# Proof that bin packing is strongly NP-complete?

Wikipedia claims that bin packing is strongly NP-complete. Unfortunately I haven't been able to find a proof. Does anyone know where to find it?

## 1 Answer

Such a proof is outlined in Garey and Johnson, 'Strong' NP-Completeness Results: Motivation, Examples, and Implications, mentioned in the Wikipedia article. The proof is by a straightforward reduction from 3-PARTITION. The problem 3-PARTITION one is given $3n$ numbers, and the task is to decide whether they can be split into triples of equal sum. There is a straightforward reduction from 3-PARTITION to BIN-PACKING: if the sum of the numbers is $nB$, ask whether the $3n$ numbers can be packed into $n$ bins of size $B$.

The strong NP-completeness of 3-PARTITION was proved in a different paper of Garey and Johnson, Complexity results for multiprocessor scheduling under resource constraints. The proof is quite long, ultimately a reduction from 3DM (3-dimensional matching), but going through many intermediate steps.