On the Complexity of Community-Aware Network Sparsification
In the NP-hard Î -Network Sparsification problem, we are given an edge-weighted graph G, a collection đ of c subsets of V(G), called communities, and two numbers đ and b, and the question is whether there exists a spanning subgraph G' of G with at most đ edges of total weight at most b such that G'[C] fulfills Î for each community C â đ. We study the fine-grained and parameterized complexity of two special cases of this problem: Connectivity NWS where Î is the connectivity property and Stars NWS, where Î is the property of having a spanning star.
First, we provide a tight 2^Î©(nÂČ+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.
parameterized complexity
hypergraph support
above guarantee parameterization
exponential-time-hypothesis
Theory of computation~Parameterized complexity and exact algorithms
Mathematics of computing~Hypergraphs
60:1-60:18
Regular Paper
https://arxiv.org/abs/2402.15494
Some of the results of this work are also contained in the first authorâs Masters thesis [Herrendorf, 2022].
Emanuel
Herrendorf
Emanuel Herrendorf
Philipps-UniversitĂ€t Marburg, Fachbereich Mathematik und Informatik, Germany
Christian
Komusiewicz
Christian Komusiewicz
Friedrich Schiller University Jena, Institute of Computer Science, Germany
https://orcid.org/0000-0003-0829-7032
Nils
Morawietz
Nils Morawietz
Friedrich Schiller University Jena, Institute of Computer Science, Germany
https://orcid.org/0000-0002-7283-4982
Partially supported by the DFG, project OPERAH (KO 3669/5-1).
Frank
Sommer
Frank Sommer
Friedrich Schiller University Jena, Institute of Computer Science, Germany
https://orcid.org/0000-0003-4034-525X
Partially supported by the DFG, project EAGR (KO 3669/6-1)
10.4230/LIPIcs.MFCS.2024.60
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Emanuel Herrendorf, Christian Komusiewicz, Nils Morawietz, and Frank Sommer
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