Almost-linear-time Weighted $\ell_p$-norm Solvers in Slightly Dense Graphs via Sparsification

TL;DR

This paper introduces nearly linear-time algorithms for constructing graph sparsifiers that preserve weighted _p norms for flow and voltage objectives, enabling efficient approximation in slightly dense graphs.

Contribution

It presents the first sparsifier constructions for mixed _2 + _p objectives and for voltage objectives, using expander decompositions, spanners, and leverage score sampling.

Findings

Achieves (1+2^{- ext{poly}( ext{log} n)}) ext{ approximations} for _p norm flows and voltages.

Runs in near-linear time for slightly dense graphs with m  n^{4/3+o(1)}.

Extends previous frameworks with new sparsifier constructions for complex objectives.

Abstract

We give almost-linear-time algorithms for constructing sparsifiers with $n\ poly(\log n)$ edges that approximately preserve weighted $(\ell^{2}_2 + \ell^{p}_p)$ flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit $\ell_p$ 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 $(1+2^{-\text{poly}(\log n)})$ approximations for weighted $\ell_p$-norm minimizing flows or voltages in $p(m^{1+o(1)} + n^{4/3 + o(1)})$ time for $p=\omega(1),$ which is almost-linear for graphs that are slightly dense ($m \ge n^{4/3 + o(1)}$).