The 2D-PACKING problem is this: Given a bounded region R of the 2D cartesian pla
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The 2D-PACKING problem is this: Given a bounded region R of the 2D cartesian plane, and a non-negative integer k, is it possible to pack the region with k or more tiles. Each tile is square (1 by 1) and can be positioned anywhere in the plane, provided its sides remain parallel to the x or y axes. The tiles in a packing may not overlap, and they must stay within the given region. A packing is maximal if there are as many tiles in it as possible. The following three illustrations give a packing region and two possible packings of it. The packing region is shown in white. Tiles are not allowed to go into the gray area. The first packing has only 2 tiles, whereas the other packing as 3 tiles. The tiles may fit "snuggly" into the packing region as they do here, but that is not required. For example, the packing region could have rounded or irregular boundaries. From these illustrations, we can see that the answers to some instances of packing problem with this region are: Yes for k=2 and Yes for k=3, but No for k > 3. The following example is somewhat similar, except that we see that a maximum packing here has only 2 tiles, and furthermore there are two possible maximal packings for this region. Prove that the 2D-PACKING problem in NP.Explanation / Answer
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