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Overlaying a hypergraph with a graph with bounded maximum degree

Abstract : Let G be a graph and H a ​​hypergraph with the same set of vertices and let F be a fixed graph. The graph GF overlays a hyperedge S of H if F is a covering subgraph of the subgraph of G induced by S. The graph GF-overlaying H if F overlays each hyperedge of H. We have analyze the complexity of the following two problems. The first problem, ($∆ ≤ k$) F-OVERLAY, consists in deciding if there is a graph of maximum degree at most k which F overlays a given hypergraph H. This is a special case of the second problem, MAX ($∆ ≤ k$) F-OVERLAY, which, given a hypergraph H and an integer s, consists in deciding whether there is a graph of maximum degree at most k which F overlays at least s hyper edges of H. We prove a polynomial /NP-complete dichotomy for the MAX ($∆ ≤ k$) -F-OVERLAY, and prove the complexity of the problem ($∆ ≤ k$) F-OVERLAY for a large number of pairs (F, k). These two problems model a central problem in computational structural biology: the determination of the contacts between the proteins of a macromolecular assembly. The vertices are the proteins, the hyperedges are the known complexes, the graph F is the generic graph whose edges correspond to the contacts between the proteins of the assembly. Determining the graph G then comes down to finding the contacts between the proteins so that the graph F is a subgraph covering in each hyperedge and so that the degree is bounded (a protein is in contact with a limited number of others proteins). Finally, these problems are of more general interest for network inference problems.
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Submitted on : Friday, June 5, 2020 - 2:00:20 PM
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Frédéric Havet, Dorian Mazauric, Viet-Ha Nguyen, Rémi Watrigant. Overlaying a hypergraph with a graph with bounded maximum degree. ALGOTEL 2020 – 22èmes Rencontres Francophones sur les Aspects Algorithmiques des Télécommunications, Sep 2020, Lyon, France. ⟨hal-02796730⟩

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