The Minimum Vertex Cover (MVC) problem is a well known combinatorial optimization problem of great importance in theory and applications. In recent years, local search has been shown to be an effective and promising approach to solve hard problems, such as MVC. In this paper, we introduce two new local search algorithms for MVC, called EWLS (Edge Weighting Local Search) and EWCC (Edge Weighting Configuration Checking). The first algorithm EWLS is an iterated local search algorithm that works with a partial vertex cover, and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima. Nevertheless, EWLS has an instance-dependent parameter. Further, we propose a strategy called Configuration Checking for handling the cycling problem in local search. This is used in designing a more efficient algorithm that has no instance-dependent parameters, which is referred to as EWCC. Unlike previous vertex-based heuristics, the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution.A detailed experimental study is carried out using the well known DIMACS and BHOSLIB benchmarks. The experimental results conclude that EWLS and EWCC are largely competitive on DIMACS benchmarks, where they outperform other current best heuristic algorithms on most hard instances, and dominate on the hard random BHOSLIB benchmarks. Moreover, EWCC makes a significant improvement over EWLS, while both EWLS and EWCC set a new record on a twenty-year challenge instance. Further, EWCC performs quite well even on structured instances in comparison to the best exact algorithm we know. We also study the run-time behavior of EWLS and EWCC which shows interesting properties of both algorithms.

The Minimum Vertex Cover (MVC) problem is a prominent NP-hard combinatorial optimization problem of great importance in both theory and application. Local search has proved successful for this problem. However, there are two main drawbacks in state-of-the-art MVC local search algorithms. First, they select a pair of vertices to exchange simultaneously, which is time-consuming. Secondly, although using edge weighting techniques to diversify the search, these algorithms lack mechanisms for decreasing the weights. To address these issues, we propose two new strategies: two-stage exchange and edge weighting with forgetting. The two-stage exchange strategy selects two vertices to exchange separately and performs the exchange in two stages. The strategy of edge weighting with forgetting not only increases weights of uncovered edges, but also decreases some weights for each edge periodically. These two strategies are used in designing a new MVC local search algorithm, which is referred to as NuMVC. We conduct extensive experimental studies on the standard benchmarks, namely DIMACS and BHOSLIB. The experiment comparing NuMVC with state-of-the-art heuristic algorithms show that NuMVC is at least competitive with the nearest competitor namely PLS on the DIMACS benchmark, and clearly dominates all competitors on the BHOSLIB benchmark. Also, experimental results indicate that NuMVC finds an optimal solution much faster than the current best exact algorithm for Maximum Clique on random instances as well as some structured ones. Moreover, we study the effectiveness of the two strategies and the run-time behaviour through experimental analysis

This paper presents and analyzes two new efficient local search strategies for the Boolean Satisfiability (SAT) problem. We start by proposing a local search strategy called configuration checking (CC) for SAT. The CC strategy results in a simple local search algorithm for SAT called Swcc, which shows promising experimental results on random 3-SAT instances, and outperforms TNM, the winner of SAT Competition 2009. However, the CC strategy for SAT is still in a nascent stage, and Swcc cannot yet compete with Sparrow2011, which won SAT Competition 2011 just after Swcc had been designed. The CC strategy seems too strict in that it forbids flipping those variables even with great scores, if they do not satisfy the CC criterion. We improve the CC strategy by adopting an aspiration mechanism, and get a new variable selection heuristic called configuration checking with aspiration (CCA). The CCA heuristic leads to an improved algorithm called Swcca, which exhibits state-of-the-art performance on random 3-SAT instances and crafted ones. The third contribution concerns improving local search algorithms for random k-SAT instances with k > 3. Although the SAT community has made great achievements in solving random 3-SAT instances, the progress lags far behind on random k-SAT instances with k > 3. This work proposes a new variable property called subscore, which is utilized to break ties in the CCA heuristic when candidate variables for flipping have the same score. The resulting algorithm CCAsubscore is very efficient for solving random k-SAT instances with k > 3, and significantly outperforms other state-of-the-art ones. Combining Swcca and CCAsubscore, we obtain a local search SAT solver called CCASat, which was ranked first in the random track of SAT Challenge 2012. Additionally, we perform theoretical analyses on the CC strategy and the subscore property, and show interesting results on these two heuristics. Particularly, our analysis indicates that the CC strategy is more effective for k-SAT with smaller k, while the subscore notion is not suitable for solving random 3-SAT.