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.

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.

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.