Suppose you're consulting for a group that manages a high-performance real-time system in which asynchronous processes make use of shared resources. Thus the system has a set of n processes and a set of m resources. At any given point in time, each process specifies a set of resources that it requests to use. Each resource might be requested by many processes at once; but it can only be used by a single process at a time. Your job is to allocate resources to processes that request them. If a process is allocated all the resources it requests, then it is active; otherwise it is blocked. You want to perform the allocation so that as many processes as possible are active. Thus we phrase the Resource Reservation Problem as follows: Given a set of processes and resources, the set of requested resources for each process, and a number k, is it possible to allocate resources to processes so that at least k processes will be active?
Consider the following list of problems, and for each problem either give a polynomial-time algorithm or prove that the problem is NP-complete.
(a) The general Resource Reservation Problem defined above.
(b) The special case of the problem when k = 2.
(c) The special case of the problem when there are two types of resources—say, people and equipment—and each process requires at most one resource of each type (In other words, each process requires one specific person and one specific piece of equipment.)
(d) The special case of the problem when each resource is requested by at most two processes.
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