Minimum cost subpartitions in graphs

Nagamochi, Hiroshi; Kamidoi, Yoko

doi:10.1016/j.ipl.2006.11.011

Cited by 8 publications

(8 citation statements)

References 11 publications

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“…Our main result in this section is the following inequalities, which are counterparts of a similar result for the minimum k-way cut problem, that has already been proved in [16].…”

Section: A Basic Inequalitymentioning

confidence: 58%

“…As another variant of these problems, one may focus on the approach through (k, b)subpartitions (see [15,16]) that can be considered as a combination of the max and the mean approach and follow the same line of study.…”

Section: Discussionmentioning

confidence: 99%

“…see [15] for details and the background). Particularly, we know that there exists a polynomial time 2(1 − 1/k)-approximation algorithm for the minimum k-way cut problem which is based on computation of the minimum k-subpartition problem [16]. In Section 2, along the same lines, we prove a couple of basic inequalities (Theorem 1) which show that the isoperimetric numbers can be considered as an approximation for the minimum normalized cuts.…”

Section: Instancementioning

confidence: 95%

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On the complexity of isoperimetric problems on trees

Daneshgar

Javadi

2012

Discrete Applied Mathematics

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This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called minimum normalized cuts/isoperimteric numbers defined through taking minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a partition/subpartition. We show that the decision problem for the case of taking k-partitions and the maximum (called the max normalized cut problem NCP M ) as well as the other two decision problems for the mean version (referred to as IPP m and NCP m ) are N P -complete problems. On the other hand, we show that the decision problem for the case of taking k-subpartitions and the maximum (called the max isoperimetric problem IPP M ) can be solved in linear time for any weighted tree and any k ≥ 2. Based on this fact, we provide polynomial time O(k)-approximation algorithms for all different versions of kth isoperimetric numbers considered. Moreover, when the number of partitions/subpartitions, k, is a fixed constant, as an extension of a result of B. Mohar (1989) for the case k = 2 (usually referred to as the Cheeger constant), we prove that max and mean isoperimetric numbers of weighted trees as well as their max normalized cut can be computed in polynomial time. We also prove some hardness results for the case of simple unweighted graphs and trees.

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“…Our main result in this section is the following inequalities, which are counterparts of a similar result for the minimum k-way cut problem, that has already been proved in [16].…”

Section: A Basic Inequalitymentioning

confidence: 58%

Section: Discussionmentioning

confidence: 99%

Section: Instancementioning

confidence: 95%

See 1 more Smart Citation

On the complexity of isoperimetric problems on trees

Daneshgar

Javadi

2012

Discrete Applied Mathematics

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“…Several algorithms that yield a 2-approximation are known for k-CUT; Saran and Vazirani's algorithm based on repeated minimum-cut computations gives (2 − 2/k)approximation [25]; the same bound can be achieved by removing the (k − 1) smallest weight edges in a Gomory-Hu tree of the graph [25]. Nagamochi and Kamidoi showed that using the concept of extreme sets, a (2 − 2/k)-approximation can be found even faster [20]. Naor and Rabani developed an LP relaxation for k-CUT [21] and this yields a 2(1 − 1/n)-approximation [3].…”

Section: Introductionmentioning

confidence: 98%

LP Relaxation and Tree Packing for Minimum $k$-cuts

Chekuri¹,

Quanrud²,

Xu³

2018

Preprint

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Karger used spanning tree packings [14] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [13,15]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [28].In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [21], and analyzed in [3]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of α-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1 − 1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [2] and Ravi and Sinha [24]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [28].

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“…1(a), where the numbers beside edges indicate their weights and each of the nontrivial extreme subsets X 1 , X 2 , X 3 ∈ X (G, w) is depicted by a dotted closed curve; (b) The tree representation for X (G, w) by Watanabe and Nakamura [28] to solve the edge-connectivity augmentation problem. Extreme subsets of graphs are an important tool to design efficient algorithms for solving graph connectivity problems such as the source location problem [16,24], the minimum k-way cut problem [21], and the dynamic minimum cut problem [17] in addition to the connectivity augmentation problem. Alia and Maestrini [1], Borgatti et al [4], and Naor et al [22] showed that extreme sets of a graph can be constructed efficiently from a Gomory-Hu tree of the graph.…”

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confidence: 99%

Minimum Degree Orderings

Nagamochi

2008

Algorithmica

Self Cite

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It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.

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Minimum cost subpartitions in graphs

Cited by 8 publications

References 11 publications

On the complexity of isoperimetric problems on trees

On the complexity of isoperimetric problems on trees

LP Relaxation and Tree Packing for Minimum $k$-cuts

Minimum Degree Orderings

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