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 $pm^{1+frac{1}{p-1}+o(1)}$ operations, for any $p ge 2,$ giving the best bound for all $pgtrsim 5.24.$ Combined with the algorithm from the work of Adil et al. (SODA 19), we can now compute such flows for any $2le ple m^{o(1)}$ in time at most $O(m^{1.24}).$ In comparison, the previous best running time was $Omega(m^{1.33})$ for large constant $p.$ For $psimdelta^{-1}log m,$ our algorithm computes a $(1+delta)$-approximate maximum flow on undirected graphs using $m^{1+o(1)}delta^{-1}$ 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 $pm^{frac{1}{3}+o(1)}log^2(1/varepsilon)$ calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted $ell_{p}$-norm minimizing flows that runs in time $o(m^{1.5})$ for some $p=m^{Omega(1)}.$ 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.
تحميل البحث