algorithm,np-complete,subset-sum , Reduce Subset Sum to Polyomino Packing


Reduce Subset Sum to Polyomino Packing

Question:

Tag: algorithm,np-complete,subset-sum

This is a homework assignment, so any help is appreciated.

I should prove that the following problem is NP-complete. The hint says that you should reduce the subset sum problem to this problem.

Given a set of shapes, like the below, and an m-by-n board, decide whether is it possible to cover the board fully with all the shapes. Note that the shapes may not rotate.

For example, for a 3-by-5 board and the following pieces, the board can be covered like this:

shapes

covered board

Now the important thing to note is that the subset sum problem we are trying to reduce should be given input length polynomial in terms of m and n.

Any ideas for using another NP-complete problem are appreciated.


Answer:

I would recommend doing this reduction in two steps.

First, reduce subset-sum to the partition problem. This problem is related to subset-sum, except that instead of being given a target k, the goal is to split the set into two exactly equal halves. This isn't too hard (I'll leave it as an exercise) and makes the next part a lot easier.

Next, reduce the partition problem to the tiling problem. Intuitively, the idea is the following. If the input numbers in the partition problem are n1, n2, ..., nk, create tiles whose sizes are 3n1 × 1, 3n2 × 1, ..., and 3nk × 1, where w is some number we'll choose later. Then, if the sum of the numbers in the input set is 2N (meaning that the goal is to split the set into two subsets that sum up to N each), create an 3N × 2 board.

The intuition is that if we can split the set into two equal-sum subsets, then we can tile this board by taking all the numbers from the first half, choosing those tiles, then laying them end-to-end to form an 3N × 1 rectangle. We can then take the numbers from the remaining half and use them to form an 3N × 1 rectangle as well, so putting the first line of tiles in the top row and the second line of tiles in the second row completely tiles the board.

To show that the other direction works, first notice that the tiles all have size at least 3 × 1, so they have to be oriented from the left to the right. Therefore, if we can tile the board, we've found a way to split the tiles into two groups whose widths are equal, so there's some way to split the set into two equal-sum pieces.

Overall, the size of the input only grows by a polynomial at most (we triple the size of each number, which adds at most two bits to each input number length), and the size of the board is polynomially as large as the input. Therefore, overall, the reduction takes only polynomial time to evaluate.

Hope this helps!


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