Given a metric space on n points, an α-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an α factor of the optimal Steiner tree cost for X . An α-approximate differentially private algorithm for the Steiner tree problem take...

In this paper some properties of k-chromatic and strong k-chromatic hypergraphs are proved. Conditions for the possible existence of blocking sets in Steiner systems are established. Mathematics Subject Classification: 05C15; 05B05

We present a new algorithm that solves the minimum Steiner tree problem on an undirected graph with n nodes, k terminals, and m edges with bounded integer weights in time Õ((2− ε)n + nm). Our algorithm beats the aesthetically appealing and seemingly inherent 2 bound of the current best algorithm for Steiner tree problem from Björklund, Husfeldt, Kaski, and Koivisto. Our algorithm is based on fa...

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v, k,1) designs, mainly for t = 2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t > 2 most of these characterizations have rema...

Tree-Embedding is one of the most powerful techniques in the area of approximation algorithms as it reduces many diffcult problems like group Steiner tree (GST) on general graphs into amenable tree intances. However, the developments on tree-embedding are pertained only to undirected graphs, thus rendering it useless against problems on directed graphs like directed Steiner tree (DST) and its g...

Given an undirected graph G = (V,E) and subset of terminals T ⊆ V , the element-connectivity κ G (u, v) of two terminals u, v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellerma...

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...

We consider the analysis of linear programming (LP) relaxations for a class of connectivity problems. The central problem in the class is the survivable network design problem the problem of designing a minimum cost undirected network satisfying prespecified connectivity requirements between every pair of vertices. This class includes a number of classical combinatorial optimization problems as...

Let f k (n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f"(n) < f"(n + 1) < with equalities holding iff there exists a Steiner-system Y(t, 2t-1, n). The bounds are approximately best possible for k < 6 and of correct order of magnitude for k > 7, as well, even if the corresponding Steiner-systems do not exist. Exponential lower and upper ...