Greedy approximation algorithms for directed multicuts

Kortsarts, Yana; Kortsarz, Guy; Nutov, Zeev

doi:10.1002/net.20066

Cited by 14 publications

(11 citation statements)

References 11 publications

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“…Consider a simple case when p has LP length l common with a single path pH of H, then the probability that the random level cut affects p ∩ H is at least l. Moreover when the cut occurs it is expected to separate Ω(n 1−2ǫ l 2 ) pairs of nodes in pH because a path of LP length l has at least n 1/2−ǫ l nodes. 4 This technique has also been used previously by [11].…”

Section: Bounding A(e)mentioning

confidence: 99%

Improved approximation for directed cut problems

Agarwal

Alon

Charikar

2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

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We present improved approximation algorithms for directed multicut and directed sparsest cut. The current best known approximation ratio for these problems is O(n 1/2 ). We obtain anÕ(n 11/23 )-approximation. Our algorithm works with the natural LP relaxation used in prior work. We use a randomized rounding algorithm with a more sophisticated charging scheme and analysis to obtain our improvement. This also implies aÕ(n 11/23 ) upper bound on the ratio between the maximum multicommodity flow and minimum multicut in directed graphs.

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Section: Bounding A(e)mentioning

confidence: 99%

Improved approximation for directed cut problems

Agarwal

Alon

Charikar

2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

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“…Cheriyan, Karloff and Rabani [12] gave an O( √ n log n)-approximation algorithm for directed multicut, and Gupta [26] subsequently improved it to an O( √ n)-approximation. Kortsarts, Kortsarz and Nutov [30] showed anÕ(n 2/3 /OPT 1/3 )-approximation, where OPT is the cost of the optimal solution, and more recently, Agarwal, Alon, and Charikar [1] have further improved the approximation ratio toÕ(n 11/23 ). On the hardness front, recently, Chuzhoy and Khanna [13] established an Ω(log n/ log log n)-hardness for directed multicut, assuming that NP is not contained in DTIME n polylog(n) .…”

Section: Directed Multicutmentioning

confidence: 99%

Polynomial flow-cut gaps and hardness of directed cut problems

Chuzhoy

Khanna

2007

Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing

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We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an n-vertex graph G along with k source-sink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all source-sink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of source-sink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LP-duality, to the well-studied maximum (fractional) multicommodity flow problem, while the standard LP-relaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also the flow-cut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem.Our first result is that the flow-cut gap between maximum multicommodity flow and minimum multicut isΩ(n 1/7 ) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a long-standing lower bound of Ω(log n) for both types of flow-cut gaps. We notice that these polynomially large flow-cut gaps are in a sharp contrast to the undirected setting where both these flow-cut gaps are known to be Θ(log n). Our second result is that both directed multicut and sparsest cut are hard to approximate to within a factor of 2 Ω(log 1− n) for any constant > 0, unless NP ⊆ ZPP. This improves upon the recent Ω(log n/ log log n)-hardness result for these problems. We also show that existence of PCP's for NP with perfect completeness, polynomially small soundness, and constant number of queries would imply a polynomial factor hardness of approximation for both these problems. All our results hold for directed acyclic graphs.

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“…[1] to anÕ(n 11/23 ) ratio approximation 2 . In [26], an O(n 2/3 /opt 1/3 ) approximation algorithm for uniform costs directed multicut, is presented. Here opt is the optimum value.…”

Section: Related Workmentioning

confidence: 99%

“…If the optimum is at least n 0.566 , the ratio of [26] is better than the one of [1]. The [26] algorithm is also the only non-trivial combinatorial approximation algorithm for Directed multicut. For more papers on cut problems see [27,6,15,13,35,10].…”

Section: Related Workmentioning

confidence: 99%

On the Advantage of Overlapping Clusters for Minimizing Conductance

2013

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Graph clustering is an important problem with applications to bioinformatics, community discovery in social networks, distributed computing, and more. While most of the research in this area has focused on clustering using disjoint clusters, many real datasets have inherently overlapping clusters. We compare overlapping and non-overlapping clusterings in graphs in the context of minimizing their conductance. It is known that allowing clusters to overlap gives better results in practice. We prove that overlapping clustering may be significantly better than non-overlapping clustering with respect to conductance, even in a theoretical setting.For minimizing the maximum conductance over the clusters, we give examples demonstrating that allowing overlaps can yield significantly better clusterings, namely, one that has much smaller optimum. In addition for the min-max variant, the overlapping version admits a simple approximation algorithm, while our algorithm for the non-overlapping version is complex and yields a worse approximation ratio due to the presence of the additional constraint. Somewhat surprisingly, for the problem of minimizing the sum of conductances, we found out that allowing overlap does not help. We show how to apply a general technique to transform any overlapping clustering into a non-overlapping one with only a modest increase in the sum of conductances. This uncrossing technique is of independent interest and may find further applications in the future.We consider this work as a step toward rigorous comparison of overlapping and nonoverlapping clusterings and hope that it stimulates further research in this area. * IBM T.J.Watson research center.

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Greedy approximation algorithms for directed multicuts

Cited by 14 publications

References 11 publications

Improved approximation for directed cut problems

Improved approximation for directed cut problems

Polynomial flow-cut gaps and hardness of directed cut problems

On the Advantage of Overlapping Clusters for Minimizing Conductance

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