[Paper Review] A Deterministic Almost-Linear Time Algorithm for Minimum-Cost Flow

Jan van den Brand, Li Chen+6 more|arXiv (Cornell University)|Sep 28, 2023

Complexity and Algorithms in GraphsComputer Science3 citations

TL;DR

This paper presents the first deterministic almost-linear time algorithm for minimum-cost flow and maximum flow in directed graphs with polynomially bounded capacities, costs, and demands. It achieves an $ m^{1+o(1)} $ runtime by introducing a deterministic dynamic graph data structure that maintains low-stretch spanning trees and sparse spanners, enabling efficient computation of approximate minimum-ratio cycles without randomness.

ABSTRACT

We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM '98]. Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS '22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

Motivation & Objective

• To close the long-standing gap between randomized and deterministic algorithms for maximum flow in graphs with polynomially bounded capacities.
• To provide a deterministic implementation of the recent randomized $m^{1+o(1)}$-time framework for minimum-cost flow by Chen et al.
• To eliminate randomness in key components—vertex sparsification and edge sparsification—previously relying on random sampling and tree embeddings.
• To develop a deterministic dynamic data structure that maintains low-stretch spanning trees with subpolynomial amortized update time.
• To construct a fully dynamic spanner with subpolynomial stretch and subpolynomial update time for use in flow computation.

Proposed method

• Design a deterministic dynamic graph data structure that maintains a sparse spanner of the underlying graph with subpolynomial stretch and subpolynomial amortized update time per edge change.
• Replace the randomized vertex sparsification in prior work with a deterministic recursive method based on tree decompositions and vertex sampling without randomization.
• Introduce a novel edge sparsification technique that maintains an embedding of the dynamic graph into a sparse spanner using deterministic tree chain constructions.
• Use shifted tree chains to maintain low-stretch forests and support efficient cycle quality analysis during flow computation.
• Apply multiplicative weights to balance stretch and update costs in the dynamic low-stretch forest data structure.
• Integrate the dynamic spanner and low-stretch forest into a cycle-approximation framework to compute minimum-ratio cycles deterministically.

Experimental results

Research questions

• RQ1 Can the randomized $m^{1+o(1)}$-time maximum flow algorithm by Chen et al. be derandomized while preserving near-linear time complexity?
• RQ2 What deterministic alternatives exist for randomized vertex and edge sparsification in dynamic graph data structures?
• RQ3 Is it possible to maintain a fully dynamic low-stretch spanning tree with subpolynomial amortized update time deterministically?
• RQ4 Can a deterministic spanner with subpolynomial stretch and update cost be constructed for use in flow algorithms?
• RQ5 What is the impact of derandomizing the cycle-approximation framework on the overall runtime of minimum-cost flow computation?

Key findings

• The paper achieves a deterministic $ m^{1+o(1)} $-time algorithm for computing exact maximum flows and minimum-cost flows in directed graphs with polynomially bounded capacities, costs, and demands.
• The algorithm improves upon the long-standing $ ilde{O}(m imes ext{min}igrace{m^{1/2}, n^{2/3}}igrace{}) $ runtime of Goldberg-Rao (1998) for deterministic maximum flow in the general case.
• A deterministic dynamic graph data structure maintains a sparse spanner with subpolynomial stretch and subpolynomial amortized update time per edge update.
• The authors construct a deterministic dynamic low-stretch forest with subpolynomial average stretch and subpolynomial amortized update cost.
• The framework enables deterministic computation of $ m^{1+o(1)} $-approximate undirected minimum-ratio cycles, which are used to solve minimum-cost flow exactly.
• The results establish that deterministic algorithms can achieve almost-linear time for minimum-cost flow, closing a major gap between randomized and deterministic complexity in this domain.