Results 1 
7 of
7
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
On Complexity, Representation and Approximation of Integral Multicommodity Flows
, 1998
"... The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1 \Gamma ffl approximation algorithms using the randomized rounding/derandomiztion scheme provided that the components o ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
The paper has two parts. In the algorithmic part integer inequality systems of packing types and their application to integral multicommodity flow problems are considered. We give 1 \Gamma ffl approximation algorithms using the randomized rounding/derandomiztion scheme provided that the components of the right hand side vector resp. the capacities are in \Omega\Gamma ffl \Gamma2 log m) where m is the number of constraints resp. the number of edges. In the complexitytheoretic part it is shown that the approximable instances above build hard problems. Extending a result of Garg, Vazirani and Yannakakis (1993), the Mazsnp hardness of the maximum integral multicommodity flow problem for trees with large capacities (in particular c 2\Omega\Gamma390 m)) is proved. Furthermore, for every fixed nonnegative integer K the problem with specified demand function r \Gamma K is NP hard even if c is any function polynomially bounded in n and if the problem with demand function r is fractionally...
An Algorithm for NodeCapacitated Ring Routing
, 2006
"... A strongly polynomial time algorithm is described to solve the nodecapacitated routing problem in an undirected ring network. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
A strongly polynomial time algorithm is described to solve the nodecapacitated routing problem in an undirected ring network.
Lecture 23
"... Consider a planar graph G =(V,E) and a set of terminal pairs R = {(si,ti) : i = 1,k}. Assume G is planar, (V,E ∪ R) is Eulerian, and all terminals lie on the outer face of G. In this lecture, we will cover the following results. • The OkamuraSeymour theorem on the equivalence between the existence ..."
Abstract
 Add to MetaCart
Consider a planar graph G =(V,E) and a set of terminal pairs R = {(si,ti) : i = 1,k}. Assume G is planar, (V,E ∪ R) is Eulerian, and all terminals lie on the outer face of G. In this lecture, we will cover the following results. • The OkamuraSeymour theorem on the equivalence between the existence of siti edgedisjoint paths and the cut condition δE(S)  ≤δR(S), ∀S ⊆ V [OS81]. • The WagnerWeihe lineartime algorithm for finding the edgedisjoint paths [WW93, RLWW95, WW95]. Chapter 74 of [Sch03] contains a proof of the OkamuraSeymour theorem, as well as a survey of related results. 1 The OkamuraSeymour Theorem We beginwiththe main theorem. Theorem 1 (OkamuraSeymour) Consider an undirected planar graph G = (V,E) and a set of terminal pairs R = {(si,ti) : si ∈ V,ti ∈ V,i = 1, ·· ·,k} s.t. the following conditions are satisfied: 1. The terminals are on the boundary of the outside face of G. 1 2. The Euler condition: (V,E ∪ R) is Eulerian.