Approximating Multicut and the Demand Graph

Chekuri, Chandra; Madan, Vivek

doi:10.1137/1.9781611974782.54

Cited by 10 publications

(16 citation statements)

References 21 publications

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“…Significantly improved approximations are known when the demand graph (determined by viewing the (s i , t i ) pairs as edges) excludes certain induced subgraphs [CM17].…”

Section: Introductionmentioning

confidence: 99%

Metric Violation Distance: Hardness and Approximation

Fan¹,

Raichel²,

Buskirk³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Metric data plays an important role in various settings, for example, in metric-based indexing, clustering, classification, and approximation algorithms in general. Due to measurement error, noise, or an inability to completely gather all the data, a collection of distances may not satisfy the basic metric requirements, most notably the triangle inequality. In this paper we initiate the study of the metric violation distance problem: given a set of pairwise distances, modify the minimum number of distances such that the resulting set forms a metric. Three variants of the problem are considered, based on whether distances are allowed to only decrease, only increase, or the general case which allows both decreases and increases. We show that while the decrease only variant is polynomial time solvable, the increase only and general variants are NP-Complete, and moreover cannot in polynomial time be approximated to any ratio better than the minimum vertex cover problem. We then provide approximation algorithms for the increase only and general variants of the problem, by proving interesting necessary and sufficient conditions on the optimal solution, which are used to approximately reduce to a purely combinatorial problem for which we provide matching asymptotic upper and lower bounds.

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“…Significantly improved approximations are known when the demand graph (determined by viewing the (s i , t i ) pairs as edges) excludes certain induced subgraphs [CM17].…”

Section: Introductionmentioning

confidence: 99%

Metric Violation Distance: Hardness and Approximation

Fan¹,

Raichel²,

Buskirk³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…It is known that the inapproximability factor under UGC for (s, r, t)-Node-Lin-3-Cut is identical to the integrality gap of a natural distancebased LP [4]. We construct a sequence of instances such that the sequence of integrality gaps of the distance-based LP converges to √ 2.…”

Section: Resultsmentioning

confidence: 99%

“…We note that {s, t}-Edge-BiCut and EdgeBiCut are extensions of min {s, t}-cut and global min cut in undirected graphs to directed graphs respectively. While {s, t}-Edge-BiCut does not admit an efficient (2 − )-approximation assuming UGC [4,9], Edge-BiCut admits an efficient (2 − 1/448)-approximation [1], thus exhibiting a dichotomy in the approximability between fixed-terminal and global variants. Intriguingly, determining whether Edge-BiCut is NP-complete is still an open problem.…”

Section: Motivationsmentioning

confidence: 99%

“…Before explaining the motivations, we note that (s, r, t)-Node-Lin-3-Cut is NP-hard and has no (4/3− )-approximation assuming UGC (by an approximationpreserving reduction from node 3-way cut 1 in undirected graphs), and admits a combinatorial 2-approximation 2 . It is a special case of directed multicut 3 , which, for constant k, admits a trivial k-approximation and does not admit a (k − )-approximation assuming UGC [4].…”

Section: Introductionmentioning

confidence: 99%

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A tight -approximation for Linear 3-Cut

Bérczi

Chandrasekaran²,

Király

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the approximability of the linear 3-cut problem in directed graphs, which is the simplest unsolved case of the linear k-cut problem. The input here is a directed graph D = (V, E) with node weights and three specified terminal nodes s, r, t ∈ V , and the goal is to find a minimum weight subset of non-terminal nodes whose removal ensures that s cannot reach r and t, and r cannot reach t. The problem is approximation-equivalent to the problem of blocking rooted in-and out-arborescences, and it also has applications in network coding and security. The approximability of linear 3-cut has been wide open until now: the best known lower bound under the Unique Games Conjecture (UGC) was 4/3, while the best known upper bound was 2 using a trivial algorithm. In this work we completely close this gap: we present a √ 2-approximation algorithm and show that this factor is tight assuming UGC. Our contributions are twofold: (1) we analyze a natural two-step deterministic rounding scheme through the lens of a single-step randomized rounding scheme with non-trivial distributions, and (2) we construct integrality gap instances that meet the upper bound of √ 2. Our gap instances can be viewed as a weighted graph sequence converging to a "graph limit structure".

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“…Regarding exact algorithms, Klein and Marx [42] have shown how to solve planar Multiway cut exactly in time 2 O(t) · n O( √ t) , coming very close to the lower bound of n Ω( √ t) mentioned above. Chekuri and Madan [14] have recently investigated the complexity of approximating Multicut with weaker constraints on the terminal pairs, in both the undirected and the directed setting.…”

mentioning

confidence: 99%

A Near-Linear Approximation Scheme for Multicuts of Embedded Graphs with a Fixed Number of Terminals

Cohen-Addad¹,

Verdière²,

Mesmay³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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For an undirected edge-weighted graph G and a set R of pairs of vertices called pairs of terminals, a multicut is a set of edges such that removing these edges from G disconnects each pair in R. We provide an algorithm computing a (1 + ε)-approximation of the minimum multicut of a graph G in time (g + t)O(g+t) · n log n, where g is the genus of G and t is the number of terminals. This is tight in several aspects, as the minimum multicut problem is both APX-hard and W[1]-hard (parameterized by the number of terminals), even on planar graphs (equivalently, when g = 0).Our result, in the field of fixed-parameter approximation algorithms, mostly relies on concepts borrowed from computational topology of graphs on surfaces. In particular, we use and extend various recent techniques concerning homotopy, homology, and covering spaces (even in the planar case). We also exploit classical ideas stemming from approximation schemes for planar graphs and low-dimensional geometric inputs. A key insight towards our result is a novel characterization of a minimum multicut as the union of some Steiner trees in the universal cover of the surface in which G is embedded.

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Approximating Multicut and the Demand Graph

Cited by 10 publications

References 21 publications

Metric Violation Distance: Hardness and Approximation

Metric Violation Distance: Hardness and Approximation

A tight -approximation for Linear 3-Cut

A Near-Linear Approximation Scheme for Multicuts of Embedded Graphs with a Fixed Number of Terminals

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