Solving the NP-complete problem in XKCD

Answer 1

If you want an optimized algorithm, it's best to try the prices in descending order. That lets you use up as much of the remaining amount first and then see how the rest can be filled in.

Also, you can use math to figure out the maximum quantity of each food item to start each time so you don't try combinations that would go over the $15.05 goal.

This algorithm only needs to try 88 combinations to get a complete answer and that looks like the lowest that's been posted so far:

public class NPComplete {
    private static final int[] FOOD = { 580, 420, 355, 335, 275, 215 };
    private static int tries;

    public static void main(String[] ignore) {
        tries = 0;
        addFood(1505, "", 0);
        System.out.println("Combinations tried: " + tries);
    }

    private static void addFood(int goal, String result, int index) {
        // If no more food to add, see if this is a solution
        if (index >= FOOD.length) {
            tries++;
            if (goal == 0)
                System.out.println(tries + " tries: " + result.substring(3));
            return;
        }

        // Try all possible quantities of this food
        // If this is the last food item, only try the max quantity
        int qty = goal / FOOD[index];
        do {
            addFood(goal - qty * FOOD[index],
                    result + " + " + qty + " * " + FOOD[index], index + 1);
        } while (index < FOOD.length - 1 && --qty >= 0);
    }
}

Here's the output showing the two solutions:

9 tries: 1 * 580 + 0 * 420 + 2 * 355 + 0 * 335 + 0 * 275 + 1 * 215
88 tries: 0 * 580 + 0 * 420 + 0 * 355 + 0 * 335 + 0 * 275 + 7 * 215
Combinations tried: 88