Editing Approximation Algorithms (section)

== <span style="color: #FFFFFF;">Applying</span> ==
'''Modeling 'The Greedy Knapsack' (Approximating the best way to pack a bag):'''
<syntaxhighlight lang="python">
def greedy_knapsack(items, capacity):
    """
    Items = [(value, weight), ...]
    Picks the 'Best Ratio' first.
    """
    # Sort by 'Value per Weight' (The Greedy Choice)
    sorted_items = sorted(items, key=lambda x: x[0]/x[1], reverse=True)
    
    bag = []
    current_weight = 0
    total_value = 0
    
    for val, wt in sorted_items:
        if current_weight + wt <= capacity:
            bag.append((val, wt))
            current_weight += wt
            total_value += val
            
    return f"Bag Value: {total_value} | Weight: {current_weight}/{capacity}"

# Case: Value/Weight items
my_items = [(100, 10), (60, 20), (120, 30)] # (Val, Wt)
print(greedy_knapsack(my_items, 50))
</syntaxhighlight>

; Approximation Landmarks
: '''The 'Bin Packing' Problem''' → How to fit "Boxes into a Truck" or "Ads onto a Webpage." Approximation algorithms ensure that "Minimum Space is Wasted."
: '''Network Design''' → How "Internet Providers" lay fiber-optic cables. They use "Steiner Tree" approximations to connect "Every City" with the "Minimum amount of cable."
: '''Deep Learning (SGD)''' → The way "AIs" learn (Stochastic Gradient Descent) is an "Approximation." We don't find the "Perfect Brain," we "Approximate" one that "Works well enough" through billions of tiny tweaks.
: '''Scheduling''' → How "Airlines" manage thousands of flights and crews. Since the "Perfect Schedule" is impossible to calculate, they use "Local Search" to "Refine" a good schedule until it is "Almost Perfect."
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