Finding k-cuts within twice the optimal

Saran, Huzur; Vazirani, Vijay V.

doi:10.1109/sfcs.1991.185443

Cited by 63 publications

(49 citation statements)

References 13 publications

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“…According to the way of definitions in [9], we can define the dual of the Min k-Cut problem [16] as follows: Given an undirected graph G = (V, E) and an integer k > 0, finding an edge subset whose removal breaks graph G into exactly k components, such that the number of remaining edges is maximized. Let's call this problem the Max k-Partition problem.…”

Section: Related Work and Relation To Other Problemsmentioning

confidence: 99%

Algorithmic aspects of homophyly of networks

Zhang

2015

Theoretical Computer Science

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We investigate the algorithmic problems of the homophyly phenomenon in networks. Given an undirected graph G = (V, E) and a vertex coloring c : V → {1, 2, · · · , k} of G, we say that a vertex v ∈ V is happy if v shares the same color with all its neighbors, and unhappy, otherwise, and that an edge e ∈ E is happy, if its two endpoints have the same color, and unhappy, otherwise. Supposing c is a partial vertex coloring of G, we define the Maximum Happy Vertices problem (MHV, for short) as to color all the remaining vertices such that the number of happy vertices is maximized, and the Maximum Happy Edges problem (MHE, for short) as to color all the remaining vertices such that the number of happy edges is maximized.Let k be the number of colors allowed in the problems. We show that both MHV and MHE can be solved in polynomial time if k = 2, and that both MHV and MHE are NP-hard if k ≥ 3. We devise a max{1/k, Ω(∆ −3 )}-approximation algorithm for the MHV problem, where ∆ is the maximum degree of vertices in the input graph, and a 1/2-approximation algorithm for the MHE problem. This is the first theoretical progress of these two natural and fundamental new problems.

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Section: Related Work and Relation To Other Problemsmentioning

confidence: 99%

Algorithmic aspects of homophyly of networks

Zhang

2015

Theoretical Computer Science

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“…min-k-cut is N P-hard when k is part of the input [5], but polynomial for every fixed k, with a O(n k 2 ) algorithm [6]. It is also W [1]-hard for the parameter k [1], and there are several approximation algorithms, with ratios smaller than 2 [13]. Even if min-k-cut and sum-max graph partitioning seem related, it is not straightforward to directly re-use exact or approximation algorithms for min-k-cut for our problem.…”

Section: Related Workmentioning

confidence: 99%

On the sum-max graph partitioning problem

Watrigant

Bougeret

Giroudeau

et al. 2014

Theoretical Computer Science

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Abstract. In this paper we consider the classical combinatorial optimization graph partitioning problem, with Sum-Max as objective function. Given a weighted graph G = (V, E) and a integer k, our objective is to find a k-partition (V1, . . . , V k ) of V that minimizes, where w(u, v) denotes the weight of the edge {u, v} ∈ E. We establish the N P-completeness of the problem and its unweighted version, and the W [1]-hardness for the parameter k. Then, we study the problem for small values of k, and show the membership in P when k = 3, but the N P-hardness for all fixed k ≥ 4 if one vertex per cluster is fixed. Lastly, we present a natural greedy algorithm with an approximation ratio better than k 2 , and show that our analysis is tight.

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“…The number of edges leaving a set in the partition is related to conductance, but the problems are very different, still. For two more problems that admit non trivial approximation and require to partition the graph into more than two parts see [12,34]. Generally speaking, it seems that approximating partitioning problems into many parts is a quite complex subject.…”

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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Finding k-cuts within twice the optimal

Cited by 63 publications

References 13 publications

Algorithmic aspects of homophyly of networks

Algorithmic aspects of homophyly of networks

On the sum-max graph partitioning problem

On the Advantage of Overlapping Clusters for Minimizing Conductance

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