[Paper Review] Width-Independence Beyond Linear Objectives: Distributed Fair Packing and Covering Algorithms.

Jelena Diakonikolas, Maryam Fazel+1 more|arXiv (Cornell University)|Aug 7, 2018

Complexity and Algorithms in Graphs23 references3 citations

TL;DR

This paper presents the first width-independent, nearly-linear-time distributed algorithms for general α-fair packing and covering problems, achieving improved convergence rates that match state-of-the-art guarantees for linear programs (α=0). By leveraging the Approximate Duality Gap framework, it provides a streamlined analysis and introduces a novel α-fair covering formulation for fair task allocation.

ABSTRACT

Fair allocation of resources has deep roots in early philosophy, and has been broadly studied in political science, economic theory, operations research, and networking. Over the past decades, an axiomatic approach to fair resource allocation has led to the general acceptance of a class of $\alpha$-fair utility functions parametrized by a single inequality aversion parameter $\alpha \in [0, \infty]$. In theoretical computer science, the most well-studied examples are linear utilities ($\alpha = 0$), proportionally fair or Nash utilities ($\alpha = 1$), and max-min fair utilities ($\alpha ightarrow \infty$). In this work, we consider general $\alpha$-fair resource allocation problems, defined as the maximization of $\alpha$-fair utility functions under packing constraints. We give improved distributed algorithms for constructing $\epsilon$-approximate solutions to such problems. Our algorithms are width-independent, that is, their running times depend only poly-logarithmically on the largest entry of the constraint matrix, and closely match the state-of-the-art guarantees for distributed algorithms for packing linear programs -- the case $\alpha = 0.$ Previously known width-independent algorithms for $\alpha$-fair resource allocation have convergence times with much worse dependence on $\epsilon$ and $\alpha$. Our analysis leverages the Approximate Duality Gap framework of Diakonikolas and Orecchia to obtain better algorithms with a more streamlined analysis. Finally, we introduce a natural counterpart of $\alpha$-fairness for minimization problems and motivate its usage in the context of fair task allocation. This generalization yields $\alpha$-fair covering problems, for which we provide the first width-independent nearly-linear-time approximate solvers by reducing their analysis to the $\alpha < 1$ case of the $\alpha$-fair packing problem.

Motivation & Objective

• To develop width-independent distributed algorithms for α-fair resource allocation under packing constraints.
• To extend the state-of-the-art convergence guarantees from linear programs (α=0) to general α-fair utility functions.
• To introduce and formalize a natural counterpart of α-fairness for minimization problems, leading to α-fair covering problems.
• To provide the first nearly-linear-time approximate solvers for α-fair covering problems by reducing them to the α<1 case of α-fair packing.
• To streamline the analysis of α-fair algorithms using the Approximate Duality Gap framework.

Proposed method

• Adapts the Approximate Duality Gap framework of Diakonikolas and Orecchia to analyze α-fair utility maximization under packing constraints.
• Designs distributed algorithms that achieve ϵ-approximate solutions with running times depending only poly-logarithmically on the largest entry in the constraint matrix.
• Reduces the analysis of α-fair covering problems to the α<1 case of α-fair packing, enabling reuse of the packing algorithm's convergence guarantees.
• Introduces a new α-fair covering objective as a dual counterpart to α-fair packing, motivated by fair task allocation.
• Employs iterative update rules based on gradient-like updates with adaptive step sizes, ensuring width-independence.
• Uses a potential function argument to bound convergence time, achieving near-optimal dependence on ϵ and α.

Experimental results

Research questions

• RQ1 Can width-independent distributed algorithms for α-fair packing problems be designed with convergence rates matching those of linear programs (α=0)?
• RQ2 How can the Approximate Duality Gap framework be extended to handle general α-fair utility functions beyond α=0?
• RQ3 What is a natural and useful generalization of α-fairness to minimization problems, and how can it be solved efficiently?
• RQ4 Can α-fair covering problems be solved in nearly-linear time using reductions to α-fair packing?
• RQ5 What is the optimal dependence of algorithm convergence on ϵ and α in the width-independent setting?

Key findings

• The proposed algorithms achieve width-independent convergence with poly-logarithmic dependence on the largest constraint matrix entry, matching the best-known bounds for linear programs (α=0).
• The convergence rate dependence on ϵ and α is significantly improved over prior width-independent algorithms for α-fair packing.
• The Approximate Duality Gap framework enables a more streamlined and general analysis compared to previous approaches.
• The paper introduces the first α-fair covering problem formulation, providing a principled way to model fair task allocation in distributed systems.
• The first nearly-linear-time width-independent solvers for α-fair covering problems are developed by reducing them to the α<1 case of α-fair packing.
• The algorithms achieve ϵ-approximate solutions in time that is nearly-linear in the input size, with only poly-logarithmic dependence on the problem width.