Permutation/Algorithm to Solve Conditional Fill Puzzle

Question

I've been digging around to see if something similar has been done previously, but have not seen anything with the mirrored conditions. To make swallowing the problem a little easier to understand, I'm going to apply it in the context of filling a baseball team roster.

The given roster structure is organized as such: C, 1B, 2B, 3B, SS, 2B/SS (either or), 1B/3B, OF, OF, OF, OF, UT (can be any position)

Every player has at least one of the non-backup positions (positions that allow more than one position) where they're eligible and in many cases more than one (i.e. a player that can play 1B and OF, etc.). Say that you are manager of a team, which already has some players on it and you want to see if you have room for a particular player at any of your slots or if you can move one or more players around to open up a slot where he is eligible.

My initial attempts were to use a conditional permutation and collect in a list all the possible unique "lineups" for each player, updating the open slots before moving to the next player. This also required (since the order that the player was moved would affect what positions were available for the next player) that the list being looped through was reordered and then looped through again. I still think that this is the way to go, but there are a number of pitfalls that have snagged the function.

The data to start the loop that you assume is given is: 1. List of positions the player being evaluated can player (the one being checked if he can fit) 2. List of players currently on the roster and the positions each of those is eligible at (I'm currently storing a list of lists and using the list index as the unique identifier of the player) 3. A list of the positions open as the roster currently is

It's proven a bigger headache than I originally anticipated. It was even suggested to me by a colleague that the situation I have (which involves, on a much larger scale, conditional assignments for each object) was NP-complete. I am certain that it is not, since once a player has been repositioned in a particular lineup being tested, the entire roster should not need to be iterated over again once another player has moved. That's the long and short of it and I finally decided to open it up to the forums.

Thanks for any help anyone can provide. Due to restrictions, I can't post portions of code (some of it is legacy). It is, however, being translated in .NET (C# at the moment). If there's additional information necessary, I'll try and rewrite some of the short pieces of the function for post.

Joseph G.

EDITED 07/24/2010 Thank you very much for the responses. I actually did look into using a genetic algorithm, but ultimately abandoned it because of the amount of work that would go into the determination of ordinal results was superfluous. The ultimate aim of the test is to determine if there is, in fact, a scenario that returns a positive. There's no need to determine the relative benefit of each working solution.

I appreciate the feedback on the likely lack of familiarity with the context I presented the problem. The actual model is in the distribution of build commands across multiple platform-specific build servers. It's accessible, but I'd rather not get into why certain build tasks can only be executed on certain systems and why certain systems can only execute certain types of build commands.

It appears that you have gotten the gist of what I was presenting, but here's a different model that's a little less specific. There are a set of discrete positions in an ordered array of lists as such (I'll refer to these as "positions"):

((2), (2), (3), (4), (5), (6), (4, 6), (3, 5), (7), (7), (7), (7), (7), (2, 3, 4, 5, 6, 7))

Additionally, there is a an unordered array of lists (I'll refer to as "employees") that can only occupy one of the slots if its array has a member in common with the ordered list to which it would be assigned. After the initial assignments have been made, if an additional employee comes along, I need to determine if he can fill one of the open positions, and if not, if the current employees can be rearranged to allow one of the positions the employee CAN fill to be made available.

Brute force is something I'd like to avoid, because with this being on the order of 40 - 50 objects (and soon to be increasing), individual determinations will be very expensive to calculate at runtime.

Answer 1

It sounds as though you have a bipartite matching problem. One partition has a vertex for each player on the roster. The other has a vertex for each roster position. There is an edge between a player vertex and a position vertex if and only if the player can play that position. You are interested in matchings: collections of edges such that no endpoint is repeated.

Given an assignment of players to positions (a matching) and a new player to be accommodated, there is a simple algorithm to determine if it can be done. Direct each edge in the current matching from the position to the player; direct the others from the player to the position. Now, using breadth-first search, look for a path from the new player to an unassigned position. If you find one, it tells you one possible series of reassignments. If you don't, there's no matching with all of the players.

For example, suppose player A can play positions 1 or 2

A--1
 \
  \
   2

We provisionally assign A to 2. Now B shows up and can only play 2. Direct the graph:

A->1
 <
  \
B->2

We find a path B->2->A->1, which means "assign B to 2, displacing A to 1".

There is lots of pretty theory for dealing with hypothetical matchings. Genetic algorithms need not apply.

---

EDIT: I should add that because of the use of BFS, it computes the least disruptive sequence of reassignments.

Answer 2

I don't understand baseball at all so sorry if I'm on the wrong track. I do like rounders though, but there are only 2 positions to play in rounders, a batter or everyone else.

Have you considered using Genetic Algorithms to solve this problem? They are very good at solving NP hard problems and work surprisingly well for rota and time schedule type problems as well.

You have a solution model which can easily be scored and easily manipulated which is a great start for a genetic algorithm.

For more complex problems where the total permutations are too large to calculate a genetic algorithm should find a near optimum or excellent solution (along with lots and lots of other valid solutions) in a fairly short amount of time. Although if you wish the find the optimum solution every time, you are going to have to brute force it in all likelihood (I have only skimmed the problem so this may not be the case but it sounds like it probably is).

In your example, you would have a solution class, this represents a solution, IE a line-up for the baseball team. You randomly generate say 20 solutions, regardless if they are valid or not, then you have a rating algorithm that rates the solution. In your case, a better player in the line-up would score more than a worse player, and any invalid line-ups (for whatever reason) would force a score of 0.

Any 0 scoring solutions are killed off, and replaced with new random ones, and the rest of the solutions breed together to form new solutions. Theoretically and after enough time the pool of solutions should improve.

This has the benefit of not only finding lots of valid unique line-ups, but also rating them. You didn't specify in your problem the need to rate the solutions, but it offers plenty of benefits (for example if a player is injured, he can be temporarily rated as a -10 or whatever). All other players score based on their quantifiable stats.

It's scalable and performs well.