Abstract
The recently introduced online Minimum Peak Appointment Scheduling (MPAS) problem is a variant of the online bin-packing problem that allows for deferred decision making. Specifically, it allows for the problem to be split into an online phase where a stream of appointment requests arrive requiring a scheduled time, followed by an offline phase where those appointments are scheduled into rooms. Similar to the bin-packing problem, the aim is to use the minimum number of rooms in the final configuration. This model more accurately captures scheduling appointments than bin packing. For example, a dialysis patient needs to know what time to arrive for an appointment, but does not need to know the assigned station ahead of time.
Previous work developed a randomized algorithm for this problem which achieved an asymptotic competitive ratio of at most 1.5, proving that online MPAS was fundamentally different from the online bin-packing problem. Our main contribution is to develop a new randomized algorithm for the problem that achieves an asymptotic competitive ratio under 1.455, indicating the potential for further progress. This improvement is attained by modifying the process for scheduling appointments to increase the density of the packing in the worst case, along with utilizing the dual of the bin-packing linear programming relaxation to perform the analysis. We also present the first known lower bound of 1.2 on the asymptotic competitive ratio of both deterministic and randomized online MPAS algorithm. These results demonstrate how deferred decision-making can be leveraged to yield improved worst-case performance, a phenomenon which should be investigated in a broader class of settings.
This research is supported in part by grants NSF/FDA SIR IIS-1935809, NSF CCF-1740822, NSF DMS-1839346, and NSF CCF-1522054.
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References
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Appendices
A Particularly Hard Input
It is possible to show that the asymptotic competitive ratio derived for the given algorithm is nearly the best possible. For any integer n, construct an input to the MPAS problem containing 2n items of size 0.3438, n items of size 0.2501, and n items of size 0.0623. Since this input has no Large or Third Items, the algorithm will pack this input in the same manner regardless of the order items arrive. The optimal way to pack these items is to put two of the items of size 0.3438 in a bin with one of each of the other items, requiring exactly n bins. However, as n goes to infinity, the algorithm will pack sets of two items of size 0.3438 to a bin, sets of 11 items of size 0.2501 to 4 bins, and sets of 54 items of size 0.0623 to 4 bins. This will result in the algorithm using roughly \(n(1 + 4/11 + 4/54) = n\frac{427}{297}\) bins, for a competitive ratio of \(\frac{427}{297} > 1.437\).
B Configuration Strategies
This section will detail a few important configuration strategies, which are used implicitly in the following section.
1.1 A.1 Alternating Sides
Every complete set will have an even number of type 2 bins, and in the following section half of them will be on either side. Further, when rematching with Large Items, half of the Large Items will be on each side.
1.2 B.2 Type 2 Small Bins
When rematching a Small Item category with Large or Third Items, any type 2 small bins within the complete set of bins will be placed in a bin with a Large Item or inner bin respectively.
1.3 C.3 Rematching with Different Large Item Categories
Several of the Small Item categories will have different rematching guidelines depending on the size of the Large Items (when rematching with only Large Items). In every such case, the smaller size will have every type 2 large bin assigned to the same bin as a Large Item, while the larger case will have type 2 large bins assigned to a bin with another type 2 large bin.
1.4 D.4 Rematching with Quarter Item Sets
When rematching Small Item sets with a Quarter Item set and some other items, any type 2 large bins in the Small Item set will be rematched with the type 2 large Quarter Item bins if possible. This will leave the Quarter Item type 2 small bins to be rematched with Large Items or inner bins.
1.5 E.5 Rematching with Large Items
The properties the above analysis uses rely on there being 0.6875 Large Items for every bin after a rematching with a Small Item category, with an average fullness of at least 0.6875. In what follows, what will be shown is how to rematch so there are at least two Large Items per 3 bins, and an average density of at least 0.7. This is an equivalent statement, since for every 15 rematched bins an extra Large Item in its own bin can be assigned, leading to 11 Large Items for 16 bins with an average fullness of at least 11/16.
C Category List and Matching Process Description
This section will be dedicated to going through each Small Item category, detailing the manner in which they are packed, and verifying all relevant properties used above for the category.
1.1 F.6 Very Large Items
Very Large Items will only have one category, and do not get rematched.
1.2 G.7 Large Items
The interior cutoffs for the Large Item categories will be:
\(\frac{2}{3}, 0.642, 0.635, 0.625, 0.6, 0.588, 0.57, \frac{6}{11}, 0.54\)
Rematching with Large Items is detailed in the relevant item category.
1.3 H.8 Medium Items
The interior cutoffs for the Medium Item categories will be:
\(0.358, 0.365, 0.375, 0.4, 0.412, 0.43, \frac{5}{11}, 0.46\)
Medium Items rematch with Large Items by being placed in the same bin as a Large Item on the opposite side, if they both fit in a bin.
Two Non-half Medium Items can rematch with a Quarter Item set and 8 Large Items. The two Medium Items are placed in a bin with the two Quarter Item type 2 large bins. The Quarter Item type 1 bin is placed in its own bin. The 4 Quarter Item type 2 small bins are placed in bins with Large Items, and the remaining 4 Large Items are placed in their own bin. This process will require 20 bins, and lead to an average fullness of at least \((0.34375 \cdot 2 + 0.25 \cdot 11 + 0.54 \cdot 8)/11 > 0.7\) over the bins. This process requires the Large Items to not fit in a bin with the relevant Medium Items remaining (otherwise this rematching would not happen in the algorithm’s execution).
1.4 I.9 Third Items
Third Items will only have one category.
Third Item sets can be rematched with 6 Large Items. Four left and right Large Items will be placed in the same bins as right and left outer bins respectively. The remaining 2 Large Items will be placed alone in a bin, as will the two inner bins. This process will require 8 bins, and lead to an average fullness of at least \(((1/3)\cdot 8 + .5 \cdot 6)/8 > 0.7\) over the bins. This process requires there to be Large Items which fit in a bin with the relevant Third Items remaining.
Third Item sets can be rematched with 1 Quarter Item set and 14 Large Items. Two left and right Quarter Item type 2 large bins will be placed in the same bins as right and left outer bins respectively. The 4 type 2 small Quarter Item bins will be placed in a bin with a Large Item. The Quarter Item type 1 bins will remain in their own bin, as will the inner bins. The remaining two outer bins will be placed in their own bin. The remaining 10 Large Items will be placed alone in a bin. This process will require 20 bins, and lead to an average fullness of at least \(((1/3)\cdot 8 + 0.25 \cdot 11 + (2/3) \cdot 14)/20 > 0.7\) over the bins. This process requires the Large Items to not fit in a bin with the relevant Third Items remaining (otherwise this rematching would not happen in the algorithm’s execution).
1.5 J.10 Small Items
Sup-Category 3 (0.3125, 1/3]. A Type 2 Large Bin for this category will contain 1 item.
A Type 1 Bin for this category will contain 3 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 3 bins for an average fullness of at least \((0.3125 \cdot 8)/3 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 2/3 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + 0.3125 \cdot 8)/6 \ge 0.7\)
6 Large Items over size 2/3 in 9 bins, for an average fullness of at least \((\frac{2}{3} \cdot 6 + 0.3125 \cdot 8)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.3125 \cdot 8)/11 \ge 0.6875\).
This category will rematch with 0 Quarter Items sets and satisfy the quarter size reallocation property.
Category 3 (0.27, 0.3125]. A Type 2 Small Bin for this category will contain 1 item.
A Type 1 Bin for this category will contain 3 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
3 Type 1 Bins Unmatched
Filling 5 bins for an average fullness of at least \((0.27 \cdot 13)/5 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
6 Large Items in 9 bins, for an average fullness of at least \((0.5 \cdot 6 + 0.27 \cdot 13)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.27 \cdot 13)/11 \ge 0.6875\).
This category will rematch with 0 Quarter Items sets and satisfy the quarter size reallocation property.
Quarter Items (0.25, 0.27]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 3 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Small Bins Matched with 2 Type 2 Large Bins Pairwise
-
1 Type 1 Bin Unmatched
Filling 4 bins for an average fullness of at least \((0.25 \cdot 11)/4 = 0.6875\).
This category rematches in a unique way with almost every other item type, explained where relevant. When matched directly with Large Items, Quarter Item and Large Item bins are grouped together, with no items being rematched into new bins.
Sup-Category 4 (0.23, 0.25]. A Type 2 Small Bin for this category will contain 1 item.
A Type 1 Bin for this category will contain 4 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \((0.23 \cdot 12)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + 0.23 \cdot 12)/6 \ge 0.7\)
2 Third Item sets (16 Third Items) in 10 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.23 \cdot 12)/10 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + 0.23 \cdot 12)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 17 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + 0.23 \cdot 12)/17 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/17 \ge 0.6875\).
Category 4 (0.215, 0.23]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
4 Type 2 Large Bins Matched Pairwise
Filling 3 bins for an average fullness of at least \((0.215 \cdot 10)/3 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
6 Large Items under size 0.54 in 6 bins, for an average fullness of at least \((0.5 \cdot 6 + 0.215 \cdot 10)/6 \ge 0.7\)
4 Large Items over size 0.54 in 6 bins, for an average fullness of at least \((0.54 \cdot 4 + 0.215 \cdot 10)/6 \ge 0.7\)
2 Third Item sets (16 Third Items) in 10 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.215 \cdot 10)/10 \ge 0.6875\)
8 Large Items & 1 Quarter Item set in 12 bins, for an average fullness of at least \((0.5 \cdot 8 + 0.25 \cdot 11 + 0.215 \cdot 10)/12 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 8)/12 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 16 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + 0.215 \cdot 10)/16 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/16 \ge 0.6875\).
Sub-category 4 (0.206, 0.215]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
6 Type 2 Large Bins Matched Pairwise
Filling 4 bins for an average fullness of at least \((0.206 \cdot 14)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
8 Large Items under size 0.57 in 8 bins, for an average fullness of at least \((0.5 \cdot 8 + 0.206 \cdot 14)/8 \ge 0.7\)
6 Large Items over size 0.57 in 9 bins, for an average fullness of at least \((0.57 \cdot 6 + 0.206 \cdot 14)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.206 \cdot 14)/11 \ge 0.6875\)
14 Large Items & 2 Quarter Item sets in 21 bins, for an average fullness of at least \((0.5 \cdot 14 + 0.25 \cdot 22 + 0.206 \cdot 14)/21 \ge 0.7\)
and satisfying \((0.3125 \cdot 22 + 0.6875 \cdot 14)/21 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 17 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + 0.206 \cdot 14)/17 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/17 \ge 0.6875\).
Sub-Sub-Category 4 (0.2, 0.206]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
6 Type 2 Large Bins Matched Pairwise
Filling 4 bins for an average fullness of at least \((0.2 \cdot 14)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
8 Large Items under size 0.588 in 8 bins, for an average fullness of at least \((0.5 \cdot 8 + 0.2 \cdot 14)/8 \ge 0.7\)
6 Large Items over size 0.588 in 9 bins, for an average fullness of at least \((0.588 \cdot 6 + 0.2 \cdot 14)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.2 \cdot 14)/11 \ge 0.6875\)
14 Large Items & 2 Quarter Item sets in 21 bins, for an average fullness of at least \((0.5 \cdot 14 + 0.25 \cdot 22 + 0.2 \cdot 14)/21 \ge 0.7\)
and satisfying \((0.3125 \cdot 22 + 0.6875 \cdot 14)/21 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 17 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + 0.2 \cdot 14)/17 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/17 \ge 0.6875\).
Sup-Category 5 (0.1825, 0.2]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 5 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
4 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 5 bins for an average fullness of at least \((0.1825 \cdot 20)/5 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
6 Large Items under size 0.6 in 8 bins, for an average fullness of at least \((0.5 \cdot 6 + 0.1825 \cdot 20)/8 \ge 0.7\)
8 Large Items over size 0.6 in 12 bins, for an average fullness of at least \((0.6 \cdot 8 + 0.1825 \cdot 20)/12 \ge 0.7\)
3 Third Item sets (24 Third Items) in 16 bins, for an average fullness of at least \((\frac{1}{3} \cdot 24 + 0.1825 \cdot 20)/16 \ge 0.6875\)
16 Large Items & 2 Quarter Item sets in 24 bins, for an average fullness of at least \((0.5 \cdot 16 + 0.25 \cdot 22 + 0.1825 \cdot 20)/24 \ge 0.7\)
and satisfying \((0.3125 \cdot 22 + 0.6875 \cdot 16)/24 \ge 0.6875\)
5 Third Item sets (40 Third Items) & 2 Quarter Item sets in 28 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 40 + 0.25 \cdot 22 + 0.1825 \cdot 20)/28 \ge 0.6875\) and satisfying \((0.3125 \cdot 22 + 0.34375 \cdot 40)/28 \ge 0.6875\).
Category 5 (0.179, 0.1825]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 5 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
10 Type 2 Large Bins Matched Pairwise
-
6 Type 1 Bins Unmatched
Filling 13 bins for an average fullness of at least \((0.179 \cdot 54)/13 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
14 Large Items under size 0.635 in 20 bins, for an average fullness of at least \((0.5 \cdot 14 + 0.179 \cdot 54)/20 \ge 0.7\)
22 Large Items over size 0.635 in 33 bins, for an average fullness of at least \((0.635 \cdot 22 + 0.179 \cdot 54)/33 \ge 0.7\)
7 Third Item sets (56 Third Items) in 39 bins, for an average fullness of at least \((\frac{1}{3} \cdot 56 + 0.179 \cdot 54)/39 \ge 0.6875\)
42 Large Items & 5 Quarter Item sets in 63 bins, for an average fullness of at least \((0.5 \cdot 42 + 0.25 \cdot 55 + 0.179 \cdot 54)/63 \ge 0.7\)
and satisfying \((0.3125 \cdot 55 + 0.6875 \cdot 42)/63 \ge 0.6875\)
12 Third Item sets (96 Third Items) & 5 Quarter Item sets in 69 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 96 + 0.25 \cdot 55 + 0.179 \cdot 54)/69 \ge 0.6875\) and satisfying \((0.3125 \cdot 55 + 0.34375 \cdot 96)/69 \ge 0.6875\).
Sub-category 5 (1/6, 0.179]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 5 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
10 Type 2 Large Bins Matched Pairwise
-
6 Type 1 Bins Unmatched
Filling 13 bins for an average fullness of at least \(((1/6) \cdot 54)/13 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
14 Large Items under size 0.642 in 20 bins, for an average fullness of at least \((0.5 \cdot 14 + (1/6) \cdot 54)/20 \ge 0.7\)
22 Large Items over size 0.642 in 33 bins, for an average fullness of at least \((0.642 \cdot 22 + (1/6) \cdot 54)/33 \ge 0.7\)
7 Third Item sets (56 Third Items) in 39 bins, for an average fullness of at least \((\frac{1}{3} \cdot 56 + (1/6) \cdot 54)/39 \ge 0.6875\)
36 Large Items under size 0.642 & 4 Quarter Item sets in 54 bins, for an average fullness of at least
\((0.5 \cdot 36 + 0.25 \cdot 44 + (1/6) \cdot 54)/54 \ge 0.7\) and satisfying \((0.3125 \cdot 44 + 0.6875 \cdot 36)/54 \ge 0.6875\)
42 Large Items over size 0.642 & 5 Quarter Item sets in 63 bins, for an average fullness of at least
\((0.642 \cdot 42 + 0.25 \cdot 55 + (1/6) \cdot 54)/63 \ge 0.7\) and satisfying \((0.3125 \cdot 55 + 0.6875 \cdot 42)/63 \ge 0.6875\)
12 Third Item sets (96 Third Items) & 5 Quarter Item sets in 69 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 96 + 0.25 \cdot 55 + (1/6) \cdot 54)/69 \ge 0.6875\) and satisfying \((0.3125 \cdot 55 + 0.34375 \cdot 96)/69 \ge 0.6875\).
Sup-Category 6 (0.15625, 1/6]. A Type 2 Small Bin for this category will contain 1 item.
A Type 2 Large Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 6 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \((0.15625 \cdot 18)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 2/3 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + 0.15625 \cdot 18)/6 \ge 0.7\)
6 Large Items over size 2/3 in 9 bins, for an average fullness of at least \(((2/3) \cdot 6 + 0.15625 \cdot 18)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + 0.15625 \cdot 18)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + 0.15625 \cdot 18)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 17 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + 0.15625 \cdot 18)/17 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/17 \ge 0.6875\).
Category 6 (1/7, 0.15625]. A Type 2 Small Bin for this category will contain 2 items.
A Type 1 Bin for this category will contain 6 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \(((1/7) \cdot 20)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + (1/7) \cdot 20)/6 \ge 0.7\)
2 Third Item sets (16 Third Items) in 10 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/7) \cdot 20)/10 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/7) \cdot 20)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
3 Third Item sets (24 Third Items) & 1 Quarter Item set in 17 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 24 + 0.25 \cdot 11 + (1/7) \cdot 20)/17 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 24)/17 \ge 0.6875\).
Category 7 (1/8, 1/7]. A Type 2 Small Bin for this category will contain 2 item.
A Type 2 Large Bin for this category will contain 3 items.
A Type 1 Bin for this category will contain 7 items.
A Complete Set of Matched Bins for this category will contain
-
4 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 5 bins for an average fullness of at least \(((1/8) \cdot 28)/5 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
6 Large Items in 9 bins, for an average fullness of at least \((0.5 \cdot 6 + (1/8) \cdot 28)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/8) \cdot 28)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/8) \cdot 28)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + (1/8) \cdot 28)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
Category 8 (1/9, 1/8]. A Type 2 Small Bin for this category will contain 2 item.
A Type 2 Large Bin for this category will contain 3 items.
A Type 1 Bin for this category will contain 8 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \(((1/9) \cdot 26)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 5/8 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + (1/9) \cdot 26)/6 \ge 0.7\)
6 Large Items over size 5/8 in 9 bins, for an average fullness of at least \(((5/8) \cdot 6 + (1/9) \cdot 26)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/9) \cdot 26)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/9) \cdot 26)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + (1/9) \cdot 26)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
Category 9 (1/10, 1/9]. A Type 2 Small Bin for this category will contain 2 item.
A Type 2 Large Bin for this category will contain 3 items.
A Type 1 Bin for this category will contain 9 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \(((1/10) \cdot 28)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 2/3 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + (1/10) \cdot 28)/6 \ge 0.7\)
6 Large Items over size 2/3 in 9 bins, for an average fullness of at least \(((2/3) \cdot 6 + (1/10) \cdot 28)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/10) \cdot 28)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/10) \cdot 28)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + (1/10) \cdot 28)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
Category 10 (1/11, 1/10]. A Type 2 Small Bin for this category will contain 3 item.
A Type 2 Large Bin for this category will contain 4 items.
A Type 1 Bin for this category will contain 10 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \(((1/11) \cdot 34)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 0.6 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + (1/11) \cdot 34)/6 \ge 0.7\)
6 Large Items over size 0.6 in 9 bins, for an average fullness of at least \((0.6 \cdot 6 + (1/11) \cdot 34)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/11) \cdot 34)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/11) \cdot 34)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + (1/11) \cdot 34)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
Category 11 (1/12, 1/11]. A Type 2 Small Bin for this category will contain 3 item.
A Type 2 Large Bin for this category will contain 5 items.
A Type 1 Bin for this category will contain 11 items.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \(((1/12) \cdot 38)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 6/11 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + (1/12) \cdot 38)/6 \ge 0.7\)
6 Large Items over size 6/11 in 9 bins, for an average fullness of at least \(((6/11) \cdot 6 + (1/12) \cdot 38)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least \((\frac{1}{3} \cdot 16 + (1/12) \cdot 38)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least \((0.5 \cdot 10 + 0.25 \cdot 11 + (1/12) \cdot 38)/15 \ge 0.7\)
and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + (1/12) \cdot 38)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
Category 12 [0, 1/12]. This category will be slightly different, since it includes all items of size at most 1/12. So each bin type will not specify a number of items, but rather a cutoff a which point items will stop being added. For example, a Type 1 bin will be until 11/12 full, which means that it will not be “full” and able to accept more items until it is 11/12ths full, at which point it may not be able to accept another item without going over the limit and is thus at capacity.
A Type 2 Small Bin for this category will be filled until 1/4 full.
A Type 2 Large Bin for this category will be filled until 113/300 full.
A Type 1 Bin for this category will be filled until 11/12 full.
A Complete Set of Matched Bins for this category will contain
-
2 Type 2 Small Bins Matched Pairwise
-
2 Type 2 Large Bins Matched Pairwise
-
2 Type 1 Bins Unmatched
Filling 4 bins for an average fullness of at least \((((1/4) + (113/300) + (11/12)) \cdot 2)/4 \ge 0.6875\).
A Complete Set of Rematched Bins can be obtained by rematching with
4 Large Items under size 0.54 in 6 bins, for an average fullness of at least \((0.5 \cdot 4 + ((1/4) + (113/300) + (11/12)) \cdot 2)/6 \ge 0.7\)
6 Large Items over size 0.54 in 9 bins, for an average fullness of at least \((0.54 \cdot 6 + ((1/4) + (113/300) + (11/12)) \cdot 2)/9 \ge 0.7\)
2 Third Item sets (16 Third Items) in 11 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 16 + ((1/4) + (113/300) + (11/12)) \cdot 2)/11 \ge 0.6875\)
10 Large Items & 1 Quarter Item set in 15 bins, for an average fullness of at least
\((0.5 \cdot 10 + 0.25 \cdot 11 + ((1/4) + (113/300) + (11/12)) \cdot 2)/15 \ge 0.7\) and satisfying \((0.3125 \cdot 11 + 0.6875 \cdot 10)/15 \ge 0.6875\)
4 Third Item sets (32 Third Items) & 1 Quarter Item set in 21 bins, for an average fullness of at least
\((\frac{1}{3} \cdot 32 + 0.25 \cdot 11 + ((1/4) + (113/300) + (11/12)) \cdot 2)/21 \ge 0.6875\) and satisfying \((0.3125 \cdot 11 + 0.34375 \cdot 32)/21 \ge 0.6875\).
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Smedira, D., Shmoys, D. (2022). Scheduling Appointments Online: The Power of Deferred Decision-Making. In: Chalermsook, P., Laekhanukit, B. (eds) Approximation and Online Algorithms. WAOA 2022. Lecture Notes in Computer Science, vol 13538. Springer, Cham. https://doi.org/10.1007/978-3-031-18367-6_5
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DOI: https://doi.org/10.1007/978-3-031-18367-6_5
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