Approximate min–max theorems for Steiner rooted-orientations of graphs and hypergraphs

Given an undirected hypergraph and a subset of vertices S subset of V with a specified root vertex r epsilon S, the STEINER ROOTFD-ORIENTATION problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the "connectivity" from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the STEINER TREE PACKING problem. Some complementary hardness results are presented at the end. (c) 2008 Elsevier Inc. All rights reserved

Approximate min-max theorems of Steiner rooted-orientations of hypergraphs

Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the STEINER ROOTED-ORIENTATION problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the "connectivity" from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: • Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. • Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the STEINER TREE PACKING problem. Some complementary hardness results are presented at the end. © 2006 IEEE

The (2,k)(2,k)-connectivity augmentation problem: Algorithmic aspects

Durand de Gevigney and Szigeti \cite{DgGSz} have recently given a min-max theorem for the $(2,k)$-connectivity augmentation problem. This article provides an $O(n^3(m+ n \textrm{ }log\textrm{ }n))$ algorithm to find an optimal solution for this problem

Approximate min-max theorems for Steiner rooted-orientations of graphs and hypergraphs

Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner Rooted-Orientation problem is to find an orientation of all the hyperedges so that in the resulting directed hypergraph the “connectivity” from the root r to the vertices in S is maximized. This is motivated by a multicasting problem in undirected networks as well as a generalization of some classical problems in graph theory. The main results of this paper are the following approximate min-max relations: • Given an undirected hypergraph H, if S is 2k-hyperedge-connected in H, then H has a Steiner rooted k-hyperarc-connected orientation. • Given an undirected graph G, if S is 2k-element-connected in G, then G has a Steiner rooted k-element-connected orientation. Both results are tight in terms of the connectivity bounds. These also give polynomial time constant factor approximation algorithms for both problems. The proofs are based on submodular techniques, and a graph decomposition technique used in the Steiner Tree Packing problem. Some complementary hardness results are presented at the end

Kombinatorikus Optimalizálás: Algoritmusok, Strukturák, Alkalmazások = Combinatorial optimization: algorithms, structures, applications

Mint azt az OTKA-pályázat munkaterve tartalmazza, a pályázatban résztvevő kutatók alkotják a témavezető irányításával működő Egerváry Jenő Kombinatorikus Optimalizálási Kutatócsoportot. A csoport a kutatási tervben szereplő több témában jelentős eredményeket ért el az elmúlt 4 évben, ezekről a pályázat résztvevőinek több mint 50 folyóiratcikke jelent meg, és számos rangos nemzetközi konferencián ismertetésre kerültek. Néhány kiemelendő eredmény: sikerült polinomiális kombinatorikus algoritmust adni irányított gráf pont-összefüggőségének növelésére; jelentős előrelépés történt a háromdimenziós térben merev gráfok jellemzésével és a molekuláris sejtéssel kapcsolatban; 2 dimenzióban sikerült bizonyítani Hendrickson sejtését; a párosításelméletben egy újdonságnak számító módszerrel számos új algoritmikus eredmény született; több, gráfok élösszefüggőségét jellemző tételt sikerült hipergráfokra általánosítani. | As the research plan indicates, the researchers participating in the project are the members of the Egerváry Research Group, led by the coordinator. The group has made important progress in the past 4 years in the research topics declared in the research plan. The results have been published in more than 50 journal papers, and have been presented at several prestigious international conferences. The most significant results are the following: a polynomial algorithm has been found for the node-connectivity augmentation problem of directed graphs; considerable progress has been made towards the characterization of 3-dimensional rigid graphs and towards the proof of the molecular conjecture; Hendrickson's conjecture has been proved in 2 dimensions; several new algorithmic results were obtained in matching theory using a novel approach; several theorems characterizing connectivity properties of graphs have been generalized to hypergraphs

Edge-connectivity augmentation of graphs over symmetric parity families

AbstractIn this note we solve the edge-connectivity augmentation problem over symmetric parity families. It provides a solution for the minimum T-cut augmentation problem. We also extend a recent result of Zhang [C.Q. Zhang, Circular flows of nearly eulerian graphs and vertex splitting, J. Graph Theory 40 (2002) 147–161]