The activity selection problem is a combinatorial optimization problem concerning the selection of non-conflicting activities to perform within a given time frame, given a set of activities each marked by a start time (si) and finish time (fi). The problem is to select the maximum number of activities that can be performed by a single person or machine, assuming that a person can only work on a single activity at a time. The activity selection problem is also known as the Interval scheduling maximization problem (ISMP), which is a special type of the more general Interval Scheduling problem.

A classic application of this problem is in scheduling a room for multiple competing events, each having its own time requirements (start and end time), and many more arise within the framework of operations research.

Formal definition

Assume there exist n activities with each of them being represented by a start time si and finish time fi. Two activities i and j are said to be non-conflicting if si ≥ fj or sj ≥ fi. The activity selection problem consists in finding the maximal solution set (S) of non-conflicting activities, or more precisely there must exist no solution set S' such that |S'| > |S| in the case that multiple maximal solutions have equal sizes.

Optimal solution

The activity selection problem is notable in that using a greedy algorithm to find a solution will always result in an optimal solution. A pseudocode sketch of the iterative version of the algorithm and a proof of the optimality of its result are included below.

Algorithm

Explanation

Line 1: This algorithm is called Greedy-Iterative-Activity-Selector, because it is first of all a greedy algorithm, and then it is iterative. There's also a recursive version of this greedy algorithm.

• is an array containing the activities.

• is an array containing the start times.

• is an array containing the finish times.

Note that these arrays are indexed starting from 1 up to the length of the corresponding array.

Line 3: Sorts in increasing order of finish times time, using for example merge sort, heap sort, or quick sort algorithms.

Line 4: to store the selected activities that has the earliest finish time.

Line 5: that keeps track of the index of the last selected activity.

Line 9: up to its last element.

Lines 10,11: If the start time) is greater or equal to the finish time of the last selected activity.

Line 12:.

Proof of optimality

is an optimal solution, also ordered by finish time; and that the index of the first activity in A, i.e., this optimal solution does not, and the activities in A are disjoint by definition, the activities in B are also disjoint. Since B has the same number of activities as A, B is also optimal.

Once the greedy choice is made, the problem reduces to finding an optimal solution for the subproblem. If A is an optimal solution to the original problem S.

Why? If this were not the case, pick a solution B′ to S′ with more activities than A′ containing the greedy choice for S′. Then, adding 1 to B′ would yield a feasible solution B to S with more activities than A, contradicting the optimality.

Weighted activity selection problem

The generalized version of the activity selection problem involves selecting an optimal set of non-overlapping activities such that the total weight is maximized. Unlike the unweighted version, there is no greedy solution to the weighted activity selection problem. However, a dynamic programming solution can readily be formed using the following approach:

Consider an optimal solution containing activity . We now have non-overlapping activities on the left and right of . We can recursively find solutions for these two sets because of optimal sub-structure. As we don't know , where is the last non-overlapping interval with solution:

References

External links

• Activity Selection Problem