# Computing optimal assignments using little memory

I have two lists where each item in the first list has a rating for each item in the second. I need to determine an optimal matching (or the best x scenarios) where items are matched, but each item can only be matched once.

For example, there are 150 movies available, 100 people, and each person has rated each movie between 0 and 100. Once a movie is given to a person, it is no longer available for another person.

In this example, I'd like to find the scenario where the person/movie with the lowest rating is the best rating possible. Then, the average of all ratings is the best possible. Ideally, no one person would be unhappy with their movie (data permitting). There might be multiple optimal solutions, so I'd like to determine and rank the top 5 scenarios.

I plan to implement this in node.js and Javascript, so the memory footprint of data structure(s) is a factor.

What would be the optimal data structure and approach to solve something like this?

• Your input is a weighted bipartite graph (vertices are people and movies; there's an edge from each person to each movie, weighted according to preference). So you want to look for algorithms to find optimal matchings according to your criterion. How to program it is likely to be off-topic here, especially at the level of detail where you're telling us what language you want to use. – David Richerby Jun 10 '15 at 14:27
• Thanks, David. I mention the programming language to show that memory consumption is a factor, not for a particular implementation. I've updated the question. – Gary Jun 10 '15 at 14:38
• 1. "the scenario where the person/movie with the lowest rating is the best rating possible" is not the same optimization criteria as "the average of all ratings is the best possible"; in general those might lead to two different solutions. So what exactly is the objective function you want to maximize? 2. What research have you done? Have you read about the assignment problem, maximum matchings in bipartite graphs, linear programming, etc.? We expect you to do a good chunk of research before asking and to show it in the question, so I suggest doing some research and editing the question. – D.W. Jun 10 '15 at 20:30
• From the way the question is phrased I think he meant you should read the optimization criteria lexicographically. I.e. an assignment $A \succ B$ if the minimum rating $A >$ min rating $B$, but if min rating $A =$ min rating $B$, $A \succ B$ iff average rating $A >$ average rating $B$. If the number of possible ratings is low compared to the number of persons this actually makes sense. @Gary If this is correct you want to put this in your question more explicitly. To help you on your way, the first criterion is bottleneck assignment program, the second is maximum weight bipartite matching. – Thomas Bosman Jun 12 '15 at 10:54
• Additionally, the question for an optimal datastructure makes no sense without an approach. Datastructure for what? So you should start with finding an approach, you can always find an optimal datastructure later. I don't think it's likely there is an off the shelf algorithm for your exact problem, but you could try to combine approaches to the aformentioned problems in sequence. No idea if this is feasible but with these pointers you should have enough to do some research and ask a more directed question if you get stuck. – Thomas Bosman Jun 12 '15 at 11:04

Your problem is closely related to the assignment problem, and can be solved efficiently using techniques derived from that literature. I'll show how to solve it, by building up a solution step by step.

## Maximizing the lowest rating

Here is how you could efficiently find the assignment that maximizes the lowest rating. First, focus on the associated decision problem:

Input: movie ratings; an integer $t$
Goal: find an assignment where each person is assigned to a movie they rate at least $t$ or higher

This decision problem is an instance of bipartite matching, and can be solved using standard algorithms for it. Namely, look for any assignment that assigns each person a movie that they related at least $t$.

Now, to find the assignment that maximizes the lowest rating, use binary search over $t$.

This doesn't solve your problem yet, because it doesn't take into account that you want to resolve ties by maximizing the average rating.

## Resolving ties by maximizing the average rating

First, find the largest value $t$ such that there exists an assignment where everyone is assigned a movie they rate $\ge t$. (See above for how to do it.) Now if there are multiple such assignments, we want to find one of them where the average rating is maximized.

How do we do that? Answer: use a standard algorithm for the Hungarian algorithm.

In more detail, we do several steps:

1. Delete any (person,movie) rating where the person has rated the movie something less than $t$. It's not OK to assign that person that movie.

2. Now, you have an instance of the assignment problem, where each person has rated some of the movies with scores that are $t$ or larger. You want to find a matching assignment) that maximizes the average of the scores of assigned movies. Since every solution must assign 1 movie to each of the 100 people, the size of every assignment is the same, so maximizing the average is equivalent to maximizing the sum. Maximizing the sum of the ratings is exactly the assignment problem. Therefore, you can use any standard algorithm for solving the assignment problem (e.g., the Hungarian algorithm).

This will find an optimal solution: i.e., an assignment that maximizes the lowest rating, and that resolves ties by maximizing the average rating.

If you want to find the top-5 solutions, you can use an algorithm for enumerating solutions to the assignment problem in a streaming/output-sensitive fashion. See, e.g., Algorithm for a list of best solutions to the Assignment problem and Does Ford-Fulkerson always produce the left-most min-cut.