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The edge dominating set problem (EDS) is to compute a minimum size edge set such that every edge is dominated by some edge in it. This paper considers a variant of EDS with extensions of multiple and connected dominations combined. In the b-EDS problem, each edge needs to be dominated b times. Connected EDS requires an edge dominating set to be connected while it has to form a tree in Tree Cover. Although each of EDS, b-EDS, and Connected EDS (or Tree Cover) has been well studied, each known to be approximable within 2 (or 8/3 for b-EDS in general), nothing is known when these extensions are imposed simultaneously on EDS unlike in the case of the (vertex) dominating set problem. We consider Connected 2-EDS and 2-Tree Cover (i.e., a combination of 2-EDS and Tree Cover), and present a polynomial algorithm approximating each within 2. Moreover, it will be shown that the single tree computed is no larger than twice the optimum for (not necessarily connected) 2-EDS, thus also approximating 2-EDS equally well. It also implies that 2-EDS with clustering properties can be approximated within 2 as well. [ABSTRACT FROM AUTHOR]

We present a greedy algorithm for the directed Steiner tree problem (DST), where any tree rooted at any (uncovered) terminal can be a candidate for greedy choice. It will be shown that the algorithm, running in polynomial time for any constant $$l$$ , outputs a directed Steiner tree of cost no larger than $$2(l-1)(\ln n+1)$$ times the cost of any $$l$$ -restricted Steiner tree, which is such a Steiner tree in which every terminal is at most $$l$$ arcs away from the root or another terminal. We derive from this result that (1) DST for a class of graphs, including quasi-bipartite graphs, in which the length of paths induced by Steiner vertices is bounded by some constant can be approximated within a factor of $$O(\log n)$$ , and (2) the tree cover problem on directed graphs can also be approximated within a factor of $$O(\log n)$$ . [ABSTRACT FROM AUTHOR]

Abstract: The set cover problem is that of computing a minimum weight subfamily , given a family of weighted subsets of a base set U, such that every element of U is covered by some subset in . The k-set cover problem is a variant in which every subset is of size at most k. It has been long known that the problem can be approximated within a factor of by the greedy heuristic, but no better bound has been shown except for the case of unweighted subsets. In this paper we consider approximation of a restricted version of the weighted 3-set cover problem, as a first step towards better approximation of general k-set cover problem, where any two distinct subset costs differ by a multiplicative factor of at least 2. It will be shown, via LP duality, that an improved approximation bound of can be attained, when the greedy heuristic is suitably modified for this case. A key to our algorithm design and analysis is the Gallai–Edmonds structure theorem for maximum matchings. [Copyright &y& Elsevier]

We present a polynomial-time algorithm approximating the minimum weight edge dominating set problem within a factor of 2. It has been known that the problem is NP-hard but, when edge weights are uniform (so that the smaller the better), it can be efficiently approximated within a factor of 2. When general weights were allowed, however, very little had been known about its approximability, and only very recently was it shown to be approximable within a factor of 21/10 by reducing to the edge cover problem via LP relaxation. In this paper we extend the approach given therein, by studying more carefully polyhedral structures of the problem, and obtain an improved approximation bound as a result. While the problem considered is as hard to approximate as the weighted vertex cover problem is, the best approximation (constant) factor known for vertex cover is 2 even for the unweighted case, and has not been improved in a long time, indicating that improving our result would be quite difficult. [Copyright &y& Elsevier]