A \(\boldsymbol{\phi }\) -Competitive Algorithm for Scheduling Packets with Deadlines.

In the online packet scheduling problem with deadlines (PacketSchD, for short), the goal is to schedule transmissions of packets that arrive over time in a network switch and need to be sent across a link. Each packet has a deadline, representing its urgency, and a nonnegative weight, which represents its priority. Only one packet can be transmitted in any time slot, so if the system is overloaded, some packets will inevitably miss their deadlines and be dropped. In this scenario, the natural objective is to compute a transmission schedule that maximizes the total weight of packets that are successfully transmitted. The problem is inherently online, with the scheduling decisions made without the knowledge of future packet arrivals. The central problem concerning PacketSchD that has been a subject of intensive study since 2001 is to determine the optimal competitive ratio of online algorithms, namely the worst-case ratio between the optimum total weight of a schedule (computed by an offline algorithm) and the weight of a schedule computed by a (deterministic) online algorithm. We solve this open problem by presenting a phi-competitive online algorithm for PacketSchD (where phi approximate to 1.618 is the golden ratio), matching the previously established lower bound.