Generalized steiner problem in outerplanar networks

The generalized Steiner problem (GSP) is concerned with the determination of a minimum cost subnetwork of a given network where some (not necessarily all) vertices satisfy certain pairwise (vertex or edge) connectivity requirements.The GSP has applications to the design of water and electricity supply networks, communication networks and other large-scale systems where connectivity requirements ensure the communication between the selected vertices when some vertices and/or edges can become inoperational due to scheduled maintenance, error, or overload.The GSP is known to beNP-complete. In this paper we show that if the subnetwork is required to be biconnected or respectively edge-biconnected, and the underlying network is outerplanar, the GSP can be solved in linear time.     

Coherent orientations and series-parallel networks

Let p be an edge of the graph G. An orientation of G is p-coherent if the set of directed circuits is exactly the set of circuits containing the edge p. Theorem: A matroidally connected graph G is a series-parallel network if and only if for every edge p of G, there exists a p-coherent orientation.     

Generalized steiner triple systems with group size five

Generalized Steiner triple systems, GS(2, 3, n, g) are used to construct maximum constant weight codes over an alphabet of size g+1 with distance 3 and weight 3 in which each codeword has length n. The existence of GS(2, 3, n, g) has been solved for g=2, 3, 4, 9. In this paper, by introducing a special kind of holey generalized Steiner triple systems (denoted by HGS(2, 3, (n, u), g)), singular indirect product (SIP) construction for GDDs is used to construct generalized Steiner systems. The numerical necessary conditions for the existence of a GS(2, 3, n, g) are shown to be sufficient for g=5.     

The steiner tree packing problem in VLSI design

In this paper we describe several versions of the routing problem arising in VLSI design and indicate how the Steiner tree packing problem can be used to model these problems mathematically. We focus on switchbox routing problems and provide integer programming formulations for routing in the knock-knee and in the Manhattan model. We give a brief sketch of cutting plane algorithms that we developed and implemented for these two models. We report on computational experiments using standard test instances. Our codes are able to determine optimum solutions in most cases, and in particular, we can show that some of the instances have no feasible solution if Manhattan routing is used instead of knock-knee routing.     

Minimum cost flow algorithms for series-parallel networks

It is shown that an acyclic multigraph with a single source and a single sink is series-parallel if and only if for arbitrary linear cost functions and arbitrary capacities the corresponding minumum cost flow problem can be solved by a greedy algorithm. Furthermore, for networks of this type with m edges and n vertices, two O(mn+logm)-algorithms. One of them is based on the greedy scheme.     

Some formulations for the group steiner tree problem

The group Steiner tree problem consists of, given a graph G, a collection R of subsets of V(G) and a cost c(e) for each edge of G, finding a minimum-cost subtree that connects at least one vertex from each R∈R. It is a generalization of the well-known Steiner tree problem that arises naturally in the design of VLSI chips. In this paper, we study a polyhedron associated with this problem and some extended formulations. We give facet defining inequalities and explore the relationship between the group Steiner tree problem and other combinatorial optimization problems.     

系列平行图上带时间约束的Steiner最小树问题

对一类特殊系列平行图上带有时间约束的Steiner最小树问题,证明了其复杂性为NPC,并给出了一个完全多项式时间近似方案.     

From steiner centers to steiner medians

The Steiner distance of set S of vertices in a connected graph G is the minimum number of edges in a connected subgraph of G containing S. For n ≥ 2, the Steiner n-eccentricity e_n(v) of a vertex v of a graph G is the maximum Steiner distance among all sets S of n vertices of G that contain v. The Steiner n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity. The Steiner n-distance of a vertex v of G is defined as . The Steiner n-median of G is the subgraph of G induced by the vertices of G of minimum Steiner n-distance. Known algorithms for finding the Steiner n-centers and Steiner n-medians of trees are used to show that the distance between the Steiner n-centre and Steiner n-median of a tree can be arbitrarily large. Centrality measures that allow every vertex on a shortest path from the Steiner n-center to the Steiner n-median of a tree to belong to the “center” with respect to one of these measures are introduced and several proeprties of these centrality measures are established. © 1995 John Wiley & Sons, Inc.     

On the core of the minimum cost steiner tree game in networks

A cost allocation problem arising from the Steiner Tree (ST) problem in networks is analyzed. This cost allocation problem is formulated as a cost cooperative game in characteristic function form, referred to as theST-game. The class ofST games generalizes the class of minimum cost spanning tree games which were used in the literature to analyze a variety of cost allocation problems. In general, the core of anST-game may be empty. We construct an efficient Core Heuristic to compute a good lower bound on the maximum fraction of the total cost that can be distributed among users while satisfying the core constraints. Based on the Core Heuristic, we also provide a sufficient condition for a givenST not to be optimal for the linear programming relaxation of an integer programming formulation of theST problem. The Core Heuristic was implemented and tested on 76 data sets from the literature (Wong's, Aneja's and Beasley's Steiner tree problems). Core points were found for 69 of these cases, and points close to the core were computed in the others.     

The robust shortest path problem in series-parallel multidigraphs with interval data

In this paper the robust shortest path problem in edge series-parallel multidigraphs with interval costs is examined. The maximal regret criterion is applied to calculate the optimal solution. It is shown that this problem is NP-hard. A pseudopolynomial algorithm for the studied problem is constructed.     

Generalized complementarity problem

A general complementarity problem with respect to a convex cone and its polar in a locally convex, vector-topological space is defined. It is observed that, in this general setting, the problem is equivalent to a variational inequality over a convex cone. An existence theorem is established for this general case, from which several of the known results for the finite-dimensional cases follow under weaker assumptions than have been required previously. In particular, it is shown that, if the given map under consideration is strongly copositive with respect to the underlying convex cone, then the complementarity problem has a solution.This research was partially supported by the Office of Naval Research, Contract No. N-00014-67-A0112-0011, by the Atomic Energy Commission, Contract No. AT[04-3]-326-PA-18, and by the National Science Foundation, Grant No. GP-9329.     

On the solution of the generalized steiner problem by the subgradient method

This paper is concerned with the problem of constructing a minimal cost weighted tree connecting a set ofn given terminal vertices on an Euclidean plane. Both theoretical and numerical aspect of the problem are considered. As regards the first ones, the convexity of the objective function is studied and the necessary and sufficient optimality conditions are deduced. As regards the numerical aspects, a subgradient type algorithm is presented.     

Generalized Feistel networks revisited

This work deals with the classification, security and efficiency of generalized Feistel networks (GFNs) with 4 lines. We propose a definition of a GFN, essentially limiting consideration to Feistel-type constructions with domain-preserving F-functions and rotation by one line between rounds. Under this definition, we demonstrate that there are only two non-contracting representatives in the class of 4-line GFNs up to equivalence, namely, the type-I and type-II GFNs that avoid obvious differential effects. We propose to instantiate the GFNs with SPS-functions (two substitution layers separated by a permutation layer) instead of single SP-functions (one substitution-permutation layer only). We prove tight lower bounds on the number of differentially and linearly active functions and S-boxes in such ciphers. We show that the instantiation with SPS-functions using MDS diffusion provides a proportion of differentially and linearly active S-boxes by up to 33 and 50 % higher than that with single SP-functions for type-I and type-II GFNs, respectively, if the same matrix is used in all rounds. Moreover, we present the upper bounds on the differential and the linear hull probability for the type-II GFNs with SPS-functions. This opens up the possibility of designing more efficient block ciphers based on GFN structure.     

On steiner spaces

A Steiner space is a Steiner triple system that is not generated by a triangle. We give new constructions of Steiner spaces and solve the existence problem of Steiner spaces of order v for all but 4 values of v.     

Orthogonal steiner systems

Aequationes mathematicae -