## The Complexity of Multiterminal Cuts (1994)

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Venue: | SIAM Journal on Computing |

Citations: | 144 - 0 self |

### BibTeX

@ARTICLE{Dahlhaus94thecomplexity,

author = {E. Dahlhaus and D. S. Johnson and C. H. Papadimitriou and P. D. Seymour and M. Yannakakis},

title = {The Complexity of Multiterminal Cuts},

journal = {SIAM Journal on Computing},

year = {1994},

volume = {23},

pages = {864--894}

}

### Years of Citing Articles

### OpenURL

### Abstract

In the Multiterminal Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2 - 2/k of the optimal cut weight.

### Citations

11781 |
Computers and Intractability: A Guide to the Theory of NP-Completeness
- Garey, Johnson
- 1979
(Show Context)
Citation Context ... as "The complexity of multiway cuts" in Proc. 24th Annual ACM Symposium on Theory of Computing (1992), pp. 241-251. - 2 - constraint imposed on the size of the components into which the gra=-=ph is cut [9,10]-=-. In this paper we ask whether the problem might be tractable without such a constraint (as it is for k = 2). Our first results concern the planar case. The restriction to planar graphs, besides its b... |

1394 |
Combinatorial Optimization: Algorithms and Complexity
- Papadimitriou, Steiglitz
- 1982
(Show Context)
Citation Context ... "min-cut/max-flow" problem, a problem of central significance in the field of combinatorial optimization due to its many applications and the fact that it can be solved in polynomial time (=-=e.g., see [7,17,18,20]). The &qu-=-ot;k-terminal cut" problem for k > 2 has been a subject of discussion in the combinatorics community for years (closely-related variants were proposed as early as 1969 by T. C. Hu [17,p.150]). A ... |

731 | Proof verification and the hardness of approximation problems
- Arora, Lund, et al.
- 1998
(Show Context)
Citation Context ...o find cuts that are within 2(k - 1)/k of optimal. Theorem 5. For any fixed ks3, the k-Terminal Cut problem is MAX SNP-hard (and hence cannot have a polynomial time approximation scheme unless P = NP =-=[1,21]-=-). The results presented here can be contrasted to those of [13,16,22], which concern what might be called the k-Cut problem. In this problem we are given G, k, and w as above (but not S), and are ask... |

587 |
Optimization, approximation and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ...o find cuts that are within 2(k - 1)/k of optimal. Theorem 5. For any fixed ks3, the k-Terminal Cut problem is MAX SNP-hard (and hence cannot have a polynomial time approximation scheme unless P = NP =-=[1,21]-=-). The results presented here can be contrasted to those of [13,16,22], which concern what might be called the k-Cut problem. In this problem we are given G, k, and w as above (but not S), and are ask... |

569 | Tarjan, A new approach to the maximum-flow problem
- Goldberg, E
- 1998
(Show Context)
Citation Context ...her idea, we might be forced to use a general algorithm, and the running time would grow to O(n 2 logn). (This can be obtained for instance by using the O(nmlog (n 2 / m) ) 2terminal cut algorithm of =-=[12]-=- and taking advantage of the fact that although our graphs need not remain planar, they do remain sparse.) This would force our overall running time up to O(n 4 logn). Fortunately, we can get around t... |

503 |
Combinatorial Optimization: Networks and Matroids
- Lawler
- 1976
(Show Context)
Citation Context ... "min-cut/max-flow" problem, a problem of central significance in the field of combinatorial optimization due to its many applications and the fact that it can be solved in polynomial time (=-=e.g., see [7,17,18,20]). The &qu-=-ot;k-terminal cut" problem for k > 2 has been a subject of discussion in the combinatorics community for years (closely-related variants were proposed as early as 1969 by T. C. Hu [17,p.150]). A ... |

401 | Some simplified NP – complete graph problems - GAREY, JOHNSON, et al. |

199 |
Planar formulae and their uses
- Lichtenstein
- 1982
(Show Context)
Citation Context ...NAL CUT is in NP is immediate. To complete the proof, we need to provide a polynomial transformation to it from some known NP-complete problem. The source problem we choose is PLANAR 3-SATISFIABILITY =-=[9,19]-=- (PLANAR 3-SAT for short). In the 3-SATISFIABILITY problem we are given a set X = {x 1 , x 2 , . . . , x n } of variables and a collection C = {c 1 , c 2 , . . . , c m } of 3-element clauses, i.e., su... |

159 |
Multiprocessor scheduling with the aid of network algorithms
- Stone
- 1977
(Show Context)
Citation Context ...were proposed as early as 1969 by T. C. Hu [17,p.150]). A variety of applications have been suggested, most having to do with the minimization of communication costs in parallel computing systems. In =-=[23]-=-, Stone points out how the problem of assigning program modules to processors can be formulated in this framework. Other applications involve partitioning files among the nodes of a network, assigning... |

150 | Approximate max-flow min-(multi)cut theorems and their applications
- GARG, VAZARANI, et al.
- 1996
(Show Context)
Citation Context ...is polynomial for fixed k. The question remains open as to whether there is a polynomialtime approximation algorithm that works for arbitrary k and provides a constant guarantee, although Garg et al. =-=[11]-=- have devised a polynomial-time algorithm that works for arbitrary k and has worst-case ratio O(logk). More recently, Erd .. os and Szekely in [5,6] proposed the following generalization of Multitermi... |

125 |
Fast algorithms for shortest paths in planar graphs, with applications
- Frederickson
- 1987
(Show Context)
Citation Context ...2 n ) invocations of the Type I procedure in Step 2.2. The all-pairs shortest path computations takes place in a planar graph, and so can be implemented to run in time O(n 2 ) using the techniques of =-=[8]-=-. The isolating cut computations reduce as noted to 2-terminal minimum cut computations, and so can be performed using standard 2-terminal cut algorithms. For planar graphs, such algorithms run in tim... |

95 |
Integer Programming and Network Flows
- Hu
- 1970
(Show Context)
Citation Context ... "min-cut/max-flow" problem, a problem of central significance in the field of combinatorial optimization due to its many applications and the fact that it can be solved in polynomial time (=-=e.g., see [7,17,18,20]). The &qu-=-ot;k-terminal cut" problem for k > 2 has been a subject of discussion in the combinatorics community for years (closely-related variants were proposed as early as 1969 by T. C. Hu [17,p.150]). A ... |

45 |
Polynomial algorithm for the kcut problem
- Goldschmidt, Hochbaum
- 1988
(Show Context)
Citation Context ...any fixed ks3, the k-Terminal Cut problem is MAX SNP-hard (and hence cannot have a polynomial time approximation scheme unless P = NP [1,21]). The results presented here can be contrasted to those of =-=[13,16,22]-=-, which concern what might be called the k-Cut problem. In this problem we are given G, k, and w as above (but not S), and are asked merely for a minimum weight set of edges Eswhose removal separates ... |

44 | Finding k-cuts within twice the optimal - Saran, Vazirani - 1991 |

19 |
The optimal multiterminal cut problem
- Cunningham
- 1991
(Show Context)
Citation Context ...ard dynamic programming algorithm. (This can be generalized to graphs of bounded tree-width for any fixed bound, by standard techniques.) The facets of the Multiterminal Cut polyhedron are studied in =-=[2,3]-=-. An interesting generalization of the Multiterminal Cut problem, about which we shall have more to say in our concluding section, is studied in [5,6]. The 1983 abstract did not contain our proofs; th... |

19 |
The Complexity of multiway cuts", extended abstract
- Dahlhaus, Johnson, et al.
- 1983
(Show Context)
Citation Context ...bliographic confusion, we should mention that, with the exception of Theorem 5, the results in the current paper were first announced in 1983 in an unpublished but widely circulated extended abstract =-=[4]-=-. The abstract has since been widely cited, both in the abovementioned work on k-Cut, and in follow-up work on the Multiterminal Cut problem itself: In [2], Chopra and Rao observe, as we failed to do ... |

11 |
Tarjan, "A New Approach to the Maximum Flow Problem
- Goldberg, E
- 1988
(Show Context)
Citation Context ...urther idea, we might be forced to use a general algorithm, and the running time would grow to O(n 2 logn). (This can be obtained for instance by using the O(nmlog(n 2 /m)) 2terminal cut algorithm of =-=[12]-=- and taking advantage of the fact that although our graphs need not remain planar, they do remain sparse.) This would force our overall running time up to O(n 4 logn). Fortunately, we can get around t... |

9 |
On the multiway cut polyhedron, Networks 21
- Chopra, Rao
- 1991
(Show Context)
Citation Context ...shed but widely circulated extended abstract [4]. The abstract has since been widely cited, both in the abovementioned work on k-Cut, and in follow-up work on the Multiterminal Cut problem itself: In =-=[2]-=-, Chopra and Rao observe, as we failed to do in our original abstract, that for trees and 2-trees, the general k-Terminal Cut problem can be solved in linear time by a straightforward dynamic programm... |

8 |
Cutting and partitioning a graph after a fixed pattern
- Yannakakis, Kanellakis, et al.
(Show Context)
Citation Context ...rating each pair of vertices u i , v i , 1sisk. This is just 2terminal Cut when k = 1. The problem is also polynomial time solvable when k = 2, by using two applications of a 2-terminal cut algorithm =-=[24]-=-. Our result for 3-Terminal Cut implies that it is NP-hard for arbitrary graphs when k = 3, even if all edge weights are equal: merely let the three pairs be (s 1 , s 2 ), (s 2 , s 3 ), and (s 3 , s 1... |

5 |
An O(jV j ) Algorithm for the Planar 3-Cut Problem
- Hochbaum, Shmoys
- 1985
(Show Context)
Citation Context ...any fixed ks3, the k-Terminal Cut problem is MAX SNP-hard (and hence cannot have a polynomial time approximation scheme unless P = NP [1,21]). The results presented here can be contrasted to those of =-=[13,16,22]-=-, which concern what might be called the k-Cut problem. In this problem we are given G, k, and w as above (but not S), and are asked merely for a minimum weight set of edges Eswhose removal separates ... |

4 |
An O(|V | 2 ) algorithm for the planar 3-cut problem
- Hochbaum, Shmoys
- 1985
(Show Context)
Citation Context ...y fixed k ≥ 3, the k-Terminal Cut problem is MAX SNP-hard (and hence cannot have a polynomial time approximation scheme unless P = NP [1,21]). The results presented here can be contrasted to those of =-=[13,16,22]-=-, which concern what might be called the k-Cut problem. In this problem we are given G, k, and w as above (but not S), and are asked merely for a minimum weight set of edges E′ whose removal separates... |

3 |
The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1
- TSCHEL, SCHRIJVER
- 1981
(Show Context)
Citation Context ... a brief digression into what at first seemed like a promising algorithmic approach to the 3-Terminal Cut problem, based on results on submodular set functions by Gr .. otschel, Lovasz, and Schrijver =-=[14]. The key -=-"gadget" in the NP-completeness proof we shall be presenting also serves as a counterexample to the applicability of this approach. (Indeed, before we discovered the gadget, we already had a... |

2 |
The complexity of multiway cuts," extended abstract
- DAHLHAUS, JOHNSON, et al.
- 1983
(Show Context)
Citation Context ...bliographic confusion, we should mention that, with the exception of Theorem 5, the results in the current paper were first announced in 1983 in an unpublished but widely circulated extended abstract =-=[4]-=-. The abstract has since been widely cited, both in the abovementioned work on k-Cut, and in follow-up work on the Multiterminal Cut problem itself: In [2], Chopra and Rao observe, as we failed to do ... |

2 |
On the planar 3-cut problem
- HE
- 1991
(Show Context)
Citation Context ... (For k = 3 and unweighted planars- graphs, the O(n 7 ) of the general k-Cut result has been beaten more directly, first by an O(n 2 ) algorithm in [16], and subsequently by an O(nlog n) algorithm in =-=[15]-=-.) Reference [22] concerns approximation results for the k-Cut problem, showing that the bounds we obtain in Theorem 4 for Multiterminal Cut can be obtained for k-Cut directly, without having to apply... |

1 |
On the multiway cut polyhedron," Networks 21
- CHOPRA, RAO
- 1991
(Show Context)
Citation Context ...shed but widely circulated extended abstract [4]. The abstract has since been widely cited, both in the abovementioned work on k-Cut, and in follow-up work on the Multiterminal Cut problem itself: In =-=[2]-=-, Chopra and Rao observe, as we failed to do in our original abstract, that for trees and 2-trees, the general k-Terminal Cut problem can be solved in linear time by a straightforward dynamic programm... |

1 |
Evolutionary trees: An integer multicommodity maxflow min-cut theorem
- S, KELY
(Show Context)
Citation Context ...f the Multiterminal Cut polyhedron are studied in [2,3]. An interesting generalization of the Multiterminal Cut problem, about which we shall have more to say in our concluding section, is studied in =-=[5,6]-=-. The 1983 abstract did not contain our proofs; these are presented here for the first time. (The 1983 abstract also used the lessdescriptive term multiway cut for what we now call a multiterminal cut... |

1 |
KELY, "Algorithms and min-max theorems for certain multiway cuts," Iin "Integer Programming and Combinatorial Optimization
- S, SZE
- 1992
(Show Context)
Citation Context ...f the Multiterminal Cut polyhedron are studied in [2,3]. An interesting generalization of the Multiterminal Cut problem, about which we shall have more to say in our concluding section, is studied in =-=[5,6]-=-. The 1983 abstract did not contain our proofs; these are presented here for the first time. (The 1983 abstract also used the lessdescriptive term multiway cut for what we now call a multiterminal cut... |

1 |
SZÉKELY, ‘‘Algorithms and min-max theorems for certain multiway cuts,’’ Iin ‘‘Integer Programming and Combinatorial Optimization
- S, A
- 1992
(Show Context)
Citation Context ...f the Multiterminal Cut polyhedron are studied in [2,3]. An interesting generalization of the Multiterminal Cut problem, about which we shall have more to say in our concluding section, is studied in =-=[5,6]-=-. The 1983 abstract did not contain our proofs; these are presented here for the first time. (The 1983 abstract also used the lessdescriptive term multiway cut for what we now call a multiterminal cut... |