Results 1  10
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14
Packing Steiner trees
"... The Steiner packing problem is to find the maximum number of edgedisjoint subgraphs of a given graph G that connect a given set of required points S. This problem is motivated by practical applications in VLSIlayout and broadcasting, as well as theoretical reasons. In this paper, we study this p ..."
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Cited by 87 (5 self)
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The Steiner packing problem is to find the maximum number of edgedisjoint subgraphs of a given graph G that connect a given set of required points S. This problem is motivated by practical applications in VLSIlayout and broadcasting, as well as theoretical reasons. In this paper, we study this problem and present an algorithm with an asymptotic approximation factor of S/4. This gives a sufficient condition for the existence of k edgedisjoint Steiner trees in a graph in terms of the edgeconnectivity of the graph. We will show that this condition is the best possible if the number of terminals is 3. At the end, we consider the fractional version of this problem, and observe that it can be reduced to the minimum Steiner tree problem via the ellipsoid algorithm.
Minimum Cuts in NearLinear Time
, 1999
"... We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that fi ..."
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Cited by 71 (10 self)
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We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log³ n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n² log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n² log³ n). Other applications of the treepacking approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
Improved Algorithms For Bipartite Network Flow
, 1994
"... In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jE ..."
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Cited by 41 (5 self)
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In this paper, we study network flow algorithms for bipartite networks. A network G = (V; E) is called bipartite if its vertex set V can be partitioned into two subsets V 1 and V 2 such that all edges have one endpoint in V 1 and the other in V 2 . Let n = jV j, n 1 = jV 1 j, n 2 = jV 2 j, m = jEj and assume without loss of generality that n 1 n 2 . We call a bipartite network unbalanced if n 1 ø n 2 and balanced otherwise. (This notion is necessarily imprecise.) We show that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a twoedge push rule that allows us to "charge" most computation to vertices in V 1 , and hence develop algorithms whose running times depend on n 1 rather than n. For example, we show that the twoedge push version of Goldberg and Tarjan's FIFO preflow push algorithm runs in O(n 1 m + n 3 1 ) time and that the analogous version of Ahuja and Orlin's excess scaling algori...
On Bipartite Drawings and the Linear Arrangement Problem
"... The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close ..."
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Cited by 17 (0 self)
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The bipartite crossing number problem is studied, and a connection between this problemand the linear arrangement problem is established. It is shown that when the arboricity is close
Fullydynamic mincut
 STOC'01
, 2001
"... We show that we can maintain up to polylogarithmic edge connectivity for a fullydynamic graph in ~ O ( p n) time per edge insertion or deletion. Within logarithmic factors, this matches the best time bound for 1edge connectivity. Previously, no o(n) bound was known for edge connectivity above 3, a ..."
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Cited by 14 (1 self)
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We show that we can maintain up to polylogarithmic edge connectivity for a fullydynamic graph in ~ O ( p n) time per edge insertion or deletion. Within logarithmic factors, this matches the best time bound for 1edge connectivity. Previously, no o(n) bound was known for edge connectivity above 3, and even for 3edge connectivity, the best update time was O(n 2=3), dating back to FOCS’92. Our algorithm maintains a concrete mincut in terms of a pointer to a tree spanning one side of the cut plus ability to list the cut edges in O(log n) time per edge. By dealing with polylogarithmic edge connectivity, we immediately get a sampling based expected factor (1 + o(1)) approximation to general edge connectivity in ~O ( p n) time per edge insertion or deletion. This algorithm also maintains a pointer to one side of a mincut, but if we want to list the cut edges in O(log n) time per edge, the update time increases to ~O ( p m).
Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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Cited by 13 (5 self)
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Constraining plane configurations in cad: combinatorics of lengths and directions
 SIAM Journal on Discrete Mathematics
, 1999
"... Abstract. Configurations of points in the plane constrained by only directions or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph w ..."
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Cited by 11 (4 self)
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Abstract. Configurations of points in the plane constrained by only directions or by lengths alone lead to equivalent theories known as parallel drawings and infinitesimal rigidity of plane frameworks. We combine these two theories by introducing a new matroid on the edge set of the complete graph with doubled edges to describe the combinatorial properties of direction–length designs. 1.
Random Sampling and Greedy Sparsification for Matroid Optimization Problems.
 Mathematical Programming
, 1998
"... Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems ..."
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Cited by 8 (2 self)
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Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be solved in roughly the same ...
Design is as easy as optimization
 In 33rd International Colloquium on Automata, Languages and Programming (ICALP
, 2006
"... We consider the class of maxmin and minmax optimization problems subject to a global budget (or weight) constraint and we undertake a systematic algorithmic and complexitytheoretic study of such problems, which we call problems design problems. Every optimization problem leads to a natural design ..."
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Cited by 7 (0 self)
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We consider the class of maxmin and minmax optimization problems subject to a global budget (or weight) constraint and we undertake a systematic algorithmic and complexitytheoretic study of such problems, which we call problems design problems. Every optimization problem leads to a natural design problem. Our main result uses techniques of FreundSchapire [FS99] from learning theory, and its generalizations, to show that for a large class of optimization problems, the design version is as easy as the optimization version. We also observe a close relationship between design problems and packing problems; this yields relationships between fractional packing of spanning and Steiner trees in a graph, the strength of the graph, and the integrality gap of the bidirected cut relaxation for the graph. 1
Experimental Study of Minimum Cut Algorithms
 M.S. DISSERTATION, MIT
, 1997
"... Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of the relative performance of thes ..."
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Cited by 6 (0 self)
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Recently, several new algorithms have been developed for the minimum cut problem that substantially improve worstcase time bounds for the problem. These algorithms are very different from the earlier ones and from each other. We conduct an experimental evaluation of the relative performance of these algorithms. In the process, we develop heuristics and data structures that substantially improve practical performance of the algorithms. We also develop problem families for testing minimum cut algorithms. Our work leads to a better understanding of practical performance of the minimum cut algorithms and produces very efficient codes for the problem.