Results 1  10
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12
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.
Treewidth: Computational Experiments
, 2001
"... Many NPhard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost ..."
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Cited by 43 (12 self)
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Many NPhard graph problems can be solved in polynomial time for graphs with bounded treewidth. Equivalent results are known for pathwidth and branchwidth. In recent years, several studies have shown that this result is not only of theoretical interest but can successfully be applied to find (almost) optimal solutions or lower bounds for many optimization problems. To apply a tree decomposition approach, the treewidth of the graph has to be determined, independently of the application at hand. Although for fixed k, linear time algorithms exist to solve the decision problem “treewidth < k”, their practical use is very limited. The computational tractability of treewidth has been rarely studied so far. In this paper, we compare four heuristics and two lower bounds for instances from applications such as the frequency assignment problem and the vertex coloring problem. Three of the heuristics are based on wellknown algorithms to recognize triangulated graphs. The fourth heuristic recursively improves a tree decomposition by the computation of minimal separating vertex sets in subgraphs. Lower bounds can be computed from maximal cliques and the minimum degree of induced subgraphs. A computational analysis shows that the treewidth of several graphs can be identified by these methods. For other graphs, however, more sophisticated techniques are necessary.
Implementing an Efficient Minimum Capacity Cut Algorithm
 MATHEMATICAL PROGRAMMING
, 1994
"... In this paper, we present an efficient implementation of the O(mn + n 2 log n) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network. To enhance computation, various ideas are added so that it can contract as many edges ..."
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Cited by 15 (0 self)
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In this paper, we present an efficient implementation of the O(mn + n 2 log n) time algorithm originally proposed by Nagamochi and Ibaraki (1992) for computing the minimum capacity cut of an undirected network. To enhance computation, various ideas are added so that it can contract as many edges as possible in each iteration. To evaluate the performance of the resulting implementation, we conducted extensive computational experiment, and compared the results with that of Padberg and Rinaldi's algorithm (1990), which is currently known as one of the practically fastest programs for this problem. The results indicate that our program is considerably faster than Padberg and Rinaldi's program, and its running time is not signicantly aected by the types of the networks being solved.
Deterministic O(nm) Time EdgeSplitting in Undirected Graphs
 J. Combinatorial Optimization
, 1997
"... This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based o ..."
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Cited by 11 (2 self)
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This paper presents a deterministic O(nm log n + n 2 log 2 n) = ~ O(nm) time algorithm for splitting o all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based on this, many graph algorithms using edgesplitting can run faster. For example, the edgeconnectivity augmentation problem in an undirected multigraph can be solved in ~ O(nm) time, which is an improvement over the previously known randomized ~ O(n 3 ) bound and deterministic ~ O(n 2 m) bound. 1 Introduction Let G = (V; E) stand for an undirected multigraph with a set V of vertices and a set E of edges, where an edge with end vertices u and v is denoted by (u; v). A singleton set fxg may be simply written as x, and \ " implies proper inclusion while \ " means \ " or \ = ". For two disjoint subsets X;Y V , we denote by EG (X; Y ) the set of edges, one of whose end vertices is i...
A Note on Minimizing Submodular Functions
 Information Processing Letters
, 1998
"... For a given submodular function f on a nite set V , we consider the problem of nding a nonempty and proper subset X of V that minimizes f(X). If the function f is symmetric, then the problem can be solved by a purely combinatorial algorithm due to Queyranne (1995). This note considers a slightly mor ..."
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Cited by 9 (2 self)
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For a given submodular function f on a nite set V , we consider the problem of nding a nonempty and proper subset X of V that minimizes f(X). If the function f is symmetric, then the problem can be solved by a purely combinatorial algorithm due to Queyranne (1995). This note considers a slightly more general condition than symmetry, i.e., f(X)+f (Y ) f(X Y )+f(Y X) for all X;Y V , and shows that a modication of Queyranne's algorithm solves the problem by using O(jV j 3 ) calls to function value oracle. In this case, all minimal optimal solutions can also be obtained by using O(jV j 3 ) calls to function value oracle. keywords: algorithms, combinatorial problems, computational complexity 1 Introduction Let V be a nite set, and f be a set function f : 2 V 7! !, where ! (! + ) is the set of reals (positive reals). A singleton set fvg may be written as v, and the union of a set X and a singleton fvg may be written as X + v. Furthermore, \ " denotes proper inclusion whil...
Packing Odd Circuits in Eulerian Graphs
 Journal of Combinatorial Theory, Series B
"... Let C be the clutter of odd circuits of a signed graph (G; ). For nonnegative integral edge{weights w, we are interested in the linear program min(w t x : x(C) 1; for C 2 C; and x 0), which we denote by (P ). Solving the related integer program is clearly equivalent to the maximum cut problem, ..."
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Cited by 8 (1 self)
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Let C be the clutter of odd circuits of a signed graph (G; ). For nonnegative integral edge{weights w, we are interested in the linear program min(w t x : x(C) 1; for C 2 C; and x 0), which we denote by (P ). Solving the related integer program is clearly equivalent to the maximum cut problem, which is NP{hard. Guenin proved that (P ) has an optimal solution that is integral so long as (G; ) does not contain a minor isomorphic to odd{K 5 . We generalize this by showing that, if (G; ) does not contain a minor isomorphic to odd{K 5 then (P ) has an integral optimal solution and its dual has a half{integral optimal solution. 1.
Designing multicommodity flow trees
 Inform. Process. Lett
, 1994
"... The traditional multicommodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multicommodity flow networkdesigu problem: given a set of multicommodity flow demands, find a network subject t ..."
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Cited by 7 (0 self)
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The traditional multicommodity flow problem assumes a given flow network in which multiple commodities are to be maximally routed in response to given demands. This paper considers the multicommodity flow networkdesigu problem: given a set of multicommodity flow demands, find a network subject to certain constraints such that the commodities can be maximally routed. This paper focuses on the case when the network is required to be a tree. The main result is an approximation algorithm for the case when the tree is required to be of constant degree. The algorithm reduces the problem to the minimumweight balancedseparator problem; the performance guarantee of the algorithm is within a factor of 4 of the performance guarantee of the balanced~parator procedure. If Leighton and P~o's balancedseparator proced'~e is used, the performance guarantee is O(logn). 1
0/1 Optimization and 0/1 Primal Separation are Equivalent
"... The 0/1 primal separation problem is: Given an extreme point x of a 0/1 polytope P and some point x , nd an inequality which is tight at x, violated by x and valid for P or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separati ..."
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Cited by 2 (0 self)
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The 0/1 primal separation problem is: Given an extreme point x of a 0/1 polytope P and some point x , nd an inequality which is tight at x, violated by x and valid for P or assert that no such inequality exists. It is known that this separation variant can be reduced to the standard separation problem for P . We show that 0/1 optimization and 0/1 primal separation are polynomial time equivalent. This implies that the problems 0/1 optimization, 0/1 standard separation, 0/1 augmentation, and 0/1 primal separation are polynomial time equivalent. We apply this result to the perfect matching problem. Here, primal separation is easier than its standard version. We present an algorithm for primal separation, which rests only on simple maxow computations. Consequently, we obtain a very simple proof that a maximum weight perfect matching of a graph can be computed in polynomial time. In contrast, the known standard separation method involves Padberg and Rao's minimum odd cut algorithm, which itself is based on the construction of a GomoryHu tree. 1
Clustering Methods Based on MinimumCut Trees
, 2002
"... In this paper we introduce a simple clustering method for undirected graphs. The clustering method uses maximum ow techniques on the linkstructure of the graph. The quality of the produced clusters is bounded by strong minimumcut and expansion criteria. We also present a framework for hierarchical ..."
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Cited by 1 (0 self)
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In this paper we introduce a simple clustering method for undirected graphs. The clustering method uses maximum ow techniques on the linkstructure of the graph. The quality of the produced clusters is bounded by strong minimumcut and expansion criteria. We also present a framework for hierarchical clustering and apply it to realworld data. We conclude that the clustering algorithms satisfy strong theoretical criteria and perform well in practice.