Multiplicative Weights for Mixed Block Constraints and Maximum Multicommodity Flow

Abstract

We study multiplicative-weights (MWU) algorithms for block-separable feasibility problems containing both packing and covering constraints. Each iteration optimizes a weighted linear objective over one block, either chosen uniformly at random or visited in a fixed cyclic order. We show that the cyclic algorithm returns an $O(\epsilon )$-approximate solution after $O(k\rho {\epsilon }^{-2}\log m)$ oracle calls where $k$ is the number of blocks and $\rho$ is the width of the oracle. The same holds for the randomized version with high probability. Previously, such bounds were shown when the coupling constraints have the same sign [Radzik'95; Villavicencio-Grigoriadis'96]. These analyses critically depended on monotonicity arguments that are not available in the mixed case. As an application, we obtain a deterministic $(1-\epsilon )$-approximation for maximum multicommodity flow (M-MCF) using $O(k{\epsilon }^{-2}\log m\log (km/\epsilon ))$ approximate single-commodity min-cost-flow calls. For planar graphs, and more generally for graph families admitting a near-linear-time min-cost-flow algorithm, this gives total running time $\tilde{O}(km{\epsilon }^{-2})$. Compared with the $O(k{m}^{1+o(1)}{\epsilon }^{-1})$ algorithm of [Chen-Graur-Sidford'25], this improves the dependence on $m$ for these graph families at the cost of a worse dependence on $\epsilon$. Finally, we give an exponentiation-free finite-precision implementation of the cyclic algorithm. It represents each internal weight by a logarithmic-bit mantissa and exponent, while every price supplied to the block oracle is rational and has a polynomially bounded common denominator.

arXiv comment15 pages