Given an undirected graph with nonnegative edge-costs, a subset of nodes of size k called the terminals, and an integer q between 1 and k, the minimum q-Steiner forest problem is to nd a forest of minimum cost with at most q trees that spans all the terminals. When q = 1, we have the classical minimum-cost Steiner tree problem on networks. In this note, we adapt a primal-dual approximation algo...

Abstract. We consider a game-theoretical variant of the Steiner forest problem in which each player j, out of a set of k players, strives to connect his terminal pair (sj , tj) of vertices in an undirected, edge-weighted graph G. In this paper we show that a natural adaptation of the primaldual Steiner forest algorithm of Agrawal, Klein and Ravi [When trees collide: An approximation algorithm f...

The minimum feature size of a crossing-free straight line drawing is the minimum distance between a vertex and a non-incident edge. This quantity measures the resolution needed to display a figure or the tool size needed to mill the figure. The spread is the ratio of the diameter to the minimum feature size. While many algorithms (particularly in meshing) depend on the spread of the input, none...

Given a directed graph G and a list (s1, t1), . . . , (sk, tk) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ k. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, . . . , tk) is known to be fixed-parameter tractable parameterized by the number of term...

We consider the k-Directed Steiner Forest (k-DSF) problem: Given a directed graph G = (V,E) with edge costs, a collection D ⊆ V × V of ordered node pairs, and an integer k ≤ |D|, find a minimum cost subgraph H of G that contains an st-path for (at least) k pairs (s, t) ∈ D. When k = |D|, we get the Directed Steiner Forest (DSF) problem. The best known approximation ratios for these problems are...

Let G = (V; E) be a graph and T V be a node set. We call an edge set S a Steiner tree with respect to T if S connects all pairs of nodes in T. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graph G = (V; E) with edge weights w e , edge capacities a Steiner tree with respect to T k , at most c e of these edge sets use edge e for e...

The purpose of this note is to establish a new version of the local Steiner formula and to give an application to convex bodies of constant width. This variant of the Steiner formula generalizes results of Hann 3] and Hug 6], who use much less elementary techniques than the methods of this paper. In fact, Hann 4] asked for a simpler proof of these results ((4], Problem 2 on p. 900). We remark t...

Given a clique-width expression of a graph G of clique-width k, we provide 2O(k) ·nO(1) time algorithms for connectivity constraints on locally checkable properties such as Connected Dominating Set, Connected Perfect Dominating Set or Node-Weighted Steiner Tree. We also propose an 2O(k) ·nO(1) time algorithm for Feedback Vertex Set. The best running times for all the considered cases were eithe...

Let P ⊂ R be a set of n points, of which k lie in the interior of the convex hull CH(P) of P . Let us call a triangulation T of P even (odd) if and only if all its vertices have even (odd) degree, and pseudo-even (pseudo-odd) if at least the k interior vertices have even (odd) degree. On the one hand, triangulations having all its interior vertices of even degree have one nice property; their v...