A Simple Distributed Algorithm for Sparse Fractional Covering and Packing Problems

TL;DR

This work presents a simple distributed algorithm in the CONGEST model that achieves a $(1+\epsilon)$-approximation for both row-sparse fractional covering and column-sparse fractional packing problems. It introduces a phase-based scheme with primal variables $x_S$, dual variables $y_e$, and reduction variables $r_e$, achieving feasibility and a tight $(1+\epsilon)$-approximation in $O(\Gamma_d \cdot \log \Gamma_p / \epsilon^2)$ rounds, outperforming prior $\epsilon$-dependence while trading off $A_{\max}$-dependence. The approach builds on and simplifies previous methods, with practical implications for distributed optimization in networks, including foundational problems like fractional set cover and fractional packing. The authors also identify open problems around constant-round, constant-factor solutions for RS-FCP/CS-FPP and discuss the nuanced landscape of lower bounds in the CONGEST model.

Abstract

This paper presents a distributed algorithm in the CONGEST model that achieves a $(1+ε)$-approximation for row-sparse fractional covering problems (RS-FCP) and the dual column-sparse fraction packing problems (CS-FPP). Compared with the best-known $(1+ε)$-approximation CONGEST algorithm for RS-FCP/CS-FPP developed by Kuhn, Moscibroda, and Wattenhofer (SODA'06), our algorithm is not only much simpler but also significantly improves the dependency on $ε$.

Theorems & Definitions (9)

• Theorem 1.1: Main Theorem
• Remark 1.2
• Theorem 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• Theorem 2.5
• proof