Tight Bounds for Online Class-constrained Packing

1. Tight Bounds for Online Class-constrained Packing Hadas Shachnai Bell Labs and The Technion IIT Tami Tamir The Technion IIT

2. Problem Definition We need to packitems into bins. • All items have the same (unit) size. Each item has a color (type). • All the bins have the same capacity. • Each bin can accommodate items from a bounded number of colors.

3. Example of class-constrained packing n=15, M=5 v = 5, c = 2 Notations n - number of items in the instance. M - number of distinct colors in the instance. v - bin’s capacity. c- number of compartments in a bin. (The bin can accommodate v items of c distinct colors)

4. Optimization Goals: Class-constrained bin-packing (CCBP): Pack all the items in a minimal number of bins. Class-constrained multiple knapsack (CCMK): Given m knapsacks (bins), pack as many items as possible in the bins. For CCBP, max( n/v , M/c ) is a lower bound for the number of required bins. Performance measure (for CCBP): For an instance I, let Noptdenote the number of bins used by an optimal algorithm to pack all the items of I. An algorithm is r-competitive if it packs all the items of I at most r·Nopt bins.

5. Applications Multimedia on Demand Systems: The system receives requests for broadcasts of M movies. The requested movie should be transmitted by a shared disk. Each disk has limited load capacity, v, and limited storage capacity, c. c=2 v=5 Production Planning: Each device possesses some amount, v, of a shared resource and can be set to c distinct configurations. There are M distinct products.

6. Online Class-constrained Packing : • The items arrive one at a time. In each step we get one unit size item of some color. • We need to pack this item without any knowledge of the subsequent items. • Formally, the instance is given as a sequence, =a1,a2,.. of length n,such that k, ak{1,..,M}. • Online CCBP: use a minimal number of bins to pack all the items in . • Online CCMK: Maximize the number of packed items in (arriving items may be rejected).

7. Possible optimal packing Online Class-constrained Packing, An Example of CCBP. First-fit Algorithm: Put an arriving item in the leftmost bin that can accommodate it.  = v = 5 c = 2 First-fit packing

8. Results :Deterministic Algorithms for CCBP • For a sequence, , of length n, let k = M/c -1 and h = n/v -1. • Sc,v(k,h) = the set of input sequences with k,h. • Lower Bound: For any deterministic algorithm, rd(Sc,v(k,h))  1+ (k+1- kc+1/v)/(h=1). (asymptotically, rd2). • This bound is achieved by first-fit. • Another simple algorithm (color sets), achieves r2. • An a-priori knowledge of n, M does not help.

9. bad bins Lower Bound for Deterministic Algorithms Theorem: for any deterministic algorithm A,rA 2. Proof:The adversary constructs the sequence, , online such that some bins include v items from few colors and some bins include c items from c distinct colors. v = 10 c = 3 The idea: repeat in  items of the same color. Switch to the next color whenever an item is packed in one of the ‘bad bins’. This color will not be repeated anymore in  .

10. The packing of the algorithm An optimal packing: The rear colors spread among the bins A closer analysis of the adversary’s strategy yields a lower bound that depends on the ratios n/v and M/c.

11. Upper Bound for First-fit Theorem:rff2. Proof: LetS1be the set of full bins (v items);S2is the set of occupied bins (c distinct colors). Claim 1:Each bin (except maybe for the last one) is either full or occupied. Claim 2: Each bin in S2 (except maybe for the last one) contains the last item of each of its c colors. (any additional appearance of a color can fit into this bin). From Claim 1, Nff = |S1|+|S2|. From Claim 2, |S2|M/c.Also,|S1|n/v. Since Nopt max( n/v , M/c ) we get a 2-approximation.

12.  = r= v/c Next-fit packing ( 15/2 bins) Optimal packing (3 bins) Next-fit Algorithm Next-fit algorithm: Put an arriving item in the currently active bin. Open a new bin if the active bin cannot accommodate this item. For traditional online bin-packing, it is known that Next-fit uses as most 2·NOPT bins. This is not the case for class-constrained packing: v=5 c=2

13. Class-constrained Packing of Temporary Items • Items are packed for a bounded time interval (that is unknown in advance) • The sequence consists of • arrival of an item a of color i{1,..,M}. • departure of an item that was packed earlier. • We consider two models: • A departure is associated with a specific item (of specific color, placed in specific bin). • A departure is associated with a color; thus, we may choose which item of that color to remove.

14. Results for CCBP of Temporary Items • For any deterministic algorithm, r  v/c. • For all any-fit algorithms, r  v/c (An any-fit algorithm opens a new bin only when no open bin can accommodate the new item). • When departure events are associated with colors, for all any fit algorithms r  min(v/c, c-1). • The color-sets algorithm has r=v (the worst possible). • An a-priori knowledge of n, M does not help.

15. The Online Class-Constrained Multiple Knapsack Problem Performance measure: number of items packed by the algorithm A number of items packed by an optimal algorithm rA() = • OurResults: • For any v  c, and any deterministic algorithm, Ad, for CCMK, rd c/v. • This ratio is achieved by any greedy algorithm (reject only items that cannot fit to any knapsack). • For any v  c, any deterministic algorithm, Ad, for temporary CCMK, and any 0, rd.

16. Related Work • Online bin packing (items with arbitrary sizes) • rff1.7 (Johnson and Demers, 1974) • rA 1.54 for deterministic algorithms (Van Vliet,1996) • Offline class-constrained packing (Shachnai and Tamir, 2001). • Data placement (Golubchik, Khanna, Khuller, Thurimella, and Zhu, 2000). • Scheduling tasks with unknown durations (Azar, Broder and Karlin, 1994).

17. Possible optimal packing The Color-sets Algorithm The algorithm: Partition (online) the M colors in  into M/ccolor-sets. Pack the items of each color-set greedily. Theorem: rcs < 2  = Color-sets v = 5 c = 2 color-sets packing