Introduction

The Subset Sum Problem is a classical decision problem in computer science and mathematics. The problem is defined as follows: given a set of integers and a target sum, determine whether there is a subset of the provided set that adds up to the target sum. A quintessential variant of this problem that arises in optimization scenarios is the Optimal Weights Subset Sum, where the goal is to find the subset whose sum is closest to, but not exceeding, a given target.

One effective technique for tackling this problem is backtracking, a methodical way of trying out different sequences, narrowing down possibilities, and pruning branches that do not lead to a solution.

The Backtracking Approach

Backtracking systematically searches for a solution through trial and error. It involves the following steps:

1. Start with an empty set: Begin the process with no elements included in the subset.
2. Add elements one-by-one: Try to build up the subset by adding elements one-by-one from the given set of integers, checking at each step if the subset still satisfies the problem requirements.
3. Check for solution: Each time an element is added, check if the current subset sum is equal to, less than, or greater than the target sum.
4. Backtrack upon failure: If the subset sum exceeds the target, remove the last added element and try the next possibility.
5. Record the best solution: If a valid subset is found, record its sum if it's better than previously recorded solutions.

Algorithm Implementation

Here's a recursive pseudocode representation for solving the Optimal Weights Subset Sum using backtracking:

• Start with an empty subset and try adding integers from the set. • Upon trying all possibilities, the optimal subset may look like ${3, 12, 5, 2}$ which sums up to 22. • Further attempts will update this optimal subset as ${5, 12, 2, 11}$ summing to 30, which might be the best fit. • Prune if the current subset sum exceeds the target. • Order elements in the set to explore larger figures first, as they potentially offer faster pruning by exceeding the target quickly.