Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree, but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.Comment: 23 pages, 6 figures An extended abstract appeared in proceedings of STACS 201

THE RESTRAINED STEINER NUMBER OF A GRAPH

For a connected graph G = (V, E) of order p, a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets. A set W of vertices of a graph G is a restrained Steiner set if W is a Steiner set, and if either W = V or the subgraph G[V − W ] induced by V − W has no isolated vertices. The minimum cardinality of a restrained Steiner set of G is the restrained Steiner number of G, and is denoted by s r (G). The restrained Steiner number of certain classes of graphs are determined. Connected graphs of order p with restrained Steiner number 2 are characterized. Various necessary conditions for the restrained Steiner number of a graph to be p are given. It is shown that, for integers a, b and p with 4 ≤ a ≤ b ≤ p, there exists a connected graph G of order p such that s(G) = a and s r (G) = b. It is also proved that for every pair of integers a, b with a ≥ 3 and b ≥ 3, there exists a connected graph G with s r (G) = a and g r (G) = b

Distance and the pattern of intra-European trade

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 Oellermann [21] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) element-disjoint Steiner forests, where h = | i Ti|. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests. • We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [12] in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

On the Steiner, geodetic and hull numbers of graphs

Given a graph G and a subset W ? V (G), a Steiner W-tree is a tree of minimum order that contains all of W. Let S(W) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W) the Steiner interval of W. If S(W) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a shortest u − v path in G is called a u − v geodesic. Let I[u, v] denote the set of all vertices in G lying on some u − v geodesic, and let J[u, v] denote the set of all vertices in G lying on some induced u − v path. Given a set S ? V (G), let I[S] = ?u,v?S I[u, v], and let J[S] = ?u,v?S J[u, v]. We call I[S] the geodetic closure of S and J[S] the monophonic closure of S. If I[S] = V (G), then S is called a geodetic set of G. If J[S] = V (G), then S is called a monophonic set of G. The minimum order of a geodetic set in G is named the geodetic number of G. In this paper, we explore the relationships both between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. We thoroughly study the relationship between the Steiner number and the geodetic number, and address the questions: in a graph G when must every Steiner set also be geodetic and when must every Steiner set also be monophonic. In particular, among others we show that every Steiner set in a connected graph G must also be monophonic, and that every Steiner set in a connected interval graph H must be geodetic

The edge-to-vertex Steiner domination number of a graph

A set W ⊆ E is said to be an edge-to-vertex Steiner dominating set of G if W is both an edge-to-vertex dominating set and a edge-to-vertex Steiner set of G. The edge-to-vertex Steiner domination number γsev(G) of G is the minimum cardinality of its edge-to-vertex Steiner dominating set of G and any edge-to-vertex Steiner dominating set of cardinality γsev(G) is a γsev-set of G. Some general properties satisfied by this concept are studied. The edge-to-vertex Steiner domination number of certain classes of graphs are determined. Connected graph of size q ≥ 3 with edge-to-vertex Steiner domination number q or q −1 are characterized. It is shown for every pair a, b of integers with 2 ≤ a ≤ b, there exists a connected graph G such that γev(G) = a and γsev(G) = b.Emerging Sources Citation Index (ESCI)MathScinetScopu

Inverse Problems Related to the Wiener and Steiner-Wiener Indices

In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of $k$ vertices and take the sum, it is called the Steiner $k$-Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value