# The Complexity of Multiterminal Cuts

@article{Dahlhaus1994TheCO,
title={The Complexity of Multiterminal Cuts},
author={Elias Dahlhaus and David S. Johnson and Christos H. Papadimitriou and Paul D. Seymour and Mihalis Yannakakis},
journal={SIAM J. Comput.},
year={1994},
volume={23},
pages={864-894}
}
In the multiterminal cut problem one is given an edge-weighted graph and a subset of the vertices called terminals, and is 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. It is shown 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… Expand
656 Citations
Parameterized Algorithm for the Multiterminal Cut Problem Yixin Cao
We study the multiterminal cut problem, which, given an n-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in aExpand
Solving (k-1)-Stable Instances of k-terminal cut with Isolating Cuts
This paper concludes that the $(2-2/k)-approximation algorithm returns the optimal solution on$(k-1)$-stable instances of the k-Terminal Cut problem, and is the first result showing that this$(1-1-\epsilon)- approximation is an exact optimization algorithm on a special class of graphs. Expand
Simple and Improved Parameterized Algorithms for Multiterminal Cuts
• Mingyu Xiao
• Mathematics, Computer Science
• Theory of Computing Systems
• 2009
This paper designs several simple and improved algorithms for Multiterminal Cut, based on a notion farthest minimum isolating cut, and shows that Edge MultiterMinal Cut can be solved in O(2lkT(n,m)) time and Vertex Multiter Minal CutCan be solvedIn O(klT( n,m) time, where T(n),m)=O(min (n2/3,m1/2)m). Expand
An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem
• Mathematics, Computer Science
• FCT
• 2013
We study the multiterminal cut problem, which, given an n-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in aExpand
The Maximum Integer Multiterminal Flow Problem
• C. Bentz
• Mathematics, Computer Science
• ICCSA
• 2006
For directed graphs, a new parameter kL ≤ k is introduced and it is proved that MaxIMTF is $\mathcal{NP}$-hard when k = kL = 2 and when k l = 1 and k = 3, and polynomial-time solvable when k L = 0 and when l = 2. Expand
An O(1.84k) parameterized algorithm for the multiterminal cut problem
• Mathematics, Computer Science
• Inf. Process. Lett.
• 2014
We study the multiterminal cut problem, which, given an n-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in aExpand
Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals
Given an undirected, edge-weighted graph G together with pairs of vertices, called pairs of terminals, the minimum multicut problem asks for a minimum-weight set of edges such that, after deletingExpand
Multicuts in Planar and Bounded-Genus Graphs with Bounded Number of Terminals
A polynomial-time algorithm is provided for the minimum multicut problem, which asks for a minimum-weight set of edges such that, after deleting these edges, the two terminals of each pair belong to different connected components of the graph. Expand
Algorithms for Multiterminal Cuts
Based on a notion farthest minimum isolating cut, some properties for Multiterminal Cuts are presented, which help shed light on the structure of optimal cut problems, and enables us to design efficient algorithms for Multitecuts, as well as some other related cut problems. Expand
An improved approximation algorithm for multiway cut
• Mathematics, Computer Science
• STOC '98
• 1998
A new linear programming relaxation for Multiway Cut is presented and a new approximation algorithm based on it achieves a performance ratio of at most 1.5?1k, which improves the previous result for every value of k. Expand

#### References

SHOWING 1-10 OF 38 REFERENCES
The complexity of multiway cuts (extended abstract)
• Mathematics, Computer Science
• STOC '92
• 1992
It is shown that the Multiway Cut problem becomes NP-hard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed <italic>k</italic>, and a simple approximation algorithm is described that is guaranteed to come within a factor of 2–2/k of the optimal cut weight. Expand
Some Simplified NP-Complete Graph Problems
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 1976
This paper shows that a number of NP - complete problems remain NP -complete even when their domains are substantially restricted, and determines essentially the lowest possible upper bounds on node degree for which the problems remainNP -complete. Expand
Polynomial algorithm for the k-cut problem
• Mathematics, Computer Science
• [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
• 1988
A polynomial algorithm for the case of a fixed k, to find a partition of an edge weighted graph into k nonempty components, such that the total edge weight between components is minimum. Expand
Evolutionary trees: An integer multicommodity max-flow-min-cut theorem
• Mathematics
• 1992
In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes theExpand
On the multiway cut polyhedron
• Mathematics, Computer Science
• Networks
• 1991
An integer programming formulation of the problem of finding a minimum-weight multiway cut that separates each pair of nodes in N and study the associated polyhedron is given. Expand
The ellipsoid method and its consequences in combinatorial optimization
• Mathematics, Computer Science
• Comb.
• 1981
The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function. Expand
Planar Formulae and Their Uses
Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography. Expand
Finding k-cuts within twice the optimal
• Mathematics, Computer Science
• [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science
• 1991
Two simple approximation algorithms are presented for the minimum k-cut problem, requiring a total of only n-1 maximum flow computations for finding a set of near-optimal k-cuts. Expand
A new approach to the maximum-flow problem
• Mathematics, Computer Science
• JACM
• 1988
An alternative method based on the preflow concept of Karzanov, which runs as fast as any other known method on dense graphs, achieving an O(n) time bound on an n-vertex graph and faster on graphs of moderate density. Expand
Approximate Max-Flow Min-(Multi)Cut Theorems and Their Applications
• Computer Science, Mathematics
• SIAM J. Comput.
• 1996
The proof technique provides a unified framework in which one can also analyse the case of flows with specified demands of Leighton and Rao and Klein et al. and thereby obtain an improved bound for the latter problem. Expand