Structured Connectivity Augmentation

Fedor Fomin 1 Petr Golovach 1 Dimitrios M. Thilikos 2, 3
3 ALGCO - Algorithmes, Graphes et Combinatoire
LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier
Abstract : We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping \phi:V(H)->V(G) if every edge uv of F is either an edge of G, or \phi^{-1}(u)\phi^{-1}(v) is an edge of H. Thus F contains both G and H as subgraphs, and the edge set of F is the union of the edge sets of G and \phi(H). We consider the following optimization problem. Given graphs G, H, and a weight function \omega assigning non-negative weights to pairs of vertices of V(G), the task is to find \phi of minimum weight \omega(\phi)=\sum_{xy\in E(H)}\omega(\phi(x)\phi(y)) such that the edge connectivity of the superposition F of G and H with respect to \phi is higher than the edge connectivity of G. Our main result is the following ``dichotomy'' complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when H\in C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when H\in C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-01890542
Contributor : Dimitrios M. Thilikos <>
Submitted on : Monday, October 8, 2018 - 5:03:48 PM
Last modification on : Wednesday, June 19, 2019 - 3:41:23 PM
Long-term archiving on : Wednesday, January 9, 2019 - 3:19:19 PM

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Fedor Fomin, Petr Golovach, Dimitrios M. Thilikos. Structured Connectivity Augmentation. MFCS: Mathematical Foundations of Computer Science, Aug 2017, Aalborg, Denmark. pp.29:1--29:13, ⟨10.4230/LIPIcs.MFCS.2017.29⟩. ⟨lirmm-01890542⟩

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