many-to-many and many-to-many intersections

Question

Say I have a database that has people, grocery stores, and items you can buy in the store, like so:

Stores               People                Foods
-----------------    ------------------    ------------------
| id |   name   |    | id |   name    |    | id |   name    |
-----------------    ------------------    ------------------
| 1  |  Giant   |    | 1  | Jon Skeet |    | 1  | Tomatoes  |
| 2  |  Vons    |    | 2  | KLee1     |    | 2  | Apples    |
| 3  | Safeway  |    ------------------    | 3  | Potatoes  |
-----------------                          ------------------

I have an additional table which keep track of which stores sell what:

Inventory
--------------------
| store_id| food_id|
--------------------
| 1       | 1      |
| 1       | 2      |
| 2       | 1      |
| 3       | 1      |
| 3       | 2      |
| 3       | 3      |
--------------------

And I have another table that has shopping lists on it

Lists
---------------------
| person_id| food_id|
---------------------
| 1        | 1      |
| 1        | 2      |
| 1        | 3      |
| 2        | 1      |
| 2        | 3      |
---------------------

My question is, given a person, or their id, what is the best way to figure out what stores they can go to so they will get everything on their list. Is there a pattern for these types of computations in MySQL?

My attempt (very ugly and messy) is something like:

-- Given that _pid is the person_id we want to get the list of stores for.

SELECT stores.name, store_id, num, COUNT(*) AS counter
FROM lists
    INNER JOIN inventory 
        ON (lists.food_id=inventory.food_id)
    INNER JOIN (SELECT COUNT(*) AS num
            FROM lists WHERE person_id=_pid 
            GROUP BY person_id) AS T
    INNER JOIN stores ON (stores.id=store_id)
WHERE person_id=_pid 
GROUP BY store_id
HAVING counter >= num;

Thanks for your time!

Edit SQL Fiddle with Data

Answer 1

If I were to solved the problem, I'll join the four tables with their linking column (specifically the foreign keys) then a subquery on the HAVING clause to count the number of items on the list for each person. Give this a try,

SET @personID := 1;

SELECT  c.name
FROM    Inventory a
        INNER JOIN Foods b
            ON a.food_id = b.id
        INNER JOIN Stores c
            ON a.store_id = c.id
        INNER JOIN Lists d
            ON d.food_id = b.id
WHERE   d.person_id = @personID
GROUP BY c.name
HAVING   COUNT(DISTINCT d.food_id) =
     (
        SELECT COUNT(*)
        FROM Lists
        WHERE person_ID = @personID
     )

SQLFiddle Demo

Answer 2

@JohnWoo: why DISTINCT?

Another one...

SET @pid=2;

SELECT store_id, name
FROM inventory 
  JOIN lists ON inventory.food_id=lists.food_id
  JOIN stores ON store_id=stores.id
WHERE person_id=@pid
GROUP BY store_id
HAVING COUNT(*)=(
  SELECT COUNT(*)
  FROM lists 
  WHERE person_id=@pid
);