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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 124 (7 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.
Computing cycle covers without short cycles
 IN PROC. 9TH ANN. EUROPEAN SYMP. ON ALGORITHMS (ESA)
, 2001
"... A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A kcycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a kcycle cover kDCC and kUCC. Given a graph with e ..."
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Cited by 18 (4 self)
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A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A kcycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a kcycle cover kDCC and kUCC. Given a graph with edge weights one and two, MinkDCC and MinkUCC are the minimization problems of finding a kcycle cover with minimum weight. We present factor 4/3 approximation algorithms for MinkDCC with running time O(n
Two approximation algorithms for 3cycle covers
 PROC. OF THE 5TH INTERNATIONAL WORKSHOP ON APPROXIMATION ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS (APPROX), LECTURE NOTES IN COMPUT. SCI. 2462
, 2002
"... A cycle cover of a directed graph is a collection of node disjoint cycles such that every node is part of exactly one cycle. A kcycle cover is a cycle cover in which every cycle has length at least k. While deciding whether a directed graph has a 2cycle cover is solvable in polynomial time, decid ..."
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Cited by 9 (3 self)
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A cycle cover of a directed graph is a collection of node disjoint cycles such that every node is part of exactly one cycle. A kcycle cover is a cycle cover in which every cycle has length at least k. While deciding whether a directed graph has a 2cycle cover is solvable in polynomial time, deciding whether it has a 3cycle cover is already NPcomplete. Given a directed graph with nonnegative edge weights, a maximum weight 2cycle cover can be computed in polynomial time, too. We call the corresponding optimization problem of finding a maximum weight 3cycle cover Max3DCC. In this paper we present two polynomial time approximation algorithms for Max3DCC. The heavier of the 3cycle covers computed by these algorithms has at least a fraction of 3 5 − ǫ, for any ǫ> 0, of the weight of a maximum weight 3cycle cover. As a lower bound, we prove that Max3DCC is APXcomplete, even if the weights fulfil the triangle inequality.
Dynamic Matchings and Quasidynamic Fractional Matchings. II
, 1983
"... Consider a directed graph G in which every edge has an associated realvalued distance and a realvalued weight. The weight of an undirected circuit of G is the sum of the weights of the edges, whereas the distance of an undirected circuit is the sum of the distances of the forward edges of the circ ..."
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Consider a directed graph G in which every edge has an associated realvalued distance and a realvalued weight. The weight of an undirected circuit of G is the sum of the weights of the edges, whereas the distance of an undirected circuit is the sum of the distances of the forward edges of the circuit minus the sum of the distances of the backward edges. A trivial circuit is a twoedge circuit in which one edge of G appears twice on the circuit. A quasidynamic fractional matching (or Qmatching) is a collection of vertexdisjoint circuits such that each circuit is either trivial or else it is an odd circuit whose distance is nonzero. The Qmatching problem is to find a Qmatching that maximizes the sum of the weights of its circuits. The Qmatching problem generalizes both the matching problem and the fractional matching problem. Moreover, the dynamic matching problem, which is a matching problem on an infinite dynamic (timeexpanded) graph, is linearly transformable to the Qmatching problem, as shown in Part I of this paper. In this paper we solve the Qmatching problem by generalizing Edmonds ' blossom algorithm. In fact, all of the major components of the blossom algorithmincluding alternating trees, augmentations, shrinking, and expandingare appropriately generalized to yield a running time that is proportional to that for the weighted matching problem. Furthermore, if all edge distances are equal to zero, this new algorithm reduces to the blossom algorithm.