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Drawing graphs by eigenvectors: Theory and practice
 Computers and Mathematics with Applications
, 2005
"... Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this ..."
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Abstract. The spectral approach for graph visualization computes the layout of a graph using certain eigenvectors of related matrices. Some important advantages of this approach are an ability to compute optimal layouts (according to specific requirements) and a very rapid computation time. In this paper we explore spectral visualization techniques and study their properties from different points of view. We also suggest a novel algorithm for calculating spectral layouts resulting in an extremely fast computation by optimizing the layout within a small vector space.
Multiple Hotlink Assignment
 In 27th Int. Workshop on GraphTheoric Concepts in Computer Science, volume 2204 of LNCS
, 2001
"... The input for the hotlink assignment problem consists of a node weighted directed acyclic graph with a designated root node r. The goal is to minimize the weighted shortest path length rooted at r by adding a restricted number of outgoing arcs (hotlinks) to each node. ..."
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The input for the hotlink assignment problem consists of a node weighted directed acyclic graph with a designated root node r. The goal is to minimize the weighted shortest path length rooted at r by adding a restricted number of outgoing arcs (hotlinks) to each node.
Online bincoloring
 In Proceedings of the 9th European Symposium on Algorithms (ESA
, 2001
"... We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly environment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size B 2 N. The packi ..."
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Cited by 6 (0 self)
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We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly environment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size B 2 N. The packing process is subject to the constraint that at any moment in time at most q 2 N bins are partially filled. Moreover, bins may only be closed if they are filled completely. An online algorithm must pack each item must be packed without knowledge of any future items. We investigate the existence of competitive online algorithms for the bin coloring problem. We prove an upper bound of 3q 1 and a lower bound of 2q for the competitive ratio of a natural greedytype algorithm, and show that surprisingly a trivial algorithm which uses only one open bin has a strictly better competitive ratio of 2q 1. Moreover, we show that any deterministic algorithm has a competitive ratio q) and that randomization does not improve this lower bound even when the adversary is oblivious.
A Fast Multi Method for Drawing Large Graphs
 Journal of Graph Algorithms and Applications
, 2001
"... We present a mu ltiscale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that aresuSVV tially larger than those we haveencou tered in prior work. For example, the paper contains a drawing of ..."
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We present a mu ltiscale layou algorithm for the aesthetic drawing ofuV#860L32 graphs with straightline edges. The algorithm is extremely fast, and is capable of drawing graphs that aresuSVV tially larger than those we haveencou tered in prior work. For example, the paper contains a drawing of a graph with over 15,000 vertices. Also we achieve "optimal" drawings of 1000 vertex graphs in ab ou 1 second. The proposed algorithm embodies a new mu ltiscale scheme for drawing graphs, which was motivated by the earlier muS8820L4# algorithm of Hadany and Harel [HH99]. In principle, it cou# significantly improve the speed of essentially any forcedirected method (regardless of that method's ability of drawing weighted graphs or the continu0 y of its costfuL436280 1 Introducti7 AgraphG(V,E) is an abstract structure that is used to model a relation over a set V ofen tities. Graph drawin is acon ven tion# tool for the visualization of relationq in]4#]Rq]#] an its usefulnRq depenV on its readability, that is, the capability of con veyin the meanfi# of the diagram quickly # A shorter version appeared in Proc. GraphD awing 2000, LNCS 1984, pp. 183196, Springer Verlag, 2000. clearly. In recen t years, man y algorithms for drawin graphs automatically were proposed (the state of the art is surveyedcomprehen]8 ely in 99, KW01]). Wecon#E trateon the problem of drawin an unF8ERqfi4 graph with straightlin edges. In this case the problem reduces to that of positionRq the vertices bydeterminfiF a mappin L : V R . A populargenrR# approach to this problem is the forcedirected technRVEF which in troduces a heuristic costfunR]F9 (an energy) of themappin L, which (hopefully) achieves its min4 umwhen the layout isn ice. Varian ts of this approach di#er in the defin9fiFR of t...
39 Most Tensor Problems are NPHard
"... We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NPhard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm ..."
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We show that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NPhard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or spectral norm; determining a best rank1 approximation to a tensor; and determining the rank of a tensor. Additionally, we prove that some of these problems have no polynomial time approximation schemes, and at least one enumerative version is #Phard. We also show that restricting these problems to symmetric tensors does not alleviate their NPhardness and that the problem of deciding nonnegative definiteness of a symmetric 4tensor is also NPhard. Except for this nonnegative definiteness problem, all our results apply to 3tensors. We shall argue that these 3tensor problems are a boundary separating the computational tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones.