In this paper, we propose a branch-and-bound algorithm to solve exactly the minimum vertex cover (MVC) problem. Since a tight lower bound for MVC has a significant influence on the efficiency of a branch-and-bound algorithm, we define two novel lower bounds to help prune the search space. One is based on the degree of vertices, and the other is based on MaxSAT reasoning. The experiment confirms that our algorithm is faster than previous exact algorithms and can find better results than heuristic algorithms.

The minimum connected dominating set (MCDS) problem is a very significant NP-hard combinatorial optimization problem, and it has been used in many fields such as wireless sensor networks and ad hoc networks. In this paper, we propose a novel multi-start local search algorithm (MSLS) to tackle the minimum connected dominating set problem. Firstly, we present the fitness mechanism to design the vertex score mechanism so that our algorithm can jump out of the local optimum. Secondly, we use the configuration checking (CC) mechanism to avoid the cycling problem. Then, we propose the vertex flipping mechanism to change the vertex state by combing the CC mechanism with the vertex score mechanism. Finally, we propose a multi-start local search framework based on these mechanisms. We compare the algorithm MSLS with other compared algorithms on extensive instances. The results of experiment show that MSLS is superior to other algorithms in solution quality and time efficiency on most instances.

The multi-objective minimum weighted vertex cover problem aims to minimize the sum of different single type weights simultaneously. In this paper, we focus on the bi-objective minimum weighted vertex cover and propose a multi-objective algorithm integrating iterated neighborhood search with decomposition technique to solve this problem. Initially, we adopt the decomposition method to divide the multi-objective problem into several scalar optimization sub-problems. Meanwhile, to find more possible optimal solutions, we design a mixed score function according to the problem feature, which is applied in initializing procedure and neighborhood search. During the neighborhood search, three operators ( $Add$ , $Delete$ , $Swap$ ) explore the search space effectively. We performed numerical experiments on many instances, and the results show the effectiveness of our new algorithm (combining decomposition and neighborhood search with mixed score) on several experimental metrics. We compared our experimental results with the classical multi-objective algorithm non-dominated sorting genetic algorithm II. It was obviously shown that our algorithm can provide much better results than the comparative algorithm considering the different metrics.

Given an undirected, connected and edge-weighted graph, the dominating tree problem consists of finding a tree with minimum total edge weight such that for each vertex is either in the tree or adjacent to a vertex in the tree. In this paper, we propose a hybrid framework combining genetic algorithm with iterated local search (GAITLS) for solving the dominating tree problem. The main components of our framework are as follows: (1) the score functions $Dscore$ and $Wscore$ applied in the initialization and local search phase; (2) the initialization procedure with restricted candidate list (RCL) by controlling the parameter to balance the greediness and randomness; (3) the iterated local search with three phases, which is used to intensify the individuals; (4) the mutation with high diversity proposed to perturb the population. The experimental results on the classical instances show that our method performs much better than the-state-of-art algorithms.

The minimum weighted connected dominating set problem is a significant NP-hard problem with wide applications, and is an extension of the classical minimum dominating set problem. In order to solve this problem, we present a restart local search algorithm with tabu method (RLS $\_$ Tabu). In our RLS $\_$ Tabu algorithm, we firstly involve the random restart initialization method to jump out of the local optimum. Meanwhile, RLS $\_$ Tabu algorithm also applies tabu method in neighborhood search procedure to mitigate the cycling problem. Secondly, we present two strategies in neighborhood search procedure for removing vertices properly, which one is greedy and random strategy, and another one is multiple deletion strategy. The two strategies are crucial to improve the solution quality. Thirdly, the solution connected vertex is important to guarantee the feasibility of solutions. Therefore, we maintain the solution connected vertex set during the neighborhood search, and select the vertex to be added from this set. Finally, in order to intensify the solution, RLS $\_$ Tabu utilizes the pruning function to delete redundant vertices in the candidate solution. In experimental section, we will compare our algorithm with the other six algorithms on three types of benchmarks. Experimental results indicate that our algorithm significantly outperforms the comparative algorithms on most benchmark instances.