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The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and ..."
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Cited by 139 (0 self)
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In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2  2/k of the optimal cut weight.
Maximizing Loop Parallelism and Improving Data Locality via Loop Fusion and Distribution
 IN LANGUAGES AND COMPILERS FOR PARALLEL COMPUTING
, 1994
"... Loop fusion is a program transformation that merges multiple loops into one. It is effective for reducing the synchronization overhead of parallel loops and for improving data locality. This paper presents three results for fusion: (1) a new algorithm for fusing a collection of parallel and seq ..."
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Cited by 121 (10 self)
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Loop fusion is a program transformation that merges multiple loops into one. It is effective for reducing the synchronization overhead of parallel loops and for improving data locality. This paper presents three results for fusion: (1) a new algorithm for fusing a collection of parallel and sequential loops, minimizing parallel loop synchronization while maximizing parallelism; (2) a proof that performing fusion to maximize data locality is NPhard; and (3) two polynomialtime algorithms for improving data locality. These techniques also apply to loop distribution, which is shown to be essentially equivalent to loop fusion. Our approach is general enough to support other fusion heuristics. Preliminary experimental results validate our approach for improving performance by exploiting data locality and increasing the granularity of parallelism.
Approximating Minimum Feedback Sets and Multicuts in Directed Graphs
 ALGORITHMICA
, 1998
"... This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights (each of which is at le ..."
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Cited by 98 (3 self)
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This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (fvs) problem, and the weighted feedback edge set problem (fes). In the fvs (resp. fes) problem, one is given a directed graph with weights (each of which is at least 1) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NPHard problems and have many applications. We also consider a generalization of these problems: subsetfvs and subsetfes, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NPHard even when X = 2. We present approximation algorithms for the subsetfvs and subsetfes problems. The first algorithm we present achieves an approximation factor of O(log2 X). The second algorithm achieves an approximation factor of O(min(log tau log log tau; log n log log n)), where tau is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the subsetfes and subsetfvs problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1 + epsilon) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.
Multiway Cuts in Directed and Node Weighted Graphs
 in Proc. 21st ICALP, Lecture Notes in Computer Science 820
, 1994
"... this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for t ..."
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Cited by 42 (4 self)
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this paper we consider node multiway cuts; the problem of computing a minimum weight node multiway cut is known to be NPhard and max SNPhard [1]. It turns out that the approximation algorithm in [2] for edge multiway cuts does not extend to the node multiway cut problem. Let us give a reason for this. Define an isolating cut for terminal s i to be a cut that separates s i from the rest of the terminals. A minimum isolating cut for s i can be computed in polynomial time by identifying the remaining terminals, and finding a minimum cut separating them from s i . The algorithm in [2] finds such cuts for each terminal, discards the heaviest cut, and picks the union of the remaining. The approximation factor is proven by observing that on doubling each edge in the optimum multiway cut, we can partition these edges into k isolating cuts, one for each Department of Computer Science and Engg., Indian Institute of Technology, New Delhi, India
Correlation Clustering in General Weighted Graphs
 Theoretical Computer Science
, 2006
"... We consider the following general correlationclustering problem [1]: given a graph with real nonnegative edge weights and a 〈+〉/〈− 〉 edge labeling, partition the vertices into clusters to minimize the total weight of cut 〈+ 〉 edges and uncut 〈− 〉 edges. Thus, 〈+ 〉 edges with large weights (represen ..."
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Cited by 20 (0 self)
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We consider the following general correlationclustering problem [1]: given a graph with real nonnegative edge weights and a 〈+〉/〈− 〉 edge labeling, partition the vertices into clusters to minimize the total weight of cut 〈+ 〉 edges and uncut 〈− 〉 edges. Thus, 〈+ 〉 edges with large weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while 〈− 〉 edges with large weights encourage the endpoints to belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both of these parameters are effectively chosen to be the best possible by the problem definition. Correlation clustering was introduced by Bansal, Blum, and Chawla [1], motivated by both document clustering and agnostic learning. They proved NPhardness and gave constantfactor approximation algorithms for the special case in which the graph is complete (full information) and every edge has the same weight. We give an O(log n)approximation algorithm for the general case based on a linearprogramming rounding and the “regiongrowing ” technique. We also prove that this linear program has a gap of Ω(log n), and therefore our approximation is tight under this approach. We also give an O(r 3)approximation algorithm for Kr,rminorfree graphs. On the other hand, we show that the problem is equivalent to minimum multicut, and therefore APXhard and difficult to approximate better than Θ(logn). 1
Correlation clustering – minimizing disagreements on arbitrary weighted graphs
 Proceedings of the 11th Annual European Symposium on Algorithms
, 2003
"... ..."
Multicommodity Flows and Approximation Algorithms
, 1994
"... This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation ..."
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Cited by 3 (0 self)
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This thesis is about multicommodity flows and their use in designing approximation algorithms for problems involving cuts in graphs. In a groundbreaking work Leighton and Rao [34] showed an approximate maxflow mincut theorem for uniform multicommodity flow and used this to obtain an approximation algorithm for the flux of a graph. We consider the multicommodity flow problem in which the object is to maximize the sum of the flows routed and prove the following approximate maxflow minmulticut theorem minmulticut O(log k) maxflow minmulticut where k is the number of commodities. Our proof is based on a rounding technique from [34]. Further, we show that this theorem is tight. For a multicommodity flow instance with specified demands, the ratio of the maximum concurrent flow to the sparsest cut was shown to be bounded by O(log 2 k) [30, 57, 17, 47]. We use ideas from our proof of the approximate maxflow minmulticut theorem and a geometric scaling technique from [1] to provi...
Correlation Clustering in . . .
, 2005
"... We consider the following general correlationclustering problem [1]: given a graph withreal nonegative edge weights and a h+i/hi edge labeling, partition the vertices into clusters to minimize the total weight of cut h+i edges and uncut hi edges. Thus, h+i edges withlarge weights (representing s ..."
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Cited by 1 (0 self)
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We consider the following general correlationclustering problem [1]: given a graph withreal nonegative edge weights and a h+i/hi edge labeling, partition the vertices into clusters to minimize the total weight of cut h+i edges and uncut hi edges. Thus, h+i edges withlarge weights (representing strong correlations between endpoints) encourage those endpoints to belong to a common cluster while hi edges with large weights encourage the endpointsto belong to different clusters. In contrast to most clustering problems, correlation clustering specifies neither the desired number of clusters nor a distance threshold for clustering; both ofthese parameters are effectively chosen to be the best possible by the problem definition. Correlation clustering was introduced by Bansal, Blum, and Chawla [1], motivated by bothdocument clustering and agnostic learning. They proved NPhardness and gave constantfactor approximation algorithms for the special case in which the graph is complete (full information)and every edge has the same weight. We give an O(log n)approximation algorithm for thegeneral case based on a linearprogramming rounding and the "regiongrowing " technique. We also prove that this linear program has a gap of \Omega (log n), and therefore our approximation istight under this approach. We also give an O(r3)approximation algorithm for Kr,rminorfreegraphs. On the other hand, we show that the problem is equivalent to minimum multicut, and