What is the largest sum that can be constructed with the given recipes?

There are $ n$ sets of distinct positive integers, $ S_1,\ldots,S_n$ . There is a set of recipes that allows us to construct tuples of integers from these sets. For example, the recipe {1,2} allows us to construct tuples of two integers: one from $ S_1$ and one from $ S_2$ . Each integer can be used in at most one tuple. What is an algorithm that, given the sets and the recipes, constructs tuples such that the sum of integers in all tuples together is maximum? Here are some simple cases:

(a) There is a single recipe, e.g. {1,2}. Then the algorithm is simple: sort the integers in $ S_1$ and in $ S_2$ in descending order; pair the largest integers in each set; pair the second-largest integers in each set; keep pairing integers as long as the sum of the next pair is positive.

(b) There are two recipes {1,2} and {1,3}. Then, $ S_2$ and $ S_3$ are substitutes: combine them into a single set $ S_4 = S_2\cup S_3$ , and proceed as in (a) with the recipe {1, 4}.

(c) There are two recipes {1,2} and {1,3,4}. Then, $ S_3$ and $ S_4$ are complements: sort each one in descending order, and construct a new set $ S_5$ in which the largest element is the sum of the largest elements in $ S_3,S_4$ , the second-largest is the sum of the two second-largest elements, and so on. Proceed as in (b) with the recipes {1,2} and {1,5}.

(d) The above two operations can be generalized to the case in which the set of recipes has a tree structure: there is a tree in which each node is a set, and each recipe is a path from root to leaf in the tree. The tree can be "contracted" as follows: a leaf which is the unique child of its parent can be combined with its parent as in (c); two leafs of the same parent can be combined as in (b).

Is there an efficient algorithm that works for any set of recipes?

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Detecting conservation, loss, or gain in a crafting game with items and recipes

Suppose we’re designing a game like Minecraft where we have lots of items $ i_1,i_2,…,i_n\in I$ and a bunch of recipes $ r_1,r_2,…,r_m\in R$ . Recipes are functions $ r:(I\times\mathbb{N})^n\rightarrow I\times\mathbb{N}$ , that is they take some items with non-negative integer weights and produce an integer quantity of another item.

For example, the recipe for cake in Minecraft is:

3 milk + 3 wheat + 2 sugar + 1 egg $ \rightarrow$ 1 cake

… and the recipe for torches is:

1 stick + 1 coal $ \rightarrow$ 4 torches

Some recipes could even be reversible, for example: 9 diamonds $ \leftrightarrow$ 1 diamond block

If there’s some combination of recipes we can repeatedly apply to get more of the items that we started with then the game is poorly balanced and this can be exploited by players. It’s more desirable that we design the game with recipes that conserve items or possibly lose some items (thermodynamic entropy in the real world – you can’t easily un-burn the toast).

Is there an efficient algorithm that can decide if a set of recipes will:

  • conserve items?
  • lose items to inefficiency?
  • gain items?

Is there an efficient algorithm that can find the problematic recipes if a game is imbalanced?

My first thoughts are that there is a graph structure / maximum flow problem here but it’s very complex, and that it resembles a knapsack problem. Or maybe it could be formulated as a SAT problem – this is what I’m considering to code it at the moment but something more efficient might exist.

We could encode recipes in a matrix $ \mathbf{R}^{m \times n}$ where rows correspond to recipes and columns correspond to items. Column entries are negative if an item is consumed by a recipe, positive if it’s produced by the recipe, and zero if it’s unused. Similar to a well known matrix method for graph cycle detection, we could raise $ \mathbf{R}$ to some high power and get sums of each row to see if item totals keep going up, stay balanced, or go negative. However, I’m not confident this always works.

Any discussion, code, or recommended reading is very appreciated.

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Expanding crafting and adding recipes

A note for the start, I am a new DM with not so much experience.

My party is far from a bigger city and running low on healing potions. One of my players has proficiency with the herbalism kit and asked me if he could collect the ingredients and brew a healing potion.

I like the idea behind it. I have similar experiences from computer games. For example I played an MMORPG online for quite a while, where you could choose a profession and on your adventures collect ingredients for your profession. If you had the recipes for them you could then craft items from these. Recipes also had to be found or bought somewhere.

So I looked up the rules but they are quite bare on this. Xanathar’s Guide to Everything states that given a herbalism kit, a common healing potion takes one workday to brew and 25 gp (also mentioned in the answers to this RPG.SE question: How does one craft Potions of Healing?). Not sure what the player is supposed to do with the gold. I doubt it is meant as an ingredient. Also the tree next to him doesn’t seem too interested in being turned into herbs.

So I looked online and found some homebrew guides like here and here. They do include rules for collecting herbs and recipes for combining them.

However for all I know that could be untested theory. Apparently it is not commonly used in campaigns. Why is that so? I am looking for experience with this kind of crafting. Have you done something similar in your campaign and how did it go?

What other good sources on this topic are out there? Is there maybe something from previous editions of D&D that I did not find?