Suppose you re acting as a consultant for the port authority of a small Pacific Rim nation. They're currently doing a multi-billion-dollar business per year, and their revenue is constrained almost entirely by the rate at which they can unload ships that arrive in the port.
Handling hazardous materials adds additional complexity to what is, for them, an already complicated task. Suppose a convoy of ships arrives in the morning and delivers a total of n cannisters, each containing a different kind of hazardous material. Standing on the dock is a set of m trucks, each of which can hold up to k containers.
Here are two related problems, which arise from different types of constraints that might be placed on the handling of hazardous materials. For each of the two problems, give one of the following two answers:
A polynomial-time algorithm to solve it; or
A proof that it is NP-complete.
(a) For each cannister, there is a specified subset of the trucks in which it may be safely carried. Is there a way to load all n cannisters into the m trucks so that no truck is overloaded, and each container goes in a truck that is allowed to carry it?
(b) In this different version of the problem, any cannister can be placed in any truck; however, there are certain pairs of cannisters that cannot be placed together in the same truck. (The chemicals they contain may react explosively if brought into contact.) Is there a way to load all n cannisters into the m trucks so that no truck is overloaded, and no two cannisters are placed in the same truck when they are not supposed to be?
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