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503_HW5_S11_Fixed

# 503_HW5_S11_Fixed - 3-colored Use the graph coloring...

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UML CS Algorithms 91.503 (section 201) Spring, 2011 1 Homework #5 Assigned: Tuesday, 3/22 Due: Tuesday, 4/5 at 5:30 p.m. This assignment covers textbook material in Chapter 34 (NP-Completeness) and the related handouts. In this assignment you may use (with an appropriate citation) any problems that are: - shown NP-hard or NP-complete in our text or Gary & Johnson; - stated NP-hard or NP-complete in an exercise in our text; - proven by you here to be NP-hard or NP-complete. 1 . (20 points) Consider the steps 1-5 on p. 1078-1079 of our textbook. Circle TRUE or FALSE for the statement below and justify your answer. Replacing step 2 with the following preserves the correctness of steps 1-5: Select a known language NP L ' .” TRUE FALSE 2 . (20 points) This problem is about the paper by Cheeseman et al . a) Prove that the graph G below is not 3-colorable. Can you reconcile this with the descriptions of the reduction operators? Why or why not? b) Now identify one edge that, if removed from G to form a graph G’, allows G’ to be

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Unformatted text preview: 3-colored. Use the graph coloring reduction operators in the paper on G’. That is, show a sequence of reductions on G’ that reduce it to a small graph that can be easily 3-colored. Show the resulting 3-coloring of the entire graph G’. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 G UML CS Algorithms 91.503 (section 201) Spring, 2011 2 3 . (20 points) Show that the following problem is in NP (source: Garey & Johnson): INSTANCE : An n x n matrix M of 0’s and 1’s, and a positive integer K . QUESTION : Is there a collection of K or fewer two-dimensional axis-aligned rectangles that covers precisely those entries in M that are 1’s? Notes : 1. Rectangles may overlap each other. 2. A matrix entry is equal to 1 if and only if that entry is inside at least one of the rectangles. Provide pseudocode as part of your answer. 4 . (40 points) Problem 34-4 (a) and (b) on p. 1104 of our textbook....
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503_HW5_S11_Fixed - 3-colored Use the graph coloring...

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