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zapproximations
 Journal of Algorithms
, 2001
"... Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ..."
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Cited by 11 (3 self)
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Approximation algorithms for NPhard optimization problems have been widely studied for over three decades. Most of these measure the quality of the solution produced by taking the ratio of the cost of the solution produced by the algorithm to the cost of an optimal solution. In certain cases, this ratio may not be very meaningful for example, if the ratio of the worst solution to the best solution is at most some constant ff, then an approximation algorithm with factor ff may in fact yield the worst solution! To overcome this hurdle (among others), several authors have independently suggested the use of a different measure which we call zapproximation. An algorithm is an ff zapproximation if it runs in polynomial time, and produces a solution whose distance from the optimal one is at most ff times the distance between the optimal solution and the worst possible solution. The results known so far about zapproximations are either of the inapproximability type or rather straightforward observations. We design polynomial time algorithms for several fundamental discrete optimization problems, in particular we obtain a zapproximation factor of 1 2 for the directed traveling salesman problem (TSP) (with no triangle inequality assumption). For the undirected TSP this improves to
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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Cited by 8 (0 self)
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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.
Most tensor problems are NP hard
 CORR
, 2009
"... The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has ..."
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Cited by 7 (0 self)
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The idea that one might extend numerical linear algebra, the collection of matrix computational methods that form the workhorse of scientific and engineering computing, to numerical multilinear algebra, an analogous collection of tools involving hypermatrices/tensors, appears very promising and has attracted a lot of attention recently. We examine here the computational tractability of some core problems in numerical multilinear algebra. We show that tensor analogues of several standard problems that are readily computable in the matrix (i.e. 2tensor) case are NP hard. Our list here includes: determining the feasibility of a system of bilinear equations, determining an eigenvalue, a singular value, or the spectral norm of a 3tensor, determining a best rank1 approximation to a 3tensor, determining the rank of a 3tensor over R or C. Hence making tensor computations feasible is likely to be a challenge.
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.
Better Multidimesional Bin Packing in Special Cases
"... Abstract. We give a modified version of the vector packing algorithm of Chekuri and Khanna [1]. The performance ratio of this algorithm is the same as that of [1] in the worst case but beats their bound in special cases. We also characterize classes of problem instances for which our algorithm beats ..."
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Abstract. We give a modified version of the vector packing algorithm of Chekuri and Khanna [1]. The performance ratio of this algorithm is the same as that of [1] in the worst case but beats their bound in special cases. We also characterize classes of problem instances for which our algorithm beats the previous best bound and the inapproximability bound respectively. 1
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...
STABILITY OF APPROXIMATION IN DISCRETE OPTIMIZATION
"... One can try to parametrize the set of the instances of an optimization problem and look for in polynomial time achievable approximation ratio with respect to this parametrization. When the approximation ratio grows with the parameter, but is independent of the size of the instances, then we speak ..."
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One can try to parametrize the set of the instances of an optimization problem and look for in polynomial time achievable approximation ratio with respect to this parametrization. When the approximation ratio grows with the parameter, but is independent of the size of the instances, then we speak about stable approximation algorithms. An interesting point is that there exist stable approximation algorithms for problems like TSP that is not approximable within any polynomial approximation ratio in polynomial time (assuming P is not equal to NP). The investigation of the stability of approximation overcomes in this way the troubles with measuring the complexity and approximation ratio in the worstcase manner, because it may success in partitioning of the set of all input instances of a hard problem into infinite many classes with respect to the hardest of the particular inputs. We believe that approaches like this will become the core of the algorithmics, because they provide a deeper insight in the hardness of specific problems and in many application we are not interested in the worstcase problem hardness, but in the hardness of forthcoming problem instances. 1.