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Approximating Multicut and the Demand Graph
[article]

2016
*
arXiv
*
pre-print

In

arXiv:1607.07200v1
fatcat:x3sbjkqao5f7tpv4zf4qz54n2y
*the*minimum*Multicut*problem,*the*input is an edge-weighted supply*graph*G=(V,E)*and*a simple*demand**graph*H=(V,F). Either G*and*H are directed (DMulC) or both are undirected (UMulC). ... Motivated by both concrete instances from applications*and*abstract considerations, we consider*the*role that*the*structure of*the**demand**graph*H plays in determining*the**approximability*of*Multicut*. ... Since Tri-Cast instances are 2K 2 -free we obtain*the*following corollary. ...##
###
Approximation Algorithms for Steiner and Directed Multicuts

1997
*
Journal of Algorithms
*

We show an O log kt

doi:10.1006/jagm.1996.0833
fatcat:7vd3zhthpbbm3nofk4vservawa
*approximation*algorithm for*the*Steiner*multicut*problem, where k is*the*number of sets*and*t is*the*maximum cardinality of a set. ... In this paper we consider*the*Steiner*multicut*problem. ...*The*traditional*multicut**approximation*algorithms that deal with separating pairs of nodes were derived using a result about*graph*decomposition. ...##
###
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications

1996
*
SIAM journal on computing (Print)
*

We prove

doi:10.1137/s0097539793243016
fatcat:ukv5rvcukvcxxgbka63teliedm
*the*following*approximate*max-flow min-*multicut*theorem: min*multicut*< max flow < min*multicut*, O(log k) where k is*the*number of commodities. ... Much of flow theory,*and**the*theory of cuts in*graphs*, is built around a single theoremmthe celebrated max-flow min-cut theorem of Fort*and*Fulkerson [FF]*and*Elias, Feinstein,*and*Shannon [EFS]. ... We wish to thank Phil Klein for simplifying*the*formulation in 6, which made*the*presentation more uniform. ...##
###
Connections Between Unique Games and Multicut
[article]

2009
*
Electronic colloquium on computational complexity
*

In MC, one is given a network

dblp:journals/eccc/SteurerV09
fatcat:5vawk3fwu5eu7ltro3x4xahwoa
*graph**and*a*demand**graph*on*the*same vertex set*and**the*goal is to remove as few edges from*the*network*graph*as possible such that every two vertices connected by a ... Specifically, we can adapt most known algorithms for U G to work for MC*and*obtain new*approximation*guarantees for MC that depend on*the*maximum degree of*the**demand**graph*. ... 1 Introduction Minimum*Multicut*. An instance of Minimum*Multicut*(MC) is specified by two*graphs*G*and*H on*the*same vertex set. We refer to G as*the*network*graph**and*to H as*the**demand**graph*. ...##
###
Approximate max-flow min-(multi)cut theorems and their applications

1993
*
Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93
*

We prove

doi:10.1145/167088.167266
dblp:conf/stoc/GargVY93
fatcat:gnuidsrasjdhhoabwvhznd5fpe
*the*following*approximate*max-flow min-*multicut*theorem: min*multicut*< max flow < min*multicut*, O(log k) where k is*the*number of commodities. ... Much of flow theory,*and**the*theory of cuts in*graphs*, is built around a single theoremmthe celebrated max-flow min-cut theorem of Fort*and*Fulkerson [FF]*and*Elias, Feinstein,*and*Shannon [EFS]. ... We wish to thank Phil Klein for simplifying*the*formulation in 6, which made*the*presentation more uniform. ...##
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ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT

2006
*
Computational Complexity
*

We show that

doi:10.1007/s00037-006-0210-9
fatcat:jdtafma54zc7tgvufttujc6jvu
*the**Multicut*, Sparsest-Cut,*and*Min-2CNF ≡ Deletion problems are NP-hard to*approximate*within every constant factor, assuming*the*Unique Games Conjecture of Khot (2002) . ... A quantitatively stronger version of*the*conjecture implies an inapproximability factor of Ω( √ log log n). ... This work was done while Ravi Kumar*and*D. Sivakumar were at*the*IBM Almaden Research Center. ...##
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Primal-dual approximation algorithms for integral flow and multicut in trees

1997
*
Algorithmica
*

We present an efficient algorithm that computes a

doi:10.1007/bf02523685
fatcat:vthvisrmcrh5nkcvwnh7kqkhay
*multicut**and*integral flow such that*the*weight of*the**multicut*is at most twice*the*value of*the*flow. ... We study*the*maximum integral multicommodity flow problem*and**the*minimum*multicut*problem restricted to trees. ... We wish to thank Clyde Momna*and*Alex Schaffer for providing references for*the*tree-representable set systems. ...##
###
Minimal multicut and maximal integer multiflow: A survey

2005
*
European Journal of Operational Research
*

We present a survey about

doi:10.1016/j.ejor.2003.10.037
fatcat:pxc4bhuvefe6di7abxmy6l57x4
*the*maximum integral multiflow*and*minimum*multicut*problems*and*their subproblems, such as*the*multiterminal cut*and**the*unsplittable flow problems. ... We recall*the*dual relationship between both problems, give complexity results*and*algorithms, firstly in unrestricted*graphs**and*secondly in several special*graphs*: trees, bipartite or planar*graphs*. ...*The*ratio of*the*values of*the*minimum*multicut**and**the*maximum multiflow in planar*graphs*is at most Oð1Þ,*and*there is a constant factor*approximation*algorithm for IMCP. ...##
###
Correlation Clustering – Minimizing Disagreements on Arbitrary Weighted Graphs
[chapter]

2003
*
Lecture Notes in Computer Science
*

We give an equivalence argument between these problems

doi:10.1007/978-3-540-39658-1_21
fatcat:byoheelyvrhq7lpbzc4aaholpm
*and**the**multicut*problem. This implies an O(log n)*approximation*algorithm for minimizing disagreements on weighted*and*unweighted*graphs*. ...*The*equivalence also implies that these problems are APX-hard*and*suggests that improving*the*upper bound to obtain a constant factor*approximation*is non trivial. ... Note that this result holds for both weighted*and*unweighted*graphs**and*that*the*reduction of*the*unweighted correlation clustering problem results in a*multicut*problem with unity capacities*and**demands*...##
###
Integer Plane Multiflow Maximisation : Flow-Cut Gap and One-Quarter-Approximation
[article]

2020
*
arXiv
*
pre-print

Planarity means here that

arXiv:2002.10927v2
fatcat:h5hbmboc7vetho4suwl4goyxpe
*the*union of*the*supply*and**demand**graph*is planar. We first prove that there exists a multiflow of value at least half of*the*capacity of a minimum*multicut*. ... In this paper, we bound*the*integrality gap*and**the**approximation*ratio for maximum plane multiflow problems*and*deduce bounds on*the*flow-cut-gap. ...*The*Gaps*and*exact*approximation*ratios are open in*the*indicated intervals*and**the*maximum half integer plane multiflow problem is also open. ...##
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Minimum multicuts and Steiner forests for Okamura-Seymour graphs
[article]

2011
*
arXiv
*
pre-print

*The*minimum-cost Steiner forest problem has a 2-

*approximation*algorithm. Hence,

*the*minimum

*multicut*problem has a 2-

*approximation*algorithm for an Okamura-Seymour instance. ... We study

*the*problem of finding minimum

*multicuts*for an undirected planar

*graph*, where all

*the*terminal vertices are on

*the*boundary of

*the*outer face. This is known as an Okamura-Seymour instance. ... We showed its relation to

*the*minimum-cost Steiner forest problem in

*the*dual

*graph*

*and*gave a 2-

*approximation*algorithm. ...

##
###
Approximation algorithms for feasible cut and multicut problems
[chapter]

1995
*
Lecture Notes in Computer Science
*

Polynomial-time

doi:10.1007/3-540-60313-1_158
fatcat:g5c74l3ntzdp5hulervetuiuei
*approximation*algorithms for solving various NP-hard problems on*graphs*involving cuts*and**multicuts*have recently attracted a great deal of research. ...*approximation*algorithm for*the*minimum-ratio feasible cut problem gives a 2 (1 + ln T )-*approximation*algorithm for*the**multicut*problem, where T denotes*the*cardinality of S i S i . ... For an instance of problem (P3), let H denote*the*set of*demand*edges;*the**graph*(V; H) is called*the**demand**graph*. ...##
###
Optimal hierarchical decompositions for congestion minimization in networks

2008
*
Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08
*

Hierarchical

doi:10.1145/1374376.1374415
dblp:conf/stoc/Racke08
fatcat:pmurvw7fozepjnwl7hpuqaqwvy
*graph*decompositions play an important role in*the*design of*approximation**and*online algorithms for*graph*problems. ... improves*the*O(log 1.5 n)*approximation*by Feige*and*Krauthgamer [17] . ... Acknowledgement*The*author would like to thank Anupam Gupta*and*Chandra Chekuri for useful discussions*and*suggestions for improving a preliminary version of this paper. ...##
###
FPT Inapproximability of Directed Cut and Connectivity Problems

2019
*
International Symposium on Parameterized and Exact Computation
*

Formally, we show

doi:10.4230/lipics.ipec.2019.8
dblp:conf/iwpec/ChitnisF19
fatcat:sjsguktegndnjfzhdojrrvumuu
*the*following results: Cutting paths between a set of terminal pairs, i.e., Directed*Multicut*: Pilipczuk*and*Wahlstrom [TOCT '18] showed that Directed*Multicut*is W[1]-hard when parameterized ...*and**the*size p of*the*solution. ... (Soundness) If val(Γ) < γ, then every network N ⊆ G that satisfies all*demands*has cost more than 2(2 − 4γ 1/5 ). (Parameter Dependency)*The*number of*demand*pairs k = |D | is 2 . Chitnis*and*A. E. ...##
###
An information-theoretic meta-theorem on edge-cut bounds

2012
*
2012 IEEE International Symposium on Information Theory Proceedings
*

*and*flows

*approximately*achieve capacity. ... We demonstrate this in

*the*case of k-unicast in undirected networks, k-pair unicast in directed networks with symmetric

*demands*i.e. for every source communicating to a destination at a certain rate,

*the*... Theorems 11

*and*12 together imply that routing flow is

*approximately*capacity-achieving for sum-rate: Corollary 13. 1 2 R

*multicut*≤ F sum−rate ≤ C sum−rate ≤ 2 R

*multicut*. ...

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