Results 1  10
of
14
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 70 (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.
Arboricity and Bipartite Subgraph Listing Algorithms
, 1994
"... In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an nvertex graph with O(n) edges and (n/ log n) k ..."
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Cited by 31 (2 self)
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In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an nvertex graph with O(n) edges and (n/ log n) k maximal complete bipartite subgraphs K k,# . # Work supported in part by NSF grant CCR9258355. 1 Introduction A number of graph algorithms depend on finding all subgraphs of a certain type in a larger graph. For instance, in interval or chordal graphs, a decomposition into maximal cliques is key; such a decomposition can be constructed in linear time [4, 17]. Optimal triangulation construction [3] and certain planar graph computations [8] require a listing of all triangles. Related subgraph isomorphism problems also occur in a wide variety of practical applications [2, 5, 12, 9, 13, 14, 19]. For planar graphs, or more generally for graphs of bounded arboricity, the problem of listing c...
Pebble Game Algorithms and Sparse Graphs
, 2007
"... A multigraph G on n vertices is (k,ℓ)sparse if every subset of n ′ ≤ n vertices spans at most kn ′ − ℓ edges. G is tight if, in addition, it has exactly kn − ℓ edges. For integer values k and ℓ ∈ [0,2k), we characterize the (k,ℓ)sparse graphs via a family of simple, elegant and efficient algori ..."
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Cited by 18 (5 self)
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A multigraph G on n vertices is (k,ℓ)sparse if every subset of n ′ ≤ n vertices spans at most kn ′ − ℓ edges. G is tight if, in addition, it has exactly kn − ℓ edges. For integer values k and ℓ ∈ [0,2k), we characterize the (k,ℓ)sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)pebble games.
Finding and Maintaining Rigid Components
"... We give the first complete analysis that the complexity of finding and maintaining rigid components of planar barandjoint frameworks and arbitrary ddimensional bodyandbar frameworks, using a family of algorithms called pebble games, is O(n 2). To this end, we introduce a new data structure prob ..."
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Cited by 9 (9 self)
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We give the first complete analysis that the complexity of finding and maintaining rigid components of planar barandjoint frameworks and arbitrary ddimensional bodyandbar frameworks, using a family of algorithms called pebble games, is O(n 2). To this end, we introduce a new data structure problem called union pairfind, which maintains disjoint edge sets and supports pairfind queries of whether two vertices are spanned by a set. We present solutions that apply to generalizations of the pebble game algorithms, beyond the original rigidity motivation.
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 9 (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 ...
Combinatorial genericity and minimal rigidity
 In SCG ’08: Proceedings of the twentyfourth annual symposium on Computational Geometry
, 2008
"... A well studied geometric problem, with applications ranging from molecular structure determination to sensor networks, asks for the reconstruction of a set P of n unknown points from a finite set of pairwise distances (up to Euclidean isometries). We are concerned here with a related problem: which ..."
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Cited by 5 (1 self)
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A well studied geometric problem, with applications ranging from molecular structure determination to sensor networks, asks for the reconstruction of a set P of n unknown points from a finite set of pairwise distances (up to Euclidean isometries). We are concerned here with a related problem: which sets of distances are minimal with the property that they allow for the reconstruction of P, up to a finite set of possibilities? In the planar case, the answer is known generically via the landmark MaxwellLaman Theorem from Rigidity Theory, and it leads to a combinatorial answer: the underlying structure of such a generic minimal collection of distances is a minimally rigid (aka Laman) graph, for which very efficient combinatorial decision algorithms exist. For nongeneric cases the situation appears to be dramatically different, with the best known algorithms relying on exponentialtime Gröbner base methods, and some specific instances known to be NPhard. Understanding what makes a point set generic emerges as an intriguing geometric question with practical algorithmic consequences. Several definitions (some but not all equivalent) of genericity appear in the rigidity literature, and they have either a measure theoretic, topologic or algebraicgeometric flavor. Some generic point sets appear to be highly degenerate. All existing proofs of Laman’s Theorem make use at some point of one or another of these geometric genericity assumptions. The main result of this paper is the first purely combinatorial proof of Laman’s theorem, together with some interesting consequences. Genericity is characterized in terms of a certain determinant being not identicallyzero as a formal polynomial. We relate its monomial expansion to certain colorings and orientations of the graph and show that these terms cannot all cancel exactly when the underlying graph is Laman. As a surprising consequence, genericity emerges as a purely combinatorial concept.
Partitions of graphs into trees
 IN PROCEEDINGS OF GRAPH DRAWING’06 (KARLSRUHE), VOLUME 4372 OF LNCS
, 2007
"... In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) ..."
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Cited by 3 (0 self)
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In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3tree partition of the inner edges. Maximal planar bipartite graphs have a 2tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NPhardness of the ktree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].
Sparsitycertifying Graph Decompositions
, 704
"... We describe a new algorithm, the (k, ℓ)pebble game with colors, and use it obtain a characterization of the family of (k, ℓ)sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and hav ..."
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Cited by 2 (1 self)
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We describe a new algorithm, the (k, ℓ)pebble game with colors, and use it obtain a characterization of the family of (k, ℓ)sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [13] and give a new proof of the TutteNashWilliams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [6], Gabow and Westermann [5] and Hendrickson [9, 10]. 1 Introduction and
Towards an optimal algorithm for recognizing Laman Graphs
, 2009
"... A graph G with n vertices and m edges is a generically minimally rigid graph (Laman graph), if m = 2n−3 and every induced subset of k vertices spans at most 2k − 3 edges. Laman graphs play a fundamental role in rigidity theory. We discuss the Verification problem: Given a graph G with n vertices, de ..."
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Cited by 2 (0 self)
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A graph G with n vertices and m edges is a generically minimally rigid graph (Laman graph), if m = 2n−3 and every induced subset of k vertices spans at most 2k − 3 edges. Laman graphs play a fundamental role in rigidity theory. We discuss the Verification problem: Given a graph G with n vertices, decide if it is Laman. We present an algorithm that recognizes Laman graphs in O(Tst(n) + n log n) time, where Tst(n) is the best time to extract two edge disjoint spanning trees from a graph with n vertices and 2n − 2 edges, or decide no such trees exist. So far, it is known that Tst(n) is O(n 3/2 √ log n).
Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures
"... Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outde ..."
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Cited by 2 (0 self)
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Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show an Õ(E(G)/ε) time algorithm3 which finds an orientation of an input graph G with outdegree at most ⌈(1 + ε)d ∗ ⌉, where d ∗ is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is ⌈d ∗ ⌉. Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2approximation algorithms by Aichholzer et al. [1] (for orientation / pseudoarboricity), by Arikati et al. [3] (for arboricity) and by Charikar [5] (for maximum density). 1