In textbook by Vazirani's textbook, chapter 9 about Bin Packing. He give the following lemma.

Lemma 9.4 Let $\epsilon >1$ be fixed, and let K be a fixed nonnegative integer. Consider the restriction of the bin packing problem to instances in which each item is of size at least $\epsilon$ and the number of distinct item sizes is K. There is a polynomial time algorithm that optimally solves this restricted problem.

Proof. The number of items in a bin is bounded by $1/\epsilon$. Denote this by M. Therefore, the number of different bin types is bounded by $R = \binom{M+K}{M}$. Which is a large constant. Clearly, the total number of bins used is at most n. Therefore, the number of possible feasible packings is bouned by $P=\binom{n+R}{R}$, which is polynomial in n. Enumerating them and picking the best packing gives the optimal answer.

Question: We know a fact which is: $\binom{n+k-1}{n}$, now in order to find the number of different bin types we it should be something like $\binom{M+K-1}{M}$ why in the proof it is different?!!

  • $\begingroup$ 1) A binomial coefficient is not a fact. 2) Maybe the term for R is wrong, but as it's constant, the exact value does not matter for the proof as a whole. $\endgroup$ – Raphael Apr 15 '17 at 10:18
  • $\begingroup$ @Raphael No, it does matter, see the textbook and see exercise 9.4 $\endgroup$ – YOUSEFY Apr 15 '17 at 10:20

The bins contain at most $M$ elements rather than exactly $M$ elements. This is why your formula doesn't work. In order to use your formula, we need to allow for one more type of element, a dummy type, and then we get $\binom{M+(K+1)-1}{M} = \binom{M+K}{M}$.

As an example, suppose that there is one type of element (so $K=1$). In that case, a bin can contain $0,\ldots,M$ elements, in total $M+1 = \binom{M+K}{M}$ choices.

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  • $\begingroup$ tricky point in the the proof I didn't notice, Thank you Yuval! $\endgroup$ – YOUSEFY Apr 15 '17 at 11:44

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