We will designate the following algorithm as FIRSTFIT. We consider an arbitrary numbers of bins,…

We will designate the following algorithm as FIRSTFIT. We consider an arbitrary numbers of bins, B1, B2, …, all initially empty. We then consider the items in their original (arbitrary) order, placing each into the lowest indexed bin in which the item fits. That is, we try to put the item in B1, otherwise into B2, and so on, using a previously empty bin if the item did not fit into any of the earlier bins. The final solution consists of only those bins that are nonempty. i) What approximation ratio is achieved by FIRSTFit on the following problem instance. There are 18 items, with the first six having weight C+1, the second six having weight CH1, and the final six having weight CulYou should assume that C is an abrbitrary large constant. ? For the remaining parts, consider general instances of the problem (not the specific instance from part i). ii) Argue that FIRSTFIT leaves at most one bin less than half full. iii) Argue that the number of bins used by FIRSTFiT is at most [20], where W = S?=1 ???. iv) Conclude that FIRSTFiT produces a 2-approximation for bin packing.

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