Mathematical Programming

, Volume 166, Issue 1–2, pp 241–271 | Cite as

On imposing connectivity constraints in integer programs

Full Length Paper Series A


In many network applications, one searches for a connected subset of vertices that exhibits other desirable properties. To this end, this paper studies the connected subgraph polytope of a graph, which is the convex hull of subsets of vertices that induce a connected subgraph. Much of our work is devoted to the study of two nontrivial classes of valid inequalities. The first are the ab-separator inequalities, which have been successfully used to enforce connectivity in previous computational studies. The second are the indegree inequalities, which have previously been shown to induce all nontrivial facets for trees. We determine the precise conditions under which these inequalities induce facets and when each class fully describes the connected subgraph polytope. Both classes of inequalities can be separated in polynomial time and admit compact extended formulations. However, while the ab-separator inequalities can be lifted in linear time, it is NP-hard to lift the indegree inequalities.


Connected subgraph polytope Maximum-weight connected subgraph Connectivity constraints Prize-collecting Steiner tree Contiguity 

Mathematics Subject Classification

90B10 90B18 90C10 90C27 90C35 90C57 68R10 



We thank very much the anonymous referee who introduced us to indegree inequalities and the book of Korte et al. [17]. This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. Partial support by AFOSR under Grants FA9550-12-1-0103 and FA8651-12-2-0011 and by NSF under Grant CMMI-1538493 is gratefully acknowledged.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.Department of Industrial and Systems EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.School of Industrial Engineering & ManagementOklahoma State UniversityStillwaterUSA

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