partitioning the vertex set of a graph into （global） defensive k-alliances. The （global） defensive k-alliance partition number of a graph Г = （V, E）, ψkgd（F）） ψkd（F）, is defined to be the maximum number of sets in a partition of V such that each set is a （global） defensive k-alliance. We obtain tight bounds on ψkd（F） and ψkgd（F） in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of F1 × F2 into （global） defensive （kl ＋ k2）-alliances and partitions of Fi into （global） defensive ki-alliances, i ∈ {1, 2}.

AHBP: An Efficient Broadcast Protocol forMobile Ad Hoc Networks

Broadcast is an important operation in many network protocols. It is utilized to discover routes to unknown nodes in mobile ad hoc networks (MANETs) and is the key factor in scaling on-demand routing protocols to large networks. This paper presents the Ad Hoc Broadcast Protocol (AHBP) and its performance is discussed. In the protocol, messages are only rebroadcast by broadcast relay gateways that constitute a connected dominating set of the network. AHBP can efficiently reduce the redundant messages which make flooding-like protocols perform badly in large dense networks. Simulations are conducted to determine the performance characteristics of the protocol. The simulation results have shown excellent reduction of broadcast redundancy with AHBP. It also contributes to a reduced level of broadcast collision and congestion.

）istance domination of generalized te Bruijn and Kautz digraphs

Let G= （V,A） be adigraph and k ≥ 1 an integer. For u,v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V ／ D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γk（G）, is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs GB（n, d） and generalized Kautz digraphs Gg（n, d） are good candidates for interconnection k networks. Denote △k ：=（∑j^k=0 d^j）^-1. F. Tian and J. Xu showed that [n△k] ≤ γk（GB（n,d）） ≤ [n/d^k] and [n△k] ≤ γk（GK（n,d）） ≤ [n/d^k]. In this paper, we prove that every generalized de Bruijn digraph GB（n, d） has the distance k- domination number [n△k] or [n△k] ＋ 1, and the distance k-domination number of every generalized Kautz digraph GK（n, d） bounded above by [n/ （d^k-1 ＋d^k）]. Additionally, we present various sufficient conditions for γk（GB（n, d）） = [n△k] and γk（GK（n, d）） = [n△k].