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Computing MinimumWeight Perfect Matchings
 INFORMS
, 1999
"... We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimumweight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the ..."
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Cited by 83 (2 self)
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We make several observations on the implementation of Edmonds’ blossom algorithm for solving minimumweight perfectmatching problems and we present computational results for geometric problem instances ranging in size from 1,000 nodes up to 5,000,000 nodes. A key feature in our implementation is the use of multiple search trees with an individual dualchange � for each tree. As a benchmark of the algorithm’s performance, solving a 100,000node geometric instance on a 200 Mhz PentiumPro computer takes approximately 3 minutes.
Unit Disk Graph Recognition is NPHard
 Computational Geometry. Theory and Applications
, 1993
"... Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have s ..."
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Cited by 78 (1 self)
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Unit disk graphs are the intersection graphs of unit diameter closed disks in the plane. This paper reduces SATISFIABILITY to the problem of recognizing unit disk graphs. Equivalently, it shows that determining if a graph has sphericity 2 or less, even if the graph is planar or is known to have sphericity at most 3, is NPhard. We show how this reduction can be extended to 3 dimensions, thereby showing that unit sphere graph recognition, or determining if a graph has sphericity 3 or less, is also NPhard. We conjecture that Ksphericity is NPhard for all fixed K greater than 1. 1 Introduction A unit disk graph is the intersection graph of a set of unit diameter closed disks in the plane. That is, each vertex corresponds to a disk in the plane, and two vertices are adjacent in the graph if the corresponding disks intersect. The set of disks is said to realize the graph. Of course, the unit of distance is not critical, since the disks realize the same graph even if the coordina...
Bridges between Geometry and Graph Theory
 in Geometry at Work, C.A. Gorini, ed., MAA Notes 53
"... Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another ..."
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Cited by 9 (4 self)
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Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.
2008) ,“An SDPbased divideandconquer algorithm for large scale noisy anchorfree graph realization
"... Abstract. We propose the DISCO algorithm for graph realization in Rd, given sparse and noisy shortrange intervertex distances as inputs. Our divideandconquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by sol ..."
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Cited by 7 (0 self)
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Abstract. We propose the DISCO algorithm for graph realization in Rd, given sparse and noisy shortrange intervertex distances as inputs. Our divideandconquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by solving a semidefinite program. The recursive step is to break a large group of vertices into two smaller groups with overlapping vertices. These two groups are solved recursively, and the subconfigurations are stitched together, using the overlapping atoms, to form a configurations for the larger group. At intermediate stages, the configurations are improved by gradient descent refinement. The algorithm is applied to the problem of determining protein molecule structure. Tests are performed on molecules taken from the Protein Data Bank database. Given 20–30 % of the interatom distances less than 6˚A that are corrupted by a high level of noise, DISCO is able to reliably and efficiently reconstruct the conformation of large molecules. In particular, given 30 % of distances with 20 % multiplicative noise, a 13000atom conformation problem is solved within an hour with an RMSD of 1.6˚A. 1. Introduction. A
Cubicity of Interval Graphs and the Claw Number
, 2009
"... Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A bdimensional cube is a Cartesian product I1 × I2 × · · · × Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G) is the minimum positive ..."
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Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A bdimensional cube is a Cartesian product I1 × I2 × · · · × Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G) is the minimum positive integer b such that the vertices in G can be mapped to axis parallel bdimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m + 1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ⌈log 2 ψ(G)⌉. In this paper, we show that, for an interval graph G ⌈log 2 ψ(G) ⌉ ≤ cub(G) ≤ ⌈log 2 ψ(G) ⌉ + 2. It is not clear whether the upper bound of ⌈log 2 ψ(G) ⌉ + 2 is tight: Till now we are unable to find any interval graph with cub(G)> ⌈log 2 ψ(G)⌉. We also show that, for an interval graph G, cub(G) ≤ ⌈log 2 α⌉, where α is the independence number of G. Therefore, in the special case of ψ(G) = α, cub(G) is exactly ⌈log 2 α⌉. The concept of cubicity can be generalized by considering boxes instead of cubes. A bdimensional box is a Cartesian product I1 ×I2 × · · ·×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of kdimensional boxes. It is clear that box(G) ≤ cub(G). From the above result, it follows that for any graph G, cub(G) ≤ box(G) ⌈log 2 α⌉.
for LargeScale Euclidean Distance Geometry Problems
, 2012
"... Abstract. The Euclidean distance geometry problem (EDGP) includes locating sensors in a sensor network and constructing a molecular configuration using given distances in the two or threedimensional Euclidean space. When the locations of some nodes, called anchors, are given, the problem can be dea ..."
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Abstract. The Euclidean distance geometry problem (EDGP) includes locating sensors in a sensor network and constructing a molecular configuration using given distances in the two or threedimensional Euclidean space. When the locations of some nodes, called anchors, are given, the problem can be dealt with many existing methods. An anchorfree problem in the threedimensional space, however, is a more challenging problem and can be handled with only a few methods. We propose an efficient and robust numerical method for largescale EDGPs with exact and corrupted distances including anchorfree threedimensional problems. The method is based on successive application of the sparse version of full semidefinite programming relaxation (SFSDP) proposed by Kim, Kojima, Waki and Yamashita, and can be executed in parallel. Numerical results on largescale anchorfree threedimensional problems with more than 10000 nodes demonstrate that the proposed method performs better than the direct application of SFSDP and the divide and conquer method of Leung and Toh in terms of efficiency and/or effectiveness measured in the root mean squared distance. Key words. Euclidean distance geometry problem, molecular conformation, the successive
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"... Using a distributed SDP approach to solve simulated protein molecular conformation problems Xingyuan Fang and KimChuan Toh Abstract This paper presents various enhancements to the DISCO algorithm (originally introduced by Leung and Toh [18] for anchorfree graph realization in R d) for applications ..."
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Using a distributed SDP approach to solve simulated protein molecular conformation problems Xingyuan Fang and KimChuan Toh Abstract This paper presents various enhancements to the DISCO algorithm (originally introduced by Leung and Toh [18] for anchorfree graph realization in R d) for applications to conformation of protein molecules in R 3. In our enhanced DISCO algorithm for simulated protein molecular conformation problems, we have incorporated distance information derived from chemistry knowledge such as bond lengths and angles to improve the robustness of the algorithm. We also designed heuristics to detect whether a subgroup is well localized and significantly improved the robustness of the stitching process. Tests are performed on molecules taken from the Protein Data Bank. Given only 20 % of the interatomic distances less than 6 ˚A that are corrupted by high level of noises (to simulate noisy distance restraints generated from nuclear magnetic resonance experiments), our improved algorithm is