CS614. Advanced Algorithms. L19 Quiz.

CS614. Advanced Algorithms.

L19 Quiz

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Problem 1. Bin Packing

Consider the bin-packing problem:

Input: \(n\) items with sizes \(a_1 \cdots a_n\) respectively, a positive integer \(B\) (bin capacity) and a positive integer \(k\) (number of bins). Question: Is there a partition of the set \(\{1 \cdots n\}\) into sets \(S_1, \ldots, S_k\) such that for each \(i \in\{1 \cdots k\}\) we have that \(\sum_{j \in S_i} a_j \leq B\)?

Show that Bin Packing is NP-complete.

Problem 2. BOX-DEPTH

Consider the following problem, called BOX-DEPTH: Given a set of \(n\) axisaligned rectangles in the plane, how big is the largest subset of these rectangles that contain a common point?

  1. Describe a polynomial-time reduction from BOX-DEPTH to MAXCLIQUE.

  2. Describe and analyze a polynomial-time algorithm for BOX-DEPTH. [Hint: \(O\left(n^3\right)\) time should be easy, but \(O(n \log n)\) time is possible.]

  3. Why don’t these two results imply that \(\mathrm{P}=\mathrm{NP}\)?