Question

Given a scenario where we have multiple lists of pairs of items, for example:

1. {12,13,14,23,24}
2. {14,15,25}
3. {16,17,25,26,36}

where 12 is a pair of items ‘1’ and ‘2’ (and thus 21 is equivalent to 12), we want to count the number of ways that we can choose pairs of items from each of the lists such that no single item is repeated. You must select one, and only one pair, from each list. The number of items in each list and the number of lists is arbitrary, but you can assume there are at least two lists with at least one pair of items per list. And the pairs are made from symbols from a finite alphabet, assume digits [1-9]. Also, a list can neither contain duplicate pairs {12,12} or {12,21} nor can it contain symmetric pairs {11}.

More specifically, in the example above, if we choose the pair of items 14 from the first list, then the only choice we have for the second list is 25 because 14 and 15 contain a ‘1’. And consequently, the only choice from the third list is 36 because 16 and 17 contain a ‘1’, and 25 and 26 contain a ‘2’.

Does anyone know of an efficient way to count the total combinations of pairs of items without going through every permutation of choices and asking “is this a valid selection?”, as the lists can each contain hundreds of pairs of items?

---

UPDATE

After spending some time with this, I realized that it is trivial to count the number of combinations when none of the lists share a distinct pair. However, as soon as a distinct pair is shared between two or more lists, the combinatorial formula does not apply.

As of now, I’ve been trying to figure out if there is a way (using combinatorial math and not brute force) to count the number of combinations in which every list has the same pairs of items. For example:

1. {12,23,34,45,67}
2. {12,23,34,45,67}
3. {12,23,34,45,67}

Answer 1

The problem is #P-complete. This is even HARDER than NP-complete. It is as hard as finding the number of satisfying assignments to an instance of SAT.

The reduction is from Perfect matching. Suppose you have the graph G = {V, E} where E, the set of edges, is a list of pairs of vertices (those pairs that are connected by an edge). Then encode an instance of “pairs of items” by having |V|/2 copies of E. In other words, have a number of copies of E equal to half of the number of vertices. Now, a “hit” in your case would correspond to |V|/2 edges with no repeated vertices, implying that all |V| vertices were covered. This is the definition of a perfect matching. And every perfect matching would be a hit — it’s a 1-1 correspondence.