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The multicommodity flow problem is a classic problem in network flow and combinatorial optimization, with applications in transportation, communication, logistics, and supply chain management, etc. Existing algorithms often focus on low-accuracy approximate solutions, while high-accuracy algorithms typically rely on general linear program solvers. In this paper, we present efficient high-accuracy algorithms for a broad family of multicommodity flow problems on undirected graphs, demonstrating improved running times compared to general linear program solvers. Our main result shows that we can solve the 𝓁_{q, p}-norm multicommodity flow problem to a (1 + ε) approximation in time O_{q, p}(m^{1+o(1)} k² log(1/ε)), where k is the number of commodities, and O_{q, p}(⋅) hides constants depending only on q or p. As q and p approach to 1 and ∞ respectively, 𝓁_{q, p}-norm flow tends to maximum concurrent flow. We introduce the first iterative refinement framework for 𝓁_{q, p}-norm minimization problems, which reduces the problem to solving a series of decomposable residual problems. In the case of k-commodity flow, each residual problem can be decomposed into k single commodity convex flow problems, each of which can be solved in almost-linear time. As many classical variants of multicommodity flows were shown to be complete for linear programs in the high-accuracy regime [Ding-Kyng-Zhang, ICALP'22], our result provides new directions for studying more efficient high-accuracy multicommodity flow algorithms. 

Given a matrix A ∈ ℝn\texttimes{}d and a vector b ∈ ℝn, we consider the regression problem with ℓ∞ guarantees: finding a vector x′ ∈ ℝd such that $$||x'-x^* ||_infty leq frac{epsilon}{sqrt{d}}cdot ||Ax^*-b||_2cdot ||A^dagger||$$, where x* = arg minx∈Rd ||Ax – b||2. One popular approach for solving such ℓ2 regression problem is via sketching: picking a structured random matrix S ∈ ℝm\texttimes{}n with m < n and S A can be quickly computed, solve the "sketched" regression problem arg minx∈ℝd ||S Ax – Sb||2. In this paper, we show that in order to obtain such ℓ∞ guarantee for ℓ2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m = ε-2d log3(n/δ) such that solving the sketched regression problem gives the ℓ∞ guarantee, with probability at least 1 – δ. Moreover, the matrix S A can be computed in time O(nd log n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in (Price et al., 2017), in which a superlinear in d rows, m = Ω(ε-2d1+γ) for γ ∈ (0, 1) is required. Moreover, we develop a novel analytical framework for ℓ∞ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in (Song \& Yu, 2021). Our analysis is much simpler and more general than that of (Price et al., 2017). Leveraging this framework, we extend the ℓ∞ guarantee regression result to dense sketching matrices for computing the fast tensor product of vectors.