Question

We have the following scheduling problem. For each of the next n days, the hospital has determined the number of doctors they want on hand; thus, on day i, they have a requirement that exactly pi doctors be present.

There are k doctors, and each is asked to provide a list of days on which s/he is willing to work. Thus doctor j provides a set Lj of days s/he is willing to work.

The scheduling system takes these lists and tries to return to each doctor j a list Lj' with the following properties:

(1) Lj' is a subset of Lj, so that doctor j only works on days s/he finds acceptable

(2) The set of lists L1', L2', ..., Lk' causes exactly pi doctors to be present on each day i, for i = 1, 2, ..., n.

Use maximum-flow to describe a polynomial-time algorithm that implements the scheduling system. More specifically, the algorithm takes as input : n, k, the numbers p1, ..., pn, the lists L1, L2, ...Lk, and produces the following output:

returns the lists L1', L2', ..., Lk' if a schedule satisfying properties (1) and (2) is possible
otherwise reports that there is no possible schedule.
Describe the algorithm. Analyze the RT in terms of n and k.

Answer 1

The maximum-flow algorithm can be used to solve the scheduling problem by assigning doctors to the required days while considering their availability constraints. The algorithm creates a flow network and determines if a valid schedule is possible by finding the maximum flow from the source node to the sink node. The time complexity of the algorithm depends on the maximum flow algorithm used.

Step-by-step explanation:

The scheduling problem described can be solved using the maximum-flow algorithm. This algorithm can determine if it is possible to assign doctors to the required days, while meeting their availability constraints. Here is a step-by-step description of the algorithm:

1. Create a source node and connect it to each day node with an edge capacity equal to the required number of doctors for that day (pi).
2. Create a sink node and connect each doctor node to it with an edge capacity equal to the number of days they are willing to work.
3. For each doctor node, create an edge from the source node with a capacity equal to the number of days they are willing to work. Connect each day node to the corresponding doctor node with an edge capacity of 1.
4. Apply the maximum-flow algorithm to find the maximum flow from the source node to the sink node. If the maximum flow equals the sum of the required number of doctors for each day, then a valid schedule is possible.
5. If a valid schedule is possible, backtrack through the flow network to determine the days assigned to each doctor.

The time complexity of this algorithm depends on the maximum flow algorithm used. In the worst case, it can be O(n * m^2), where n is the number of days and m is the number of doctors. However, with efficient maximum flow algorithms like the push-relabel algorithm, the time complexity can be reduced to O(n^3), making it polynomial-time.