Minimum k-cut

Combinatorial optimization graph problem

In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.

Formal definition

Given an undirected graph G = (V, E) with an assignment of weights to the edges w: EN and an integer k { 2 , 3 , , | V | } , {\displaystyle k\in \{2,3,\ldots ,|V|\},} partition V into k disjoint sets F = { C 1 , C 2 , , C k } {\displaystyle F=\{C_{1},C_{2},\ldots ,C_{k}\}} while minimizing

i = 1 k 1   j = i + 1 k v 1 C i v 2 C j w ( { v 1 , v 2 } ) {\displaystyle \sum _{i=1}^{k-1}\ \sum _{j=i+1}^{k}\sum _{\begin{smallmatrix}v_{1}\in C_{i}\\v_{2}\in C_{j}\end{smallmatrix}}w(\left\{v_{1},v_{2}\right\})}

For a fixed k, the problem is polynomial time solvable in O ( | V | k 2 ) . {\displaystyle O{\bigl (}|V|^{k^{2}}{\bigr )}.} [1] However, the problem is NP-complete if k is part of the input.[2] It is also NP-complete if we specify k vertices and ask for the minimum k-cut which separates these vertices among each of the sets.[3]

Approximations

Several approximation algorithms exist with an approximation of 2 2 k . {\displaystyle 2-{\tfrac {2}{k}}.} A simple greedy algorithm that achieves this approximation factor computes a minimum cut in each of the connected components and removes the lightest one. This algorithm requires a total of n − 1 max flow computations. Another algorithm achieving the same guarantee uses the Gomory–Hu tree representation of minimum cuts. Constructing the Gomory–Hu tree requires n − 1 max flow computations, but the algorithm requires an overall O(kn) max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm.[4][5] Moreover, under the small set expansion hypothesis (a conjecture closely related to the unique games conjecture), the problem is NP-hard to approximate to within (2 − ε) factor for every constant ε > 0,[6] meaning that the aforementioned approximation algorithms are essentially tight for large k.

A variant of the problem asks for a minimum weight k-cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed k if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality.[7] More recently, polynomial time approximation schemes (PTAS) were discovered for those problems.[8]

While the minimum k-cut problem is W[1]-hard parameterized by k,[9] a parameterized approximation scheme can be obtained for this parameter.[10]

See also

Notes

  1. ^ Goldschmidt & Hochbaum 1988.
  2. ^ Garey & Johnson 1979
  3. ^ [1], which cites [2]
  4. ^ Saran & Vazirani 1991.
  5. ^ Vazirani 2003, pp. 40–44.
  6. ^ Manurangsi 2017
  7. ^ Guttmann-Beck & Hassin 1999, pp. 198–207.
  8. ^ Fernandez de la Vega, Karpinski & Kenyon 2004
  9. ^ G. Downey, Rodney; Estivill-Castro, Vladimir; Fellows, Michael; Prieto, Elena; Rosamund, Frances A. (2003-04-01). "Cutting Up Is Hard To Do: The Parameterised Complexity of k-Cut and Related Problems". Electronic Notes in Theoretical Computer Science. CATS'03, Computing: the Australasian Theory Symposium. 78: 209–222. doi:10.1016/S1571-0661(04)81014-4. hdl:10230/36518. ISSN 1571-0661.
  10. ^ Lokshtanov, Daniel; Saurabh, Saket; Surianarayanan, Vaishali (2022-04-25). "A Parameterized Approximation Scheme for Min $k$-Cut". SIAM Journal on Computing: FOCS20–205. arXiv:2005.00134. doi:10.1137/20M1383197. ISSN 0097-5397.

References

  • Goldschmidt, O.; Hochbaum, D. S. (1988), Proc. 29th Ann. IEEE Symp. on Foundations of Comput. Sci., IEEE Computer Society, pp. 444–451
  • Garey, M. R.; Johnson, D. S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, ISBN 978-0-7167-1044-8
  • Saran, H.; Vazirani, V. (1991), "Finding k-cuts within twice the optimal", Proc. 32nd Ann. IEEE Symp. on Foundations of Comput. Sci, IEEE Computer Society, pp. 743–751
  • Vazirani, Vijay V. (2003), Approximation Algorithms, Berlin: Springer, ISBN 978-3-540-65367-7
  • Guttmann-Beck, N.; Hassin, R. (1999), "Approximation algorithms for minimum k-cut" (PDF), Algorithmica, pp. 198–207
  • Comellas, Francesc; Sapena, Emili (2006), "A multiagent algorithm for graph partitioning. Lecture Notes in Comput. Sci.", Algorithmica, 3907 (2): 279–285, CiteSeerX 10.1.1.55.5697, doi:10.1007/s004530010013, ISSN 0302-9743, S2CID 25721784, archived from the original on 2009-12-12
  • Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (2000), "Minimum k-cut", A Compendium of NP Optimization Problems
  • Fernandez de la Vega, W.; Karpinski, M.; Kenyon, C. (2004). "Approximation schemes for Metric Bisection and partitioning". Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete Algorithms. pp. 506–515.
  • Manurangsi, P. (2017). "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis". 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017. pp. 79:1–79:14. doi:10.4230/LIPIcs.ICALP.2017.79.