We consider scheduling of colored packets with transition costs which form a general metric space. We design a @math competitive algorithm. Our main result is a hardness result of @math 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 @math competitive algorithm and a nearly-matching hardness result. In proving our hardness results we use some interesting non-standard embedding.
The closest model to our model is the colored packets with deadline problem @cite_17 . In that model we are given a sequence of incoming packets. Each packet is characterized by a color and a deadline (all packets have the same value). The goal is to find a schedule that maximizes the number of packets that were transmitted before the deadline, such that there is a transition time-slot between the transmission of packets of different colors. An algorithm with competitive ratio of @math and hardness result of @math are shown in @cite_17 , when @math is the number of colors and @math is the minimum of the difference between the expiration and the arrival time of packets.
@cite_17
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We study the online problem of multiplexing packets with arbitrary deadlines in bounded multi-buffer switch. In this model, a switch consists of m input buffers each with bounded capacity B and one output port. Each arriving packet is associated with a value and a deadline that specifies the time limit till the packet can be transmitted. At each time step the switch can select any non-empty buffer and transmit one packet from that buffer. In the preemptive model, stored packets may be preempted from their buffers due to lack of buffer space or discarded due to the violation of the deadline constraints. If preemption is not allowed, every packet accepted and stored in the buffer must be transmitted before its deadline has expired. The goal is to maximize the benefit of the packets transmitted by their deadlines. To date, most models for packets with deadlines assumed a single buffer. To the best of our knowledge this is the first time a bounded multi-buffer switch is used with arbitrary deadline constraints Our main result is a 9.82-competitive deterministic algorithm for packets with arbitrary values and deadlines. Note that the greedy algorithm is not competitive. For the non-preemptive model we present a 2-competitive deterministic algorithm for the unit value packets. For arbitrary values we present a randomized algorithm whose competitiveness is logarithmic in the ratio between the largest and the smallest value of the packets in the sequence