eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-07-02
9:1
9:15
10.4230/LIPIcs.ICALP.2021.9
article
Almost-Linear-Time Weighted π_p-Norm Solvers in Slightly Dense Graphs via Sparsification
Adil, Deeksha
1
Bullins, Brian
2
Kyng, Rasmus
3
Sachdeva, Sushant
1
University of Toronto, Canada
Toyota Technological Institute at Chicago, IL, USA
ETH Zurich, Switzerland
We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (πΒ²β + π^p_p) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit π_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 π_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = Ο(1), which is almost-linear for graphs that are slightly dense (m β₯ n^{4/3 + o(1)}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol198-icalp2021/LIPIcs.ICALP.2021.9/LIPIcs.ICALP.2021.9.pdf
Weighted π_p-norm
Sparsification
Spanners
Iterative Refinement