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Approximating Multicut and the Demand Graph [article]

Chandra Chekuri, Vivek Madan
2016 arXiv   pre-print
In 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.  ... 
arXiv:1607.07200v1 fatcat:x3sbjkqao5f7tpv4zf4qz54n2y

Approximation Algorithms for Steiner and Directed Multicuts

Philip N Klein, Serge A Plotkin, Satish Rao, Éva Tardos
1997 Journal of Algorithms  
We show an O log kt 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.  ... 
doi:10.1006/jagm.1996.0833 fatcat:7vd3zhthpbbm3nofk4vservawa

Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications

Naveen Garg, Vijay V. Vazirani, Mihalis Yannakakis
1996 SIAM journal on computing (Print)  
We prove 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.  ... 
doi:10.1137/s0097539793243016 fatcat:ukv5rvcukvcxxgbka63teliedm

Connections Between Unique Games and Multicut [article]

David Steurer, Nisheeth K. Vishnoi
2009 Electronic colloquium on computational complexity  
In MC, one is given a network 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 MC and obtain new approximation guarantees for MC that depend on the maximum degree of the demand graph.  ...  1 Introduction Minimum Multicut. An instance of Minimum Multicut (MC) 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.  ... 
dblp:journals/eccc/SteurerV09 fatcat:5vawk3fwu5eu7ltro3x4xahwoa

Approximate max-flow min-(multi)cut theorems and their applications

Naveen Garg, Vijay V. Vazirani, Mihalis Yannakakis
1993 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing - STOC '93  
We prove 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.  ... 
doi:10.1145/167088.167266 dblp:conf/stoc/GargVY93 fatcat:gnuidsrasjdhhoabwvhznd5fpe

ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT

Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, D. Sivakumar
2006 Computational Complexity  
We show that 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.  ... 
doi:10.1007/s00037-006-0210-9 fatcat:jdtafma54zc7tgvufttujc6jvu

Primal-dual approximation algorithms for integral flow and multicut in trees

N. Garg, V. V. Vazirani, M. Yannakakis
1997 Algorithmica  
We present an efficient algorithm that computes a 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.  ... 
doi:10.1007/bf02523685 fatcat:vthvisrmcrh5nkcvwnh7kqkhay

Minimal multicut and maximal integer multiflow: A survey

Marie-Christine Costa, Lucas Létocart, Frédéric Roupin
2005 European Journal of Operational Research  
We present a survey about 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.  ... 
doi:10.1016/j.ejor.2003.10.037 fatcat:pxc4bhuvefe6di7abxmy6l57x4

Correlation Clustering – Minimizing Disagreements on Arbitrary Weighted Graphs [chapter]

Dotan Emanuel, Amos Fiat
2003 Lecture Notes in Computer Science  
We give an equivalence argument between these problems 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  ... 
doi:10.1007/978-3-540-39658-1_21 fatcat:byoheelyvrhq7lpbzc4aaholpm

Integer Plane Multiflow Maximisation : Flow-Cut Gap and One-Quarter-Approximation [article]

Naveen Garg, Nikhil Kumar, András Sebő
2020 arXiv   pre-print
Planarity means here that 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.  ... 
arXiv:2002.10927v2 fatcat:h5hbmboc7vetho4suwl4goyxpe

Minimum multicuts and Steiner forests for Okamura-Seymour graphs [article]

Arindam Pal
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.  ... 
arXiv:1102.5478v1 fatcat:dejshsrhezdqrn3uyqk65n237m

Approximation algorithms for feasible cut and multicut problems [chapter]

Bo Yu, Joseph Cheriyan
1995 Lecture Notes in Computer Science  
Polynomial-time 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.  ... 
doi:10.1007/3-540-60313-1_158 fatcat:g5c74l3ntzdp5hulervetuiuei

Optimal hierarchical decompositions for congestion minimization in networks

Harald Räcke
2008 Proceedings of the fourtieth annual ACM symposium on Theory of computing - STOC 08  
Hierarchical 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.  ... 
doi:10.1145/1374376.1374415 dblp:conf/stoc/Racke08 fatcat:pmurvw7fozepjnwl7hpuqaqwvy

FPT Inapproximability of Directed Cut and Connectivity Problems

Rajesh Chitnis, Andreas Emil Feldmann, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
Formally, we show 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.  ... 
doi:10.4230/lipics.ipec.2019.8 dblp:conf/iwpec/ChitnisF19 fatcat:sjsguktegndnjfzhdojrrvumuu

An information-theoretic meta-theorem on edge-cut bounds

Sudeep Kamath, Pramod Viswanath
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 .  ... 
doi:10.1109/isit.2012.6283557 dblp:conf/isit/KamathV12 fatcat:22vyj5s5qngyvpo6ez3qbulp3q
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