The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to either be in the dominating set, or adjacent to some node in the dominating set. We focus on the question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of O(H (\Delta)) are presented, where \Delta is the maximum degree, and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited, or has at least one of its neighbors visited. We study a generalization of the problem when the vertices have weights, and give an algorithm which achieves a performance ratio of 3 ln n. We also consider the ...

We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)-approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i \Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasi-polynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...

Abstract- Many graph problems seek subsets of their vertices that maximize or minimize objective functions on the vertices. Among these are the capacitated p-median problem, the geometric connected dominating set problem, the capacitated k-center problem, and the traveling tourist problem. Prior genetic algorithms research in this area applied a simple mutation of an allele by random replacement. Recently an enhanced operator called hypermutation was developed, proving to be very effective for solving the capacitated p-median problem. We propose a GA with a new heuristic called the nearest four neighbors heuristic (N4N) for solving graph problems requiring a subset of vertices It is an extension of the hypermutation operator. Genetic algorithms that use each of these three mutation operators (simple, hypermutation, N4N) are applied to instances of the four graph-subset problems listed above. Results show that our N4N heuristic obtained superior results compared to the hypermutation and the simple mutation operators in every test case. I.

Abstract. The dominating set problem in graphs asks for a minimum size subset of vertices with the following property: each vertex is required to be either in the dominating set, or adjacent to some vertex in the dominating set. We focus on the related question of finding a connected dominating set of minimum size, where the graph induced by vertices in the dominating set is required to be connected as well. This problem arises in network testing, as well as in wireless communication. Two polynomial time algorithms that achieve approximation factors of 2H(�) + 2 and H(�) + 2 are presented, where � is the maximum degree and H is the harmonic function. This question also arises in relation to the traveling tourist problem, where one is looking for the shortest tour such that each vertex is either visited or has at least one of its neighbors visited. We also consider a generalization of the problem to the weighted case, and give an algorithm with an approximation factor of (cn + 1) ln n where cn ln k is the approximation factor for the node weighted Steiner tree problem (currently cn = 1.6103). We also consider the more general problem of finding a connected dominating set of a specified subset of vertices and provide a polynomial time algorithm with a (c + 1)H(�) + c − 1 approximation factor, where c is the Steiner approximation ratio for graphs (currently c = 1.644).