Colored Packets with Deadlines and Metric Space Transition Cost

TL;DR

This paper introduces a new competitive algorithm for scheduling colored packets with transition costs in metric spaces, matching the hardness bounds and extending to weighted directed graphs, advancing understanding of the problem's complexity.

Contribution

The paper presents a novel competitive algorithm for packet scheduling with metric space transition costs and establishes matching hardness bounds, improving previous results for uniform metrics.

Findings

Achieves a $1 - O( oot{MST(G)/L)}$ competitive ratio.

Proves a matching hardness of $1 - ext{Omega}( oot{MST(G)/L)}$ for general metric spaces.

Extends results to weighted directed graphs with a $1 - O( oot{TSP(G)/L)}$ competitive ratio.

Abstract

We consider scheduling of colored packets with transition costs which form a general metric space. We design a $1 - O(\sqrt{MST(G) / L})$ competitive algorithm. Our main result is a hardness result of $1 - \Omega(\sqrt{MST(G) / L})$ which matches the competitive ratio of the algorithm for each metric space separately. In particular, we improve the hardness result of Azar at el. 2009 for uniform metric spaces. We also extend our result to weighted directed graphs which obey the triangular inequality and show a $1 - O(\sqrt{TSP(G) / L})$ competitive algorithm and a nearly-matching hardness result. In proving our hardness results we use some interesting non-standard embedding.