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Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification

Authors Deeksha Adil, Brian Bullins, Rasmus Kyng, Sushant Sachdeva

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Author Details

Deeksha Adil
  • University of Toronto, Canada
Brian Bullins
  • Toyota Technological Institute at Chicago, IL, USA
Rasmus Kyng
  • ETH Zurich, Switzerland
Sushant Sachdeva
  • University of Toronto, Canada

Funding

  • Adil, Deeksha: Supported by a Post Graduate Doctoral Scholarship awarded by NSERC (Natural Sciences and Engineering Research Council of Canada) and SS’s Discovery Grant.
  • Sachdeva, Sushant: Supported by a Discovery Grant awarded by NSERC (Natural Sciences and Engineering Research Council of Canada).

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Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva. Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.9

Abstract

We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (𝓁²₂ + 𝓁^p_p) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit 𝓁_p weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find (1+2^{-poly(log n)}) approximations for weighted 𝓁_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = ω(1), which is almost-linear for graphs that are slightly dense (m ≥ n^{4/3 + o(1)}).

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Network flows
Keywords
  • Weighted 𝓁_p-norm
  • Sparsification
  • Spanners
  • Iterative Refinement

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