Results 1  10
of
23
Planar Orientations with Low OutDegree and Compaction of Adjacency Matrices
 Theoretical Computer Science
, 1991
"... We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounde ..."
Abstract

Cited by 34 (3 self)
 Add to MetaCart
We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3bounded orientation, which can be constructed in linear time. ffl A 6bounded acyclic orientation, and a 3bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...
Applications Of Submodular Functions
, 1993
"... Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization. ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.
Strong formulations for network design problems with connectivity requirements
 NETWORKS
, 1999
"... The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edgeconnectivity ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
The network design problem with connectivity requirements (NDC) models a wide variety of celebrated combinatorial optimization problems including the minimum spanning tree, Steiner tree, and survivable network design problems. We develop strong formulations for two versions of the edgeconnectivity NDC problem: unitary problems requiring connected network designs, and nonunitary problems permitting nonconnected networks as solutions. We (i) present a new directed formulation for the unitary NDC problem that is stronger than a natural undirected formulation, (ii) project out several classes of valid inequalitiespartition inequalities, oddhole inequalities, and combinatorial design inequalitiesthat generalize known classes of valid inequalities for the Steiner tree problem to the unitary NDC problem, and (iii) show how to strengthen and direct nonunitary problems. Our results provide a unifying framework for strengthening formulations for NDC problems, and demonstrate the strength and power of flowbased formulations for network design problems with connectivity requirements.
Constructive Characterizations for Packing and Covering With Trees
, 2002
"... We give a constructive characterization of undirected graphs which contain k spanning trees after adding any new edge. This is a generalization of a theorem of Henneberg and Laman who gave the characterization for k = 2. We also give a constructive characterization of graphs which have k edgedisjoi ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
We give a constructive characterization of undirected graphs which contain k spanning trees after adding any new edge. This is a generalization of a theorem of Henneberg and Laman who gave the characterization for k = 2. We also give a constructive characterization of graphs which have k edgedisjoint spanning trees after deleting any edge of them.
A new ILP formulation for 2rootconnected prizecollecting Steiner networks
 In 15th Annual European Symposium on Algorithms (ESA’07), volume 4698 of LNCS
, 2007
"... Abstract. We consider the realworld problem of extending a given infrastructure network in order to connect new customers. By representing the infrastructure by a single root node, this problem can be formulated as a 2rootconnected prizecollecting Steiner network problem in which certain custome ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Abstract. We consider the realworld problem of extending a given infrastructure network in order to connect new customers. By representing the infrastructure by a single root node, this problem can be formulated as a 2rootconnected prizecollecting Steiner network problem in which certain customer nodes require two nodedisjoint paths to the root, and other customers only a simple path. Herein, we present a novel ILP approach to solve this problem to optimality based on directed cuts. This formulation becomes possible by exploiting a certain orientability of the given graph. To our knowledge, this is the first time that such an argument is used for a problem with nodedisjointness constraints. We prove that this formulation is stronger than the wellknown undirected cut approach. Our experiments show its efficiency over the other formulations presented for this problem, i.e., the undirected cut approach and a formulation based on multicommodity flow. 1
Parity Constrained kEdgeConnected Orientations
, 1999
"... Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected g ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V; E) having a kedgeconnected T odd orientation for every subset T ` V with jEj + jT j even. (T odd orientation: the indegree of v is odd precisely if v is in T .) As a corollary, we obtain that every (2k + 2)edgeconnected graph with jV j + jEj even has a kedgeconnected orientation in which the indegree of every node is odd. Along the way, a structural characterization will be given for digraphs with a rootnode s having k + 1 edgedisjoint paths from s to every node and k edgedisjoint paths from every node to s. 1. INTRODUCTION The notion of parity plays an important role in describing combinatorial structures. The prime example is W. Tutte's theorem [T47] on the exi...
Graph Orientation Algorithms to Minimize the Maximum Outdegree
, 2006
"... We study the problem of orienting the edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be generally. In this paper we first give optimal orientation algorithms wh ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We study the problem of orienting the edges of a weighted graph such that the maximum weighted outdegree of vertices is minimized. This problem, which has applications in the guard arrangement for example, can be shown to be generally. In this paper we first give optimal orientation algorithms which run in polynomial time for the following special cases: (i) the input is an unweighted graph, or more generally, a graph with identically weighted edges, and (ii) the input graph is a tree. Then, by using those algorithms as subprocedures, we provide a simple, combinatorial, min{ , (2#)}approximation algorithm for the general case, where wmax and w min are the maximum and the minimum weights of edges, respectively, and # is some small positive real number that depends on the input.
Approximate MinMax Theorems for Steiner RootedOrientations of Graphs and Hypergraphs
, 2006
"... Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner RootedOrientation 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. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Given an undirected hypergraph and a subset of vertices S ⊆ V with a specified root vertex r ∈ S, the Steiner RootedOrientation 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 minmax relations: • Given an undirected hypergraph H, if S is 2khyperedgeconnected in H, then H has a Steiner rooted khyperarcconnected orientation. • Given an undirected graph G, if S is 2kelementconnected in G, then G has a Steiner rooted kelementconnected 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.