Faster p-norm minimizing flows, via smoothed q-norm problems
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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-1+o(1) operations, for any p > 2, giving the best bound for all p≳ 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 Ω(m^1.33) for large constant p. For p∼δ^-1log m, our algorithm computes a (1+δ)-approximate maximum flow on undirected graphs using m^1+o(1)δ^-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/ε) calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted ℓ_p-norm minimizing flows that runs in time o(m^1.5) for some p=m^Ω(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.
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