Journal of Combinatorial Optimization

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

Extended cuts

  • Walid Ben-Ameur
  • Mohamed Didi Biha


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 


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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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