Faster p-norm minimizing flows, via smoothed q-norm problems

Abstract: We present faster high-accuracy algorithms for computing $\ell_p$-norm minimizing flows. On a graph with $m$ edges, our algorithm can compute a $(1+1/\text{poly}(m))$-approximate unweighted $\ell_p$-norm minimizing flow with $pm<sup>{1+\frac{1}{p-1}+o(1)}$ operations, for any $p \ge 2,$ giving the best bound for all $p\gtrsim 5.24.$ Combined with the algorithm from the work of Adil et al. (SODA '19), we can now compute such flows for any $2\le p\le m<sup>{o(1)}$ in time at most $O(m<sup>{1.24}).$ In comparison, the previous best running time was $\Omega(m<sup>{1.33})$ for large constant $p.$ For $p\sim\delta^{-1}\log m,$ our algorithm computes a $(1+\delta)$-approximate maximum flow on undirected graphs using $m<sup>{1+o(1)}\delta<sup>{-1}$ operations, matching the current best bound, albeit only for unit-capacity graphs. We also give an algorithm for solving general $\ell_{p}$-norm regression problems for large $p.$ Our algorithm makes $pm<sup>{\frac{1}{3}+o(1)}\log<sup>2(1/\varepsilon)$ calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted $\ell_{p}$-norm minimizing flows that runs in time $o(m<sup>{1.5})$ for some $p=m<sup>{\Omega(1)}.$ Our key technical contribution is to show that smoothed $\ell_p$-norm problems introduced by Adil et al., are interreducible for different values of $p.$ No such reduction is known for standard $\ell_p$-norm problems.