Simple, non-trivial bin-packing instance

Question

Bin packing problem is to find the minimal number of bins of size v, which can contain all objects of size [s_1, s_2, s_3, ..., s_n]

I'm searching for a simple, non-trivial instance of the bin-packing problem.

A simple instance is an instance which can be solved with no more than 5 bins.

A non-trivial instance is an instance, which can't be solved by the best-fit-decreasing heuristic algorithm, but can be solved with complete search.

For example, the instance v = 20, objects = [15, 7, 14, 3, 14, 7, 9] is simple, but not non-trivial, because complete search proves that the minimal number of bins is 5:

[[15, 3], [7, 7], [14], [14], [9]]

however, best-fit heuristic also produces a 5-bin packing:

[[15], [14], [14], [9, 7, 3], [7]]

Does a simple, non-trivial instance of bin packing exist?

Why not write a program that implements both algorithms and do an exhaustive search for it? Otherwise this sounds more like a Mathematics proof problem, in which case you should post it on https://math.stackexchange.com — Dijkgraaf, Nov 20 '17 at 23:20
thanks @Dijkgraaf, that's what I've ended up doing (exhaustive search over randomised inputs). — Adam Kurkiewicz, Nov 21 '17 at 00:32
Ask yourself what would be needed in your example to make a simple algorithm miss the solution. I assume that 2 items per bin is too easy; if you have a 15 and a 3, there's no reason not to put them together. A larger example, where you have 3 items or more per bin, is more likely to be problematic; say (14,10,10,7,7,3,3,2,2,2); if you add the 3's to the 14 instead of the 2's, you can't do 3 bins. — m69's been on strike for years, Nov 21 '17 at 00:35
@m69 your example is invalid. Both best-fit-decrasing and optimal solutions are 3 bin solutions: `[[14, 3, 3], [10, 10], [7, 7, 2, 2, 2]]` `[[14, 2, 2, 2], [10, 10], [7, 7, 3, 3]]` — Adam Kurkiewicz, Nov 21 '17 at 00:38
Yes, that's the optimal solution, found using complete search. But this is a trivial instance, best-fit-decreasing also finds an optimal packing: `[[14, 3, 3], [10, 10], [7, 7, 2, 2, 2]]`` — Adam Kurkiewicz, Nov 21 '17 at 00:41
You're right. I though b-f-d would do [14,3,3] [10,7,2] [10,7,2] and be stuck with the last 2. — m69's been on strike for years, Nov 21 '17 at 00:41
@m69 I've found it's generally hard to find an example by hand — Adam Kurkiewicz, Nov 21 '17 at 00:47
Indeed, I was. How about 16,14,12,5,3,3,3,2,2 ? BFD will do 14+5, 16+3, and then be left with 12+3+3+2 and a seperate 2. It's not that hard to force BFD to create small, unusable leftover spaces. — m69's been on strike for years, Nov 21 '17 at 00:50
It very well can be a tight lowest bound on number of elements. My search procedure doesn't terminate for k<6. — Adam Kurkiewicz, Nov 21 '17 at 00:56

score 2 · Accepted Answer · answered Nov 21 '17 at 00:35

2

Indeed, such instance exists, namely:

v = 20, objects = [11, 7, 7, 6, 5, 3, 1]

Best-fit-decreasing heuristic gives: [[11, 7], [7, 6, 5, 1], [3]]

Optimal packing is: [[11, 6, 3], [7, 7, 5, 1]]

answered Nov 21 '17 at 00:35

Adam Kurkiewicz

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v = 7, objects = [3, 3, 2, 2, 2, 2] works nicely too – Adam Kurkiewicz Nov 23 '17 at 14:38

Simple, non-trivial bin-packing instance

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