Beep-And-Sleep: Message and Energy Efficient Set Cover

TL;DR

This paper introduces message and energy-efficient distributed algorithms for the Set Cover problem, achieving near-optimal time and message complexity in the $KT_0$ model and a beeping model variant with trade-offs.

Contribution

It presents the first near-optimal distributed algorithms for Set Cover in both the $KT_0$ and beeping models, improving efficiency and energy consumption.

Findings

Achieves $ ilde{O}( ext{log}^2( ext{Delta}))$-time and $O( ext{sqrt}( ext{Delta})(n+m))$-message algorithm in $KT_0$ model.

Provides a Beeping model algorithm with carrier-sensing that balances runtime and approximation ratio.

Demonstrates near-optimality of the proposed algorithms in their respective models.

Abstract

We observe message-efficient distributed algorithms for the Set Cover problem. Given a ground set $U$ of $n$ elements and $m$ subsets of $U$, we aim to find the minimal number of these subsets that contain all elements. In the default distributed setup of this problem, each set has a bidirected communication link with each element it contains. Our first result is a $\tilde{O}(\log^2(\Delta))$-time and $O(\sqrt{\Delta)}(n+m))$-message algorithm with expected approximation ration of $O(\log(\Delta))$ in the $KT_0$ model. The value $\Delta$ denotes the maximal cardinality of each subset. Our algorithm is \emph{almost} optimal with regard to time and message complexity. Further, we present Set Cover algorithm in the Beeping model that only relies on carrier-sensing and can trade runtime for approximation ratio similar to the celebrated algorithm by Kuhn and Wattenhofer [PODC '03].