A Constant Approximation Algorithm for Scheduling Packets on Line   Networks
  In this paper we improve the approximation ratio for the problem of scheduling packets on line networks with bounded buffers, where the aim is that of maximizing the throughput. Each node in the network has a local buffer of bounded size $B$, and each edge (or link) can transmit a limited number, $c$, of packets in every time unit. The input to the problem consists of a set of packet requests, each defined by a source node, a destination node, and a release time. We denote by $n$ the size of the network. A solution for this problem is a schedule that delivers (some of the) packets to their destinations without violating the capacity constraints of the network (buffers or edges). Our goal is to design an efficient algorithm that computes a schedule that maximizes the number of packets that arrive to their respective destinations.   We give a randomized approximation algorithm with constant approximation ratio for the case where $B=\Theta(c)$. This improves over the previously best result of $O(\log^* n)$ (R\"acke and Ros\'en, Theory Comput. Syst., 49(4), 2011). Our improvement is based on a new combinatorial lemma that we prove, stating, roughly speaking, that if packets are allowed to stay put in buffers only a limited number of time steps, $2d$, where $d$ is the longest source-destination distance of any input packet, then the cardinality of the optimal solution is decreased by only a constant factor. This claim was not previously known in the directed integral (i.e., unsplittable, zero-one) case, and may find additional applications for routing and scheduling algorithms. 
Guy Even, Moti Medina, Adi Ros\'en
cs.DS