Balaji Raghavachari

Landmarks in graphs

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks. Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robots position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robots position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(log n), and some properties of graphs with metric dimension two.

Balancing minimum spanning and shortest path trees

We give a simple algorithm to find a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous tradeoff: given the two trees and aγ>0, the algorithm returns a spanning tree in which the distance between any vertex and the root of the shortest-path tree is at most 1+√2γ times the shortest-path distance, and yet the total weight of the tree is at most 1+√2/γ times the weight of a minimum spanning tree. Our algorithm runs in linear time and obtains the best-possible tradeoff. It can be implemented on a CREW PRAM to run a logarithmic time using one processor per vertex.

Approximating the minimum-degree Steiner tree to within one of optimal

Abstract The problem of constructing a spanning tree for a graph G = (V, E) with n vertices whose maximal degree is the smallest among all spanning trees of G is considered. This problem is easily shown to be NP-hard. In the Steiner version of this problem, along with the input graph, a set of distinguished vertices D ⊆ V is given. A minimum-degree Steiner tree is a tree of minimum degree which spans at least the set D. Iterative polynomial time approximation algorithms for the problems are given. The algorithms compute trees whose maximal degree is at most Δ* + 1, where Δ* is the degree of some optimal tree for the respective problems. Unless P = NP, this is the best bound achievable in polynomial time.

DOMINE: a comprehensive collection of known and predicted domain-domain interactions

DOMINE is a comprehensive collection of known and predicted domain–domain interactions (DDIs) compiled from 15 different sources. The updated DOMINE includes 2285 new domain–domain interactions (DDIs) inferred from experimentally characterized high-resolution three-dimensional structures, and about 3500 novel predictions by five computational approaches published over the last 3 years. These additions bring the total number of unique DDIs in the updated version to 26 219 among 5140 unique Pfam domains, a 23% increase compared to 20 513 unique DDIs among 4346 unique domains in the previous version. The updated version now contains 6634 known DDIs, and features a new classification scheme to assign confidence levels to predicted DDIs. DOMINE will serve as a valuable resource to those studying protein and domain interactions. Most importantly, DOMINE will not only serve as an excellent reference to bench scientists testing for new interactions but also to bioinformaticans seeking to predict novel protein–protein interactions based on the DDIs. The contents of the DOMINE are available at http://domine.utdallas.edu.

Improved Approximation Algorithms for Uniform Connectivity Problems

The problem of finding minimum-weight spanning subgraphs with a given connectivity requirement is considered. The problem is NP-hard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. The following results are presented:1. For the unweightedk-edge-connectivity problem an approximation algorithm that achieves a performance ratio of 1.85 is described. This is the first polynomial-time algorithm that achieves a constant less than 2, for allk.2. For the weightedk-vertex-connectivity problem, a constant factor approximation algorithm is given assuming that the edge-weights satisfy the triangle inequality. This is the first constant factor approximation algorithm for this problem.3. For the case of biconnectivity, with no assumptions about the weights of the edges, an algorithm that achieves a factor asymptotically approaching 2 is described. This matches the previous best bound for the corresponding edge connectivity problem.

Approximation algorithms with bounded performance guarantees for the clustered traveling salesman problem

Abstract. Let G=(V,E) be a complete undirected graph with vertex set V , edge set E , and edge weights l(e) satisfying triangle inequality. The vertex set V is partitioned into clustersV1, . . ., Vk . The clustered traveling salesman problem is to compute a shortest Hamiltonian cycle (tour) that visits all the vertices, and in which the vertices of each cluster are visited consecutively. Since this problem is a generalization of the traveling salesman problem, it is NP-hard. In this paper we consider several variants of this basic problem and provide polynomial time approximation algorithms for them.

Low-Degree Spanning Trees of Small Weight

Given

Approximation algorithms for the capacitated minimum spanning tree problem and its variants in network design

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The finite capacity dial-a-ride problem

points in the plane, the degree-

A network-flow technique for finding low-weight bounded-degree spanning trees

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