Fall 2006 CMSC 651: Homework 4 Due: Due Nov 1
(This is a WRITTEN HW)
1. (25 points)
(a) Define the 3-processor scheduling problem in a way analogous to the 2-processor schedul-
ing problem
(b) Define a kind of matching problem such that an optimal solution to the 3-processor
scheduling problem yields a maximum matching. (HINT: think about defining and using
matchings in hypergraphs.)
2. (25 points) For an infinite number of ngive an example of a connected bipartite graph on
Θ(n) vertices such that here there exists a maximal matching Mand a maximum matching
M∗such that |M∗| ≥ 2|M|.
3. (25 points) Show how to multiply two n×nmatrices over Z4in O(n3/log n) steps in a way
similar to the method showed in class for Z2.
4. (25 points) Let Pbe an instance of the 2-proc schedling problem and let Gbe the compatability
graph. We showed that (1) a solution to Pyields a matching in G, and (2) a matching in G
yields a solution in P. Show that (1) an optimal solution to Pyields a maximum matching
in G, and (2) a maximum matching in Gyields an optimal solution in P.
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