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[SOLVED] Csci 570 hw 9 1. at a dinner party, there are n families a1, a2, …, an and m tables b1, b2, …, bm

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1. At a dinner party, there are n families a1, a2, …, an and m tables b1, b2, …, bm.
The i
th family ai has gi members and the j
th table bj has hj seats. Everyone is interested in making new friends and the dinner party planner
wants to seat people such that no two members of the same family are
seated in the same table.Design an algorithm that decides if there exists
a seating assignment such that everyone is seated and no two members of
the same family are seated at the same table.2. There is a precious diamond that is on display in a museum at m disjoint
time intervals. There are n security guards who can be deployed to protect
the precious diamond. Each guard has a list of intervals for which he/she
is available to be deployed. Each guard can be deployed to at most A time
slots and has to be deployed to at least B time slots. Design an algorithm
that decides if there is a deployment of guards to intervals such that each
interval has either exactly one or exactly two guards deployed.3. Solve Kleinberg and Tardos, Chapter 7, Exercise 28.4. The computer science department course structure is represented as a directed acyclic graph G = (V, E) where the vertices correspond to courses
and a directed edge (u, v)E exists if and only if the course u is a prerequisite of the course v. By taking a course w ∈ V , you gain a benefit of
bw which could be a positive or negative number. Design an algorithm
that picks a subset
P
A ⊂ V of courses to take such that the total benefit
w∈A bw is maximized. Remember that if v ∈ A and (u, v) ∈ E, then
u has to be in A. That is, to take a course, you have to take all its
prerequisites. The running time should be polynomial in |V |.