2 Search Results for "Choudhari, Jayesh"

Document
DOI: 10.4230/LIPIcs.STACS.2025.25
Local Density and Its Distributed Approximation

Authors: Aleksander Bjørn Christiansen, Ivor van der Hoog, and Eva Rotenberg

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
The densest subgraph problem is a classic problem in combinatorial optimisation. Graphs with low maximum subgraph density are often called "uniformly sparse", leading to algorithms parameterised by this density. However, in reality, the sparsity of a graph is not necessarily uniform. This calls for a formally well-defined, fine-grained notion of density. Danisch, Chan, and Sozio propose a definition for local density that assigns to each vertex v a value ρ^*(v). This local density is a generalisation of the maximum subgraph density of a graph. I.e., if ρ(G) is the subgraph density of a finite graph G, then ρ(G) equals the maximum local density ρ^*(v) over vertices v in G. They present a Frank-Wolfe-based algorithm to approximate the local density of each vertex with no theoretical (asymptotic) guarantees. We provide an extensive study of this local density measure. Just as with (global) maximum subgraph density, we show that there is a dual relation between the local out-degrees and the minimum out-degree orientations of the graph. We introduce the definition of the local out-degree g^*(v) of a vertex v, and show it to be equal to the local density ρ^*(v). We consider the local out-degree to be conceptually simpler, shorter to define, and easier to compute. Using the local out-degree we show a previously unknown fact: that existing algorithms already dynamically approximate the local density for each vertex with polylogarithmic update time. Next, we provide the first distributed algorithms that compute the local density with provable guarantees: given any ε such that ε^{-1} ∈ O(poly n), we show a deterministic distributed algorithm in the LOCAL model where, after O(ε^{-2} log² n) rounds, every vertex v outputs a (1 + ε)-approximation of their local density ρ^*(v). In CONGEST, we show a deterministic distributed algorithm that requires poly(log n,ε^{-1}) ⋅ 2^{O(√{log n})} rounds, which is sublinear in n. As a corollary, we obtain the first deterministic algorithm running in a sublinear number of rounds for (1+ε)-approximate densest subgraph detection in the CONGEST model.
Cite as

Aleksander Bjørn Christiansen, Ivor van der Hoog, and Eva Rotenberg. Local Density and Its Distributed Approximation. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{christiansen_et_al:LIPIcs.STACS.2025.25,
  author =	{Christiansen, Aleksander Bj{\o}rn and van der Hoog, Ivor and Rotenberg, Eva},
  title =	{{Local Density and Its Distributed Approximation}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.25},
  URN =		{urn:nbn:de:0030-drops-228502},
  doi =		{10.4230/LIPIcs.STACS.2025.25},
  annote =	{Keywords: Distributed graph algorithms, graph density computation, graph density approximation, network analysis theory}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.135
Saving Critical Nodes with Firefighters is FPT

Authors: Jayesh Choudhari, Anirban Dasgupta, Neeldhara Misra, and M. S. Ramanujan

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters. We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters. We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between the firefighting problem and the notions of important separators and tight separator sequences. Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts our FPT result for the non-spreading model.
Cite as

Jayesh Choudhari, Anirban Dasgupta, Neeldhara Misra, and M. S. Ramanujan. Saving Critical Nodes with Firefighters is FPT. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 135:1-135:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{choudhari_et_al:LIPIcs.ICALP.2017.135,
  author =	{Choudhari, Jayesh and Dasgupta, Anirban and Misra, Neeldhara and Ramanujan, M. S.},
  title =	{{Saving Critical Nodes with Firefighters is FPT}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{135:1--135:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.135},
  URN =		{urn:nbn:de:0030-drops-74968},
  doi =		{10.4230/LIPIcs.ICALP.2017.135},
  annote =	{Keywords: firefighting, cuts, FPT, kernelization}
}
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