Low-Degree Spanning Trees of Small Weight

The degree-d spanning tree problem asks for a minimum-weight spanning tree in which the degree of each vertex is at most d. When d=2 the problem is TSP, and in this case, the well-known Christofides algorithm provides a 1.5-approximation algorithm (assuming the edge weights satisfy the triangle inequality). In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of finding an algorithm with performance guarantee less than 2 for Euclidean graphs (points in R^n) and d > 2. This paper gives the first answer to that challenge, presenting an algorithm to compute a degree-3 spanning tree of cost at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2 and the algorithm can also find a degree-4 spanning tree of cost at most 5/4 times the MST.Comment: conference version in Symposium on Theory of Computing (1994

Landmarks in graphs

AbstractNavigation 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 robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's 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

Designing Multi-Commodity Flow Trees

The traditional multi-commodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multi-commodity flow network-design problem: given a set of multi-commodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimum-weight balanced-separator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced-separator procedure. If Leighton and Rao's balanced-separator procedure is used, the performance guarantee is O(log n). This improves the O(log^2 n) approximation factor that is trivial to obtain by a direct application of the balanced-separator method.Comment: Conference version in WADS'9

Improved Approximation Algorthmsor 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 unweighted k-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 all k. 2. For the weighted 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. (Also cross-referenced as UMIACS-TR-95-21

Localization 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 robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's 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 {\em new\/} result is that the metric dimension can be approximated in polynomial time within a factor of $O(\log n)$; we also establish some properties of graphs with metric dimension 2. (Also cross-referenced as UMIACS-TR-94-92

Approximating the Minimum Equivalent Digraph

The MEG (minimum equivalent graph) problem is, given a directed graph, to find a small subset of the edges that maintains all reachability relations between nodes. The problem is NP-hard. This paper gives an approximation algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its analysis are based on the simple idea of contracting long cycles. (This result is strengthened slightly in ``On strongly connected digraphs with bounded cycle length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms (1994

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

DOMINE: a database of protein domain interactions

DOMINE is a database of known and predicted protein domain interactions compiled from a variety of sources. The database contains domain–domain interactions observed in PDB entries, and those that were predicted by eight different computational approaches. DOMINE contains a total of 20 513 unique domain–domain interactions among 4036 Pfam domains, out of which 4349 are inferred from PDB entries and 17 781 were predicted by at least one computational approach. This database will serve as a valuable resource to those working in the field of protein and domain interactions. DOMINE may not only serve as a reference to experimentalists who test for new protein and domain interactions, but also offers a consolidated dataset for analysis by bioinformaticians who seek to test ideas regarding the underlying factors that control the topological structure of interaction networks. DOMINE is freely available at http://domine.utdallas.edu

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

Abstract. Given an undirected graph G = (V, E) with nonnegative costs on its edges, a root node r ∈ V,aset of demands D ⊆ V with demand v ∈ D wishing to route w(v) units of flow (weight) to r, and a positive number k, the Capacitated Minimum Steiner Tree (CMStT) problem asks for a minimum Steiner tree, rooted at r, spanning the vertices in D ∪{r}, inwhich the sum of the vertex weights in every subtree connected to r is at most k. When D = V, this problem is known as the Capacitated Minimum Spanning Tree (CMST) problem. Both CMsT and CMST problems are NP-hard. In this article, we present approximation algorithms for these problems and several of their variants in network design. Our main results are the following: —We present a (γρST + 2)-approximation algorithm for the CMStT problem, where γ is the inverse Steiner ratio, and ρST is the best achievable approximation ratio for the Steiner tree problem. Our ratio improves the current best ratio of 2ρST + 2 for this problem. —Inparticular, we obtain (γ +2)-approximation ratio for the CMST problem, which is an improvement over the current best ratio of 4 for this problem. For points in Euclidean and rectilinear planes, our result translates into ratios of 3.1548 and 3.5, respectively. —For instances in the plane, under the L p norm, with the vertices in D having uniform weights, we present a nontrivial ( 7 5ρST + 3)-approximation algorithm for the CMStT problem. This translate

Degree-Bounded Minimum Spanning Trees

Given n points in the Euclidean plane, the degree-- MST problem asks for a spanning tree of minimum weight in which the degree of each node is at most