Online Packet Scheduling with Bounded Delay and Lookahead
  We study the online bounded-delay packet scheduling problem (BDPS), where packets of unit size arrive at a router over time and need to be transmitted over a network link. Each packet has two attributes: a non-negative weight and a deadline for its transmission. The objective is to maximize the total weight of the transmitted packets. This problem has been well studied in the literature, yet its optimal competitive ratio remains unknown: the best upper bound is $1.828$, still quite far from the best lower bound of $\phi \approx 1.618$.   In the variant of BDPS with $s$-bounded instances, each packet can be scheduled in at most $s$ consecutive slots, starting at its release time. The lower bound of $\phi$ applies even to the special case of $2$-bounded instances, and a $\phi$-competitive algorithm for $3$-bounded instances was given in Chin et al. Improving that result, and addressing a question posed by Goldwasser, we present a $\phi$-competitive algorithm for $4$-bounded instances.   We also study a variant of BDPS where an online algorithm has the additional power of $1$-lookahead, knowing at time $t$ which packets will arrive at time $t+1$. For BDPS with $1$-lookahead restricted to $2$-bounded instances, we present an online algorithm with competitive ratio $(\sqrt{13} - 1)/2 \approx 1.303$ and we prove a nearly tight lower bound of $(1 + \sqrt{17})/4 \approx 1.281$. 
Martin B\"ohm, Marek Chrobak, {\L}ukasz Je\.z, Fei Li, Ji\v{r}\'i   Sgall, Pavel Vesel\'y
cs.DS