The q-analogs of covering designs, Steiner systems, and Turán designs are studied. It is shown that q-covering designs and q-Turán designs are dual notions. A strong necessary condition for the existence of Steiner structures (the q-analogs of Steiner systems) over F2 is given. No Steiner structures of strength 2 or more are currently known, and our condition shows that their existence would im...

We study several network design problems with degree constraints. For the minimum-cost Degree-Constrained 2-Node-Connected Subgraph problem, we obtain the first non-trivial bicriteria approximation algorithm, with factor 6 violation for the degrees and a 4-approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2...

A partial Steiner system Sp(t, k, n) is a collection of k-subsets (i.e. subsets of size k) of n element set such that every t-subset is contained in at most one k-subset. To avoid trivial cases, we assume 2 ≤ t < k < n. It is easy to see that the size of a partial Steiner system Sp(t, k, n) is at most ( n t )

We design combinatorial approximation algorithms for the Capacitated Steiner Network (Cap-SN) problem and the Capacitated Multicommodity Flow (Cap-MCF) problem. These two problems entail satisfying connectivity requirements when edges have costs and hard capacities. In Cap-SN, the flow has to be supported separately for each commodity while in Cap-MCF, the flow has to be sent simultaneously for...

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V , and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only ac...

The minimum weight codewords in the Preparata code of length n = 4 m are utilized for the construction of an innnite family of Steiner S(4; f5; 6g; 4 m + 1) designs for any m 2. A t-wise balanced design with parameters t-(v; K;) is a pair (X; B) where X is a set of v points and B is a collection of subsets of X (called blocks) with sizes from the set K, such that every t-subset of X is containe...

For a graph G and a positive integer k, the k-power of G is the graph G with V (G) as its vertex set and {(u, v)|u, v ∈ V (G), dG(u, v) ≤ k} as its edge set where dG(u, v) is the distance between u and v in graph G. The k-Steiner root problem on a graph G asks for a tree T with V (G) ⊆ V (T ) and G is the subgraph of T k induced by V (G). If such a tree T exists, we call it a k-Steiner root of ...

We study a bottleneck Steiner tree problem: given a set P = {p1,p2, . . . , pn} of n terminals in the Euclidean plane and a positive integer k, find a Steiner tree with at most k Steiner points such that the length of the longest edges in the tree is minimized. The problem has applications in the design of wireless communication networks. We give a ratio-1.866 approximation algorithm for the pr...

This paper addresses the competitiveness of online algorithmsfor two Steiner Tree problems. In the online setting, requests for k ter-minals appear sequentially, and the algorithm must maintain a feasible,incremental solution at all times. In the first problem, the underlyinggraph is directed and has bounded asymmetry, namely the maximumweight of antiparallel links in the gr...

In this note, we prove that there does not exist a Steiner (v, k, 2) trade of volume m, where m is odd, 2k + 3 ~ m ~ 3k 4, and k ~ 7. This completes the spectrum problem for Steiner (v, k, 2) trades.