Assigning people to buildings while respecting preferences?

Question

A friend asked me a question today about an assignment problem. I found a quite straightforward solution, but I feel that it can be made simpler and faster. Your help would be a

Answer 1

This problem is known to be NP-hard by a reduction from the NP-complete problem of decomposing a graph into cliques (the clique cover problem).

First, some terminology and discussion. A clique in a graph is a set of k different nodes such that each node is connected to each other node in the clique. An independent set in the graph is a set of k different nodes such that no two nodes are connected to one another. If you take any graph G and then introduce edges whenever an edge is missing and remove all edges that previously existed, you get the complement graph of the original graph. This means that the problem of finding a clique in an initial graph is equivalent to finding an independent set in the complement graph.

The reason this is interesting is that in the problem you're describing, you are given a graph of people where each edge indicates "these two people can't be housed together." Consequently, you can think of the problem you're describing as trying to find a way to break the graph up into k subgraphs, each of which is an independent set. If you construct the complement of this graph, you get the graph of all people who are okay being placed together. In that case, you want to break the group up into k groups that are all cliques.

It is known that the following problem is NP-complete:

Given a graph and a number k, can you break the nodes in the graph apart into k cliques?

We can reduce this problem to your problem in polynomial time, showing that your problem is NP-hard. The construction is simple - given the initial graph G and number k, first construct the complement of G. This means that if you can break apart the complement into k independent sets, then the original graph G can be broken apart into k cliques (because of the duality between cliques and independent sets). Now, take this new graph and convert it into a set of people, one per node, where two people cannot be placed into the same room as one another if their nodes were connected in the original graph. Now, there is a way to distribute these people into k houses iff the complement of G can be broken down into k independent sets iff G can be broken down into k cliques. Consequently, the known NP-complete problem of clique cover can be reduced to your problem in polynomial time, so your problem is NP-hard. To ensure that any independent set can be fit into a house, just set the maximum capacity of each room to n, the number of nodes in the graph. This allows any independent set to be housed in any room.

Since your problem is NP-hard, there cannot be a polynomial-time solution to it unless P = NP. Consequently, as best as anyone knows, there is no polynomial time algorithm for it and your backtracking recursion very well might be the optimal solution.

Hope this helps!