To address Problem B as described in your document, we'll start by identifying a counterexample to the provided greedy algorithm ("largest increment in mark"), then design an efficient algorithm to solve the homework allocation problem optimally, and finally analyze its time and space complexity.
Let's consider the greedy algorithm provided in the problem statement, which prioritizes homework assignments based on the largest increase in marks for each additional hour spent. We need to find a scenario where this approach does not yield the maximum total marks when the total available hours (T) are limited, with (n \leq 2) and (T \leq 3).
Given:
- (n = 2) homework assignments.
- (T = 3) hours available.