Complexity of Finding Maximum Locally Irregular Induced Subgraphs
| Autor: | Fioravantes, Foivos, Melissinos, Nikolaos, Triommatis, Theofilos |
|---|---|
| Přispěvatelé: | Combinatorics, Optimization and Algorithms for Telecommunications (COATI), COMmunications, Réseaux, systèmes Embarqués et Distribués (Laboratoire I3S - COMRED), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL), Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision (LAMSADE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), School of Electrical Engineering, Electronics and Computer Science University of Liverpool, Liverpool, L69-3BX, UK, Inria, I3S, Université Côte d'Azur, Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-COMmunications, Réseaux, systèmes Embarqués et Distribués (Laboratoire I3S - COMRED), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), University of Liverpool |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
Mathematics of computing → Approximation algorithms
approximability Digital Object Identifier 10.4230/LIPIcs.SWAT.2022.23 FPT Mathematics of computing → Graph algorithms treewidth approximability phrases Locally irregular largest induced subgraph FPT treewidth W-hardness approximability Digital Object Identifier 10.4230/LIPIcs.SWAT.2022.23 Theory of computation → Parameterized complexity and exact algorithms Locally irregular phrases Locally irregular [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] W-hardness largest induced subgraph |
| Zdroj: | [Research Report] Inria; I3S; Université Côte d'Azur. 2021 Scandinavian Symposium and Workshops on Algorithm Theory (SWAT) 2022 Scandinavian Symposium and Workshops on Algorithm Theory (SWAT) 2022, Jun 2022, Torshavn, Faroe Islands. ⟨10.4230/LIPIcs.SWAT.2022.23⟩ |
| DOI: | 10.4230/lipics.swat.2022.24 |
| Popis: | If a graph G is such that no two adjacent vertices of G have the same degree, we say that G is locally irregular. In this work we introduce and study the problem of identifying a largest induced subgraph of a given graph G that is locally irregular. Equivalently, given a graph G, find a subset S of V(G) with minimum order, such that by deleting the vertices of S from G results in a locally irregular graph; we denote with I(G) the order of such a set S. We first examine some easy graph families, namely paths, cycles, trees, complete bipartite and complete graphs. However, we show that the decision version of the introduced problem is NP-Complete, even for restricted families of graphs, such as subcubic planar bipartite, or cubic bipartite graphs. We then show that we can not even approximate an optimal solution within a ratio of 𝒪(n^{1-1/k}), where k ≥ 1 and n is the order the graph, unless 𝒫=NP, even when the input graph is bipartite. Then, looking for more positive results, we turn our attention towards computing I(G) through the lens of parameterised complexity. In particular, we provide two algorithms that compute I(G), each one considering different parameters. The first one considers the size of the solution k and the maximum degree Δ of G with running time (2Δ)^kn^{𝒪(1)}, while the second one considers the treewidth tw and Δ of G, and has running time Δ^{2tw}n^{𝒪(1)}. Therefore, we show that the problem is FPT by both k and tw if the graph has bounded maximum degree Δ. Since these algorithms are not FPT for graphs with unbounded maximum degree (unless we consider Δ + k or Δ + tw as the parameter), it is natural to wonder if there exists an algorithm that does not include additional parameters (other than k or tw) in its dependency. We answer negatively, to this question, by showing that our algorithms are essentially optimal. In particular, we prove that there is no algorithm that computes I(G) with dependence f(k)n^{o(k)} or f(tw)n^{o(tw)}, unless the ETH fails. LIPIcs, Vol. 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022), pages 24:1-24:20 |
| Databáze: | OpenAIRE |
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