Activity selection problem

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The activity selection problem is a mathematical 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.

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[edit]

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 sifj or sjfi. 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[edit]

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 algorithm and a proof of the optimality of its result are included below.

Algorithm[edit]

Sort the set of activities by finishing time (f[i])
S = {1}
f = f[1]
for i=2 to n
if s[i] ≥ f
S = S U i
f = f[i]
endfor

Proof of optimality[edit]

Let S = {1, 2, . . ., n} be the set of activities ordered by finish time. Thus activity 1 has the earliest finish time.

Suppose, A is a subset of S is an optimal solution and let activities in A be ordered by finish time. Suppose, the first activity in A is k.

If k = 1, then A begins with greedy choice and we are done (or to be very precise, there is nothing to prove here).

If k not=1, we want to show that there is another solution B that begins with greedy choice, activity 1.

Let B = (A - {k}) U {1}. Because f[1] =< f[k], the activities in B are disjoint and since B has same number of activities as A, i.e., |A| = |B|, 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, then A` = A - {1} is an optimal solution to the activity-selection problem S` = {i in S: s[i] >= f[1]}.

Why? If we could find a solution B` to S` with more activities then A`, adding 1 to B` would yield a solution B to S with more activities than A, there by contradicting the optimality.

External links[edit]