An improved approximation algorithm for multiway cut

Călinescu, Gruia; Karloff, Howard; Rabani, Yuval

doi:10.1145/276698.276711

Cited by 119 publications

(199 citation statements)

References 22 publications

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“…Let x uv be any such vertex. Since x uv is happy, the number of neighbors of x uv that have the color as that of x uv is at least ρ(h + k + 2) > ρ h + h − ρh − 2ρ + 2 ρ + 2 = h + 2, where the first inequality is due to inequality (2). This shows that the number of neighbors of x uv having color c ′ * (x uv ) is at least h + 3.…”

Section: −ρmentioning

confidence: 91%

“…The Multiway Cut problem [5,3,2,10] should be the traditional optimization problem that is most related to MHV and MHE. Given an undirected graph G = (V, E) with costs defined on edges and a terminal set S ⊆ V , the Multiway Cut problem asks for a set of edges (called a multiway cut, or simply a cut) with the minimum total cost such that its removal from graph G separates all terminals in S from one another.…”

Section: Related Work and Relation To Other Problemsmentioning

confidence: 99%

“…Note that deg G ′ (x uv ) = h + k + 2. So, inequality (1) guarantees that if all vertices in {u, v, y uv 1 , · · · , y uv 2 } have the same color as that of x uv , x uv will be happy, and, inequality (2) guarantees that if there is one vertex in {u, v, y uv 1 , · · · , y uv 2 } having different color to that of x uv , x uv will be unhappy.…”

Section: B2 Np-hardness Of Softmhvmentioning

confidence: 99%

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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: −ρmentioning

confidence: 91%

Section: Related Work and Relation To Other Problemsmentioning

confidence: 99%

Section: B2 Np-hardness Of Softmhvmentioning

confidence: 99%

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Algorithmic aspects of homophyly of networks

Zhang

2015

Theoretical Computer Science

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“…Our algorithm is inspired by the algorithms of Calinescu, Karloff and Rabani for multiway-cut [8] and 0-extension [9]; the underlying idea has seen several applications subsequently. Let x be a feasible solution to the LP.…”

Section: -Route Multicutmentioning

confidence: 99%

Algorithms for 2-Route Cut Problems

Chekuri

Khanna

2008

Automata, Languages and Programming

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In this paper we study approximation algorithms for multi-route cut problems in undirected graphs. In these problems the goal is to find a minimum cost set of edges to be removed from a given graph such that the edge-connectivity (or node-connectivity) between certain pairs of nodes is reduced below a given threshold K. In the usual cut problems the edge connectivity is required to be reduced below 1 (i.e. disconnected). We consider the case of K = 2 and obtain poly-logarithmic approximation algorithms for fundamental cut problems including single-source, multiway-cut, multicut, and sparsest cut. These cut problems are dual to multi-route flows that are of interest in fault-tolerant networks flows. Our results show that the flow-cut gap between 2-route cuts and 2-route flows is poly-logarithmic in undirected graphs with arbitrary capacities. 2-route cuts are also closely related to well-studied feedback problems and we obtain results on some new variants. Multi-route cuts pose interesting algorithmic challenges. The new techniques developed here are of independent technical interest, and may have applications to other cut and partitioning problems.

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“…Dahlhaus et al [6] initiated the study of the Graph Multiway Cut problem; they showed that the problem is MAX SNP-hard even for k = 3 and they gave a combinatorial algorithm that achieves a (2 − 2/k) approximation. In a breakthrough result, Calinescu, Karloff, and Rabani [3] obtained an (1.5 − 1/k) approximation via a novel geometric relaxation. Since the work of [3], the approximation factor has been improved in a series of papers [7,2,10], culminating with the 1.30217 approximation of [10].…”

Section: Introductionmentioning

confidence: 99%

From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

Ene

Nguyêݱn

2014

Algorithms - ESA 2014

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We consider the Hypergraph Multiway Partition problem (Hyper-MP). The input consists of an edge-weighted hypergraph G = (V, E) and k vertices s 1 , . . . , s k called terminals. A multiway partition of the hypergraph is a partition (or labeling) of the vertices of G into k setsis the hypergraph cut function. The Hyper-MP problem asks for a multiway partition of minimum cost.Our main result is a 4/3 approximation for the Hyper-MP problem on 3-uniform hypergraphs, which is the first improvement over the (1.5−1/k) approximation of [5]. The algorithm combines the single-threshold rounding strategy of Calinescu et al. [3] with the rounding strategy of Kleinberg and Tardos [8], and it parallels the recent algorithm of Buchbinder et al. [2] for the Graph Multiway Cut problem, which is a special case.On the negative side, we show that the KT rounding scheme [8] and the exponential clocks rounding scheme [2] cannot break the (1.5 − 1/k) barrier for arbitrary hypergraphs. We give a family of instances for which both rounding schemes have an approximation ratio bounded from below by Ω( √ k), and thus the Graph Multiway Cut rounding schemes may not be sufficient for the Hyper-MP problem when the maximum hyperedge size is large. We remark that these instances have k = Θ(log n).

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An improved approximation algorithm for multiway cut

Cited by 119 publications

References 22 publications

Algorithmic aspects of homophyly of networks

Algorithmic aspects of homophyly of networks

Algorithms for 2-Route Cut Problems

From Graph to Hypergraph Multiway Partition: Is the Single Threshold the Only Route?

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