Abstract Dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set. We propose a new order-based randomised local search (RLS o ) algorithm to solve minimum dominating set problem in large graphs. Experimental evaluation is presented for multiple types of problem instances. These instances include unit disk graphs, which represent a model of wireless networks, random scale-free networks, as well as samples from two social networks and real-world graphs studied in network science. Our experiments indicate that RLS o performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search. The order-based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices. In addition, we propose a multi-start variant of RLS o that is suitable for solving the minimum weight dominating set problem. The application of RLS o in graph mining is also briefly demonstrated.
Exact algorithms require exponential time to solve the problem. To the best of our knowledge, the currently best exact algorithm was proposed by , and finds MDS in @math time @cite_37 .
@cite_37
2962745760 2567804199 1727475799 2089718638
Author(s): Impagliazzo, R; Lovett, S; Paturi, R; Schneider, S | Abstract: We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time @math where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algorithm for that subproblem. An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. If there are n variables, then an obvious classical deterministic algorithm checks out all 2n truth assignments in about 2n steps, while his quantum search algorithm can find a satisfying truth assignment in about 2n 2 steps. For several NP-complete problems, many sophisticated classical algorithms have been designed. They are still exponential, but much faster than the brute force algorithms. The question arises whether their running time can still be decreased from T(n) to O (√T(n)) by using a quantum computer. Isolated positive examples are known, and some speed-up has been obtained for wider classes. Here, we present a simple method to obtain the full T(n) to O(√T(n)) speed-up for most of the many nontrivial exponential time algorithms for NP-hard problems. The method works whenever the widely used technique of recursive decomposition is employed. This included all currently known algorithms for which such a speedup has not yet been known. For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis. In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.