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33
Approximating MinimumSize kConnected Spanning Subgraphs via Matching
 SIAM J. Comput
, 1998
"... Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spann ..."
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Cited by 35 (3 self)
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Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spanning subgraph of an undirected graph 1+[1=k], minimumsize knode connected spanning subgraph of a directed graph 1+[1=k], minimumsize kedge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimumsize kedge connected spanning subgraph of a directed graph 1+[4= p k].
Recent Developments in Maximum Flow Algorithms
 in Proceedings of Scandinavian Workshop on Algorithm Theory (SWAT
, 1998
"... Introduction The maximum flow problem is a classical optimization problem with many applications; see e.g. [1, 18, 39]. Algorithms for this problem have been studied for over four decades. Recently, significant improvements have been made in theoretical performance of maximum flow algorithms. In t ..."
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Cited by 23 (1 self)
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Introduction The maximum flow problem is a classical optimization problem with many applications; see e.g. [1, 18, 39]. Algorithms for this problem have been studied for over four decades. Recently, significant improvements have been made in theoretical performance of maximum flow algorithms. In this survey we put these results in perspective and provide pointers to the literature. We assume that the reader is familiar with basic flow algorithms, including Dinitz' blocking flow algorithm [13]. 2 Preliminaries The maximum flow problem is to find a flow of the maximum value given a graph G with arc capacities, a source s, and a sink t, Here a flow is a function on arcs that satisfies capacity constraints for all arcs and conservation constraints for all vertices except the source and the sink. For more details, see [1, 18, 39]. We distinguish between directed
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely comb ..."
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Cited by 20 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
A Better Approximation Ratio for the Minimum Size kEdgeConnected Spanning Subgraph Problem
, 1998
"... Consider the minimum size kedgeconnected spanning subgraph problem: given a positive integer k and a kedgeconnected graph G, nd a kedgeconnected spanning subgraph of G with the minimum number of edges. This problem is known to be NPcomplete. Khuller and Raghavachari presented the rst algorit ..."
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Cited by 15 (0 self)
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Consider the minimum size kedgeconnected spanning subgraph problem: given a positive integer k and a kedgeconnected graph G, nd a kedgeconnected spanning subgraph of G with the minimum number of edges. This problem is known to be NPcomplete. Khuller and Raghavachari presented the rst algorithm which, for all k, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2=(k+ 1), achieved by a slower algorithm of Cheriyan and Thurimella. In this paper, we improve Khuller and Raghavachari's analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enough k, and that it is at most 1.75 for all k. Second, we show that the minimum size 2edgeconnected spanning subgraph problem is MAX SNPhard. A preliminary version of this paper appeared in the Proceedings of the 8 th Annual ACM...
Fast Algorithms for kShredders and kNode Connectivity Augmentation
, 1996
"... A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counti ..."
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Cited by 15 (0 self)
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A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of kseparators is #Pcomplete. However, we present an O(k )time (deterministic) algorithm for finding all the kshredders. This solves an open question: efficiently find a kseparator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing knode connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing knode connectivity.
Flows in Undirected Unit Capacity Networks
 In Proceedings of the 30 th Annual Symposium on the Foundations of Computer Science
, 1997
"... We describe an O(min(m; n 3=2 )m 1=2 )time algorithm for finding maximum flows in undirected networks with unit capacities and no parallel edges. This improves upon the previous bound of Karzanov and Even and Tarjan when m = !(n 3=2 ), and upon a randomized bound of Karger when v = \Omega\Gam ..."
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Cited by 15 (1 self)
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We describe an O(min(m; n 3=2 )m 1=2 )time algorithm for finding maximum flows in undirected networks with unit capacities and no parallel edges. This improves upon the previous bound of Karzanov and Even and Tarjan when m = !(n 3=2 ), and upon a randomized bound of Karger when v = \Omega\Gamma n 7=4 =m 1=2 ). (Here v is the maximum flow value.) 1 Introduction In this paper we consider the undirected maximum flow problem in a network with unit capacities and no parallel edges. Until recently, the fastest known way to solve this problem was using a reduction to the directed problem with unit capacities and no parallel arcs. Karzanov [7] and Even and Tarjan [2] have shown that Dinitz's blocking flow algorithm [1], applied to the directed problem, runs in O(min(m 1=2 ; n 2=3 )m) time. (Here n and m is the number of input vertices and edges, respectively.) Recently, Karger [6] developed two randomized algorithms for the undirected problem, with running times of O (m 5...
OutputSensitive Reporting of Disjoint Paths
, 1996
"... A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For ..."
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Cited by 11 (2 self)
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A kpath query on a graph consists of computing k vertexdisjoint paths between two given vertices of the graph, whenever they exist. In this paper, we study the problem of performing kpath queries, with k < 3, in a graph G with n vertices. We denote with the total length of the paths reported. For k < 3, we present an optimal data structure for G that uses O(n) space and executes kpath queries in outputsensitive O() time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs.
Using expander graphs to find vertex connectivity
 Proc. 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract The (vertex) connectivity ^ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known algorithm for finding ^. For a digraph with n vertices, m edges and connectivity ^ the time bound is O((n + minf^5=2; ^n3=4g)m). Thi ..."
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Cited by 10 (0 self)
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Abstract The (vertex) connectivity ^ of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. We present the fastest known algorithm for finding ^. For a digraph with n vertices, m edges and connectivity ^ the time bound is O((n + minf^5=2; ^n3=4g)m). This improves the previous best bound of O((n + minf^3; ^ng)m). For an undirected graph both of these bounds hold with m replaced by ^n. Our approach uses expander graphs to exploit nesting properties of certain separation triples. 1 Introduction The (vertex) connectivity ^ of a graph is the smallest number of vertices whose deletion separates or trivializes the graph. (Other basic terminology is defined at the end of this section.) This is a central concept of graph theory [11]. Computing the connectivity is posed as Research Problem 5.30 in [1] where in fact a lineartime algorithm is conjectured. Yet relatively little progress has been made on computing connectivity.1 If the conjectured lineartime algorithm exists it involves techniques that are radically different from the known ones.
A General Framework for Graph Sparsification
, 2011
"... We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results ..."
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Cited by 9 (0 self)
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We present a general framework for constructing cut sparsifiers in undirected graphs — weighted subgraphs for which every cut has the same weight as the original graph, up to a multiplicative factor of (1 ± ǫ). Using this framework, we simplify, unify and improve upon previous sparsification results. As simple instantiations of this framework, we show that sparsifiers can be constructed by sampling edges according to their strength (a result of Benczúr and Karger), effective resistance (a result of Spielman and Srivastava), edge connectivity, or by sampling random spanning trees. Sampling according to edge connectivity is the most aggressive method, and the most challenging to analyze. Our proof that this method produces sparsifiers resolves an open question of Benczúr and Karger. While the above results are interesting from a combinatorial standpoint, we also prove new algorithmic results. In particular, we develop techniques that give the first (optimal) O(m)time sparsification algorithm for unweighted graphs. Our algorithm has a running time of O(m) + Õ(n/ǫ²) for weighted graphs, which is also linear unless the input graph is very sparse itself. In both cases, this improves upon the previous best running times (due to Benczúr and Karger) of O(m log² n) (for the unweighted case) and O(m log³ n) (for the weighted case) respectively. Our algorithm constructs sparsifiers that contain O(n log n/ǫ²) edges in expectation; the only known construction of sparsifiers with fewer edges is by a substantially slower algorithm running in O(n 3 m/ǫ 2) time. A key ingredient of our proofs is a natural generalization of Karger’s bound on the number of small cuts in an undirected graph. Given the numerous applications of Karger’s bound, we suspect that our generalization will also be of independent interest.
Circumference of Graphs with Bounded Degree
"... Karger, Motwani and Ramkumar have shown that there is no constant approximation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured this is the case even for graphs with bounded degree. On the other hand,Feder, Motwani and Subi have shown that there is a polynomial time a ..."
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Cited by 5 (2 self)
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Karger, Motwani and Ramkumar have shown that there is no constant approximation algorithm to find a longest cycle in a Hamiltonian graph, and they conjectured this is the case even for graphs with bounded degree. On the other hand,Feder, Motwani and Subi have shown that there is a polynomial time algorithm for finding a cycle of length nlog3 2 in a 3connected cubic nvertex graph. In this paper,we show that if G is a 3connected nvertex graph with maximum degree at most d, then one can find, in O(n3) time, a cycle in G of length at least \Omega (nlogb 2), where b = 2(d 1)² + 1.