问题
I have a matching problem, which I think is np-hard:
We need to arrange a dinner for a group of n people, some of the people are friends with eachother, some are not. We need to sit everyone at a table, but they should never sit with someone that they are not friends with. There are *k* tables K = r1+r2+···+rk=n.
**Input**
input is formatted as one first line k, then follow one line of k numbers, where each number represents a table and it's capacity. Then comes n lines of people, where we can see friendships of person i. All friendships are mutual.
**Output**
Output the formations of people that can be seated together, without having to sit with someone that they are not friends with
example:
Input:
2
3, 3
Alice: Bob, Claire, Eric, Juliet, Romeo
Bob: Alice, Claire, Juliet, Romeo
Claire: Alice, Bob, Juliet
Eric: Alice, Romeo
Juliet: Alice, Bob, Claire
Romeo: Alice, Bob, Eric
Output:
Romeo, Eric, Alice
Bob, Claire, Juliet
Im fairly certain that this problem is np-complete, but I am having some problems finding a proper reduction. The example corresponds to the following (badly drawn)graph:
I have a loose idea around using a complimentary graph to reduce to independent set. but i would be very gratefull for any ideas for solutions
回答1:
Clique problem reduction
First off, note that NP is a class for decision problems, so we'll adjust the question to "is there a table arrangement?" instead of "output the table arrangement". In practice there is of course no real difference.
Now, given a graph, let's say we want to know if there is a clique of at least size k. This is the (decision) clique problem, which is one of the famous NP-complete problems.
This graph will have at least one clique of size k if and only if your matching problem has a solution for the same graph, with a table of size k. The seating for all the others should be unconstrained, so we have n-k one-seat tables.
Thus, we can create an instance of the matching problem that is equivalent to any instance of a known NP-complete problem. This instance is roughly the same size (no exponential blow-up), so this constitutes a reduction, proving that the matching is NP-hard. As it is also (clearly?) in NP, it is also NP-complete.
来源：https://stackoverflow.com/questions/65133548/how-do-i-construct-np-reduction-for-this-matching-problem