Lecture 6 , 7 ( Sept 27 and 29 , 2011 ) : Bin Packing , MAX - SAT

Input: set of items S = {1, ..., n} with size (0 ≤ si ≤ 1) ∈ Q. Question: Can we partition S into two parts S′ and S − S′ such that ∑ i∈S Si = ∑ j∈S−S′ Sj? Let I be an instance of partition. Scale all Si’s such that ∑ Si = 2 and let this instance I ′ be the input to Bin Packing. If all items of I ′ fit into 2 bins, since their total sum is 2, both bins must be full and therefore I is a yes instance (the partition is given by the items in 2 bins for I ′). On the other hand, if I is a Yes instance, then the corresponding partition implies that the set of items in each part can be fit into one bin for the corresponding instance I ′. Therefore, the set of items of I ′ can be fit into 2 bins if and only if I is a Yes instance. So if we can distinguish between 2 and ≥ 3 for I ′ then we can decide between Yes and No for I. Therefore, there is no better than 3 2 -approximation for bin packing unless P=NP.