Generalized Flow in Nearly-linear Time on Moderately Dense Graphs

TL;DR

The authors address the problem of solving generalized flow problems on graphs with edge losses by reducing to two-sparse linear programs and solving them with an interior-point method augmented by spectral tools for lossy graphs. A key technical advance is dynamic data structures for heavy hitters, samplers, and inverse maintenance tailored to two-sparse matrices, built atop spectral approximations of lossy Laplacians and balanced lossy expanders. The main result is a randomized nearly linear-time algorithm, tilde-O((m+n^{1.5}) polylog(W/δ)), for solving two-sparse LPs to δ-accuracy, which yields corollaries for generalized maximum flow and generalized minimum-cost flow with the same complexity; in lossy max flow, exact feasibility is achievable via a known reduction. The work leverages expander decompositions, spectral sparsification, and a carefully maintained approximate eigenvector of the lossy Laplacian to enable efficient updates and IPM iterations, offering a path to broader spectral methods in optimization. The findings significantly narrow the gap between generalized and standard flow problems on moderately dense graphs and provide tools with potential applicability to a wider class of LPs and network-optimization tasks.

Abstract

In this paper we consider generalized flow problems where there is an $m$-edge $n$-node directed graph $G = (V,E)$ and each edge $e \in E$ has a loss factor $γ_e >0$ governing whether the flow is increased or decreased as it crosses edge $e$. We provide a randomized $\tilde{O}( (m + n^{1.5}) \cdot \mathrm{polylog}(\frac{W}δ))$ time algorithm for solving the generalized maximum flow and generalized minimum cost flow problems in this setting where $δ$ is the target accuracy and $W$ is the maximum of all costs, capacities, and loss factors and their inverses. This improves upon the previous state-of-the-art $\tilde{O}(m \sqrt{n} \cdot \log^2(\frac{W}δ) )$ time algorithm, obtained by combining the algorithm of [Daitch-Spielman, 2008] with techniques from [Lee-Sidford, 2014]. To obtain this result we provide new dynamic data structures and spectral results regarding the matrices associated to generalized flows and apply them through the interior point method framework of [Brand-Lee-Liu-Saranurak-Sidford-Song-Wang, 2021].

Figures (3)

• Figure 1: An illustration of the chain of reductions for the heavy hitter data structures.
• Figure 2: Illustration of $v$, $v^{\mathop{\mathrm{\mathrm{ext}}}\limits}$, and $v^{\mathop{\mathrm{\mathrm{old}}}\limits}$.
• Figure 3: An example of a two-sparse matrix $\mathbf{A} \in \mathop{\mathrm{\mathbb{R}}}\limits^{m \times n}$ with $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \beta_1, \beta_2, \beta_3 \geq 0$, and the matrix $\widehat{\mathbf{A}} \in \mathop{\mathrm{\mathbb{R}}}\limits^{m \times 2n}$ corresponding to it.

Theorems & Definitions (75)

• Definition 1.1: Two-sparse matrix and its magnitude parameter
• Theorem 1.2: Two-sparse LP
• Theorem 1.3: Generalized maxflow
• Theorem 1.4: Generalized min-cost flow
• Theorem 2.1: Heavy hitter for two-sparse matrices, informal version of \ref{['thm:heavy_hitter_two_sparse_general']}
• Lemma 2.2: Informal version of \ref{['lem:eta_to_not', 'lem:N_approx_Ivvt']}
• Theorem 2.3: Spectral approximation from low Rayleigh quotient, informal version of \ref{['thm:spectral_sparsification']}
• Theorem 2.4: Uniformity of $v_{\min}$, informal version of \ref{['thm:unifomity_v']}
• Theorem 4.1: Spectral approximation from low Rayleigh quotient
• Corollary 4.2