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

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 $p=ω(1),$ which is almost-linear for graphs that are slightly dense ($m \ge n^{4/3 + o(1)}$).