eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
60:1
60:18
10.4230/LIPIcs.MFCS.2024.60
article
On the Complexity of Community-Aware Network Sparsification
Herrendorf, Emanuel
1
Komusiewicz, Christian
2
https://orcid.org/0000-0003-0829-7032
Morawietz, Nils
2
https://orcid.org/0000-0002-7283-4982
Sommer, Frank
2
https://orcid.org/0000-0003-4034-525X
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
Friedrich Schiller University Jena, Institute of Computer Science, Germany
In the NP-hard Π-Network Sparsification problem, we are given an edge-weighted graph G, a collection 𝒞 of c subsets of V(G), called communities, and two numbers 𝓁 and b, and the question is whether there exists a spanning subgraph G' of G with at most 𝓁 edges of total weight at most b such that G'[C] fulfills Π for each community C ∈ 𝒞. We study the fine-grained and parameterized complexity of two special cases of this problem: Connectivity NWS where Π is the connectivity property and Stars NWS, where Π is the property of having a spanning star.
First, we provide a tight 2^Ω(n²+c)-time running time lower bound based on the ETH for both problems, where n is the number of vertices in G even if all communities have size at most 4, G is a clique, and every edge has unit weight. For the connectivity property, the unit weight case with G being a clique is the well-studied problem of computing a hypergraph support with a minimum number of edges. We then study the complexity of both problems parameterized by the feedback edge number t of the solution graph G'. For Stars NWS, we present an XP-algorithm for t answering an open question by Korach and Stern [Discret. Appl. Math. '08] who asked for the existence of polynomial-time algorithms for t = 0. In contrast, we show for Connectivity NWS that known polynomial-time algorithms for t = 0 [Korach and Stern, Math. Program. '03; Klemz et al., SWAT '14] cannot be extended to larger values of t by showing NP-hardness for t = 1.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.60/LIPIcs.MFCS.2024.60.pdf
parameterized complexity
hypergraph support
above guarantee parameterization
exponential-time-hypothesis