Journal of Combinatorial Optimization

, Volume 31, Issue 3, pp 1034–1044 | Cite as

Extended cuts



Given a directed graph \(G=(V,A)\), three positive integers \(r\), \(k\) and \(k'\), and a partition \(\{V_0, V_1,\ldots ,V_r\}\) of \(V\) such that there are no arcs from \(V_i\) to \(V_j\) when \(j-i>k\) or \(j-i<-k^{\prime }\), the set of direct arcs from \(V_i\) to \(V_j\), \(i <j\), is called an \((r,k,k')\)-extended cut. Given a weight vector \((c_a)_{a \in A}\), we aim to compute a minimum-weight \((r,k,k')\)-extended cut. Some special vertices may be required to belong to a particular subset \(V_i\). We study this problem, and for special cases we provide polynomial-time algorithms, linear formulations and polyhedral descriptions. We also review some NP-hard variants.


Graph partitioning Cut Polyhedra Compact formulation 

Mathematics Subject Classification

90C27 90C57 


  1. Alpert CJ, Kahng AB (1995) Directions in netlist partitioning: a survey. Integration 19:1–81MATHGoogle Scholar
  2. Ben-Ameur W, Didi Biha M (2012) On the minimum cut separator problem. Networks 59:30–36MathSciNetCrossRefMATHGoogle Scholar
  3. Ben-Ameur W, Didi Biha M (2012) Extended Cuts, Université de Caen, Technical report, LMNO 05.Google Scholar
  4. Bui TN, Jones C (1992) Finding good approximate vertex and edge partitions is NP-hard. Inf Process Lett 42:153–159MathSciNetCrossRefMATHGoogle Scholar
  5. Călinescu G, Karloff H, Rabani Y, An improved approximation algorithm for MULTIWAY CUT, in Proceedings of symposium on theory of computing, ACM, 48–52, Dalas, Texas, 1998.Google Scholar
  6. Cheung KH, Cunningham WH, Tang L (2006) Optimal 3-terminal cuts and linear programming. Math. Program. 106:1–23MathSciNetCrossRefMATHGoogle Scholar
  7. Costa MC, Létocart L, Roupin F (2005) Minimal multicut and maximal integer multiflow: a survey. Eur J Op Res 162:55–69MathSciNetCrossRefMATHGoogle Scholar
  8. Dahlhaus E, Johnson D, Papadimitriou C, Seymour P, Yannakakis M (1994) The complexity of multiterminal cuts. SIAM J. Comput. 23:864–894MathSciNetCrossRefMATHGoogle Scholar
  9. Edmonds J, Giles R (1977) A min-max relation for submodular functions on graphs. Stud Integer Program, Ann Discret Math 1:185–204MathSciNetCrossRefMATHGoogle Scholar
  10. Garey MR, Johnson DS (1979) Computers and intractabiliy: a guide to the theory of NP-completeness. W.H. Freeman and Company, New YorkMATHGoogle Scholar
  11. Garg N, Vazirani V, Yannakakis M (2004) Multiway cuts in node weighted graphs. J Algorithms 50:49–61MathSciNetCrossRefMATHGoogle Scholar
  12. Klein P, Plotkin S, Rao S, Tardos E (1997) Approximation algorithms for steiner and directed multicuts. J Algorithms 22:241–269MathSciNetCrossRefMATHGoogle Scholar
  13. Shmoys D (1997) Cut problems and their applications to divide-and-conquer. In: Hochba DS (ed) Approximation algorithms for NP-hard problem. PWS Publishing Company, Pacific Grove, pp 192–235Google Scholar
  14. Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell. 22:888–905CrossRefGoogle Scholar
  15. Stone HS (1977) Multiprocessor scheduling with the aid of network flow algorithms. IEEE Trans Softw Eng 3:85–93MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Laboratoire SamovarTélécom SudParisEvryFrance
  2. 2.LMNOUniversité de CaenCaen CedexFrance

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