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

We present faster high-accuracy algorithms for computing l(p)-norm minimizing flows. On a graph with m edges, our algorithm can compute a (1 + 1=poly(m))-approximate unweighted l(p)-norm minimizing flow with pm(1+1)/p+o(1) operations, for any p >= 2; giving the best bound for all p greater than or similar to 5.24: Combined with the algorithm from the work of Adil et al. (SODA '19), we can now compute such flows for any 2 <= p <= 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 p similar to delta(-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 l(p)-norm regression problems for large p: Our algorithm makes pm(1/3+o(1)) log(2) (1/epsilon) calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted `pnorm 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 l(p)-norm problems introduced by Adil et al., are interreducible for different values of p: No such reduction is known for standard l(p)-norm problems.