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55
Shock Graphs and Shape Matching
, 1997
"... We have been developing a theory for the generic representation of 2D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a ..."
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Cited by 268 (35 self)
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We have been developing a theory for the generic representation of 2D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint.
When trees collide: An approximation algorithm for the generalized Steiner problem on networks
, 1994
"... We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the a ..."
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Cited by 256 (39 self)
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We give the first approximation algorithm for the generalized network Steiner problem, a problem in network design. An instance consists of a network with linkcosts and, for each pair fi; jg of nodes, an edgeconnectivity requirement r ij . The goal is to find a minimumcost network using the available links and satisfying the requirements. Our algorithm outputs a solution whose cost is within 2dlog 2 (r + 1)e of optimal, where r is the highest requirement value. In the course of proving the performance guarantee, we prove a combinatorial minmax approximate equality relating minimumcost networks to maximum packings of certain kinds of cuts. As a consequence of the proof of this theorem, we obtain an approximation algorithm for optimally packing these cuts; we show that this algorithm has application to estimating the reliability of a probabilistic network.
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 142 (5 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
Improved Approximation Algorithms for Network Design Problems
, 1994
"... We consider a class of network design problems in which one needs to find a minimumcost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edgedisjoint paths between each pai ..."
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Cited by 102 (11 self)
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We consider a class of network design problems in which one needs to find a minimumcost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edgedisjoint paths between each pair of vertices v and w. We present an approximation algorithm with a performance guarantee of 2H(fmax ) = 2(1 + 2 + 3 + \Delta \Delta \Delta + fmax ) where fmax is the maximum requirement. This improves upon the best previously known performance guarantee of 2fmax . We also show
RANDOM SAMPLING IN CUT, FLOW, AND NETWORK DESIGN PROBLEMS
, 1999
"... We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for pro ..."
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Cited by 102 (12 self)
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We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cutapproximation algorithms extend unchanged to weighted graphs while our weightedgraph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a nearlinear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cutbased problems, including approximating the best balanced cut of a graph, finding a kconnected orientation of a 2kconnected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum kconnected subgraph problem from 1.85 to 1 � O(�log n)/k).
Improved Approximation Algorithms for Uniform Connectivity Problems
 J. Algorithms
"... The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighted connectivity problems are given. Th ..."
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Cited by 80 (3 self)
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The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NPhard 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 kedgeconnectivity problem an approximation algorithm that achieves a performance ratio of 1.85 is described. This is the first polynomialtime algorithm that achieves a constant less than 2, for all k. 2. For the weighted kvertexconnectivity problem, a constant factor approximation algorithm is given assuming that the edgeweights 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...
ViewBased Object Recognition Using Saliency Maps
, 1998
"... We introduce a novel viewbased object representation, called the saliency map graph (SMG), which captures the salient regions of an object view at multiple scales using a wavelet transform. This compact representation is highly invariant to translation, rotation (image and depth), and scaling, and ..."
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Cited by 52 (11 self)
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We introduce a novel viewbased object representation, called the saliency map graph (SMG), which captures the salient regions of an object view at multiple scales using a wavelet transform. This compact representation is highly invariant to translation, rotation (image and depth), and scaling, and offers the locality of representation required for occluded object recognition. To compare two saliency map graphs, we introduce two graph similarity algorithms. The first computes the topological similarity between two SMG's, providing a coarselevel matching of two graphs. The second computes the geometrical similarity between two SMG's, providing a finelevel matching of two graphs. We test and compare these two algorithms on a large database of model object views.
The Access Network Design Problem
, 1998
"... We consider the problem of designing a minimum costaccess network to carry traffic from a set of endnodes to a core network. A set of trunks of K differing types areavailable for leasing or buying. Some trunktypes have a high initial overhead cost but a low cost per unit bandwidth. Others have a ..."
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Cited by 45 (1 self)
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We consider the problem of designing a minimum costaccess network to carry traffic from a set of endnodes to a core network. A set of trunks of K differing types areavailable for leasing or buying. Some trunktypes have a high initial overhead cost but a low cost per unit bandwidth. Others have a low overhead cost but a high cost per unit bandwidth. When the central core is given, we show how to construct an access network whose cost is within O(KÂ²) of optimal,under weak assumptions on the cost structure. In contrast with previous bounds, this bound is independent of the network and the traffic. Typically, the value of K is small.Our approach uses a linear programming relaxation and is motivated by a rounding technique of Shmoys, Tardos andAardal [15]. Our techniques extend to a more complex situation inwhich the core is not given a priori. In this case we aim to minimize the switch cost of the core in addition to the trunkcost of the access network. We provide the same performance bound.
Computing NearOptimal Solutions to Combinatorial Optimization Problems
 IN COMBINATORIAL OPTIMIZATION, DIMACS SERIES IN DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
, 1995
"... In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the ..."
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Cited by 32 (0 self)
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In the past few years, there has been significant progress in our understanding of the extent to which nearoptimal solutions can be efficiently computed for NPhard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations.