Parameterized Approximation Scheme for the Multiple Knapsack Problem Yan Lu Klaus Jansen SODA 2009 CPSC669 Term Project—Paper Reading

1. Problem Definition 2. Approximation Scheme – 2.1 Instances with similar capacities – 2.2 General cases Outline

1 Problem Definition

1. Problem Definition 2. Approximation Scheme – 2.1 Instances with similar capacities – 2.2 General instances Outline

2.1 Instances with similar capacities Let be the different capacities in an instance of MKP. Suppose for each capacity,, there are. Approximation algorithm: – (1) Linear program relaxation – (2) Rounding the LP solution – (3) Selecting the Items – (4) Strip packing – (5) Shifting Technique 2 Approximation Scheme

(1) Linear Program Relaxation Main idea: select a fractional x i piece of each item a i, distribute these pieces as even smaller pieces among all bins (fractional). The LP is a relaxation of MKP Could be approximately solved as a max-min resource sharing problem [9], and solution has an objective value of at least 1-3α times OPT(LP), or OPT(A,B)

(2) Rounding the LP solution Define as the piece of item a i assigned to group l. Think large pieces () ) as rectangles with width size(a i ) and height Stack all these rectangles ordered by widths, stack height Divide each stack into parts Let K l be the set of pieces lying in more than 1 parts, notice Remove items corresponding to K l

(2) Rounding the LP solution (cont’d) Let S l,j be the set of items a i that have a piece in part j of stack l. For small pieces, compute the total area allocated to group l Area(l) = Let S 0,j be the set of a i with a small piece in the sum above Set a LP to round over groups: This LP has a feasible solution

(2) Rounding the LP solution (cont’d) Lemma 2.1 The solution can be rounded into another solution where each set has at most items. Note L l is the set of items with fractional pieces in more than 1 groups After rounding, we have a unique assignment of items to groups, i.e. each item is assigned to exactly one group l and one part j.

(3) Selecting the Items In the rounding result, items selected are generally fractional. Select complete items with near optimum profit, by solving classical fractional knapsack problems for each group and each part: – take as size of an item the value 1 and as profit the original profit profit(a i ) – the capacity of the knapsack equals to, i.e. height of a part in Stack l In overall solution A, at most one fractional item per group and part. Let M l be the set of selected fractional items in group l. All other items are selected completely.

(4) Strip packing Lemma 2.2 The set can be packed into bins of capacity c l for each l. Using strip packing algorithm by Kenyon and Remila [10] This implies (using the shifting technique described below) that most of the items (with near optimum profit) can be packed into m l bins. (5) Shifting technique Recall : items removed from LP solution. Let. Then Lemma 2.3 We can select a subset with profit at least that can be packed into m l bins.

Entire algorithm (1) Solve the LP approximately whose objective value is at least (1−3α) times the optimum LP value (2) Build t stacks of wide rectangles and sets with narrow rectangles, split the stacks into 1/δ 2 parts and round the rectangles over the groups. Then select the items via solving fractional knapsack problems and store the fractional items (3) Use the strip packing algorithm by Kenyon and Remila for each group to pack the items into bins (4) Apply the shifting strategy to select subsets that can be packed into m l bins.

2.2 General instances When number of bins smaller than, they presented an approximation scheme with running time From now on, suppose number of bins larger than (1) Modify structure of bins (2) Modify structure of high profit items (3) Overall algorithm 2 Approximation Scheme

(1) Modify the bins Order the bins according to capacities k groups with bins, with One group with bins Lemma 2.4 We can transform the optimum solution for an instance (A, B) such that the k-th group of bins is not used and the profit loss is at most

(2) Modify the high profit items Let APP(A, B) be the profit of the greedy algorithm, Consider only items with large profit where γ is the number of bins in B 2 – Case 1:, at most items in the instance. We could pack all of them in B 2. Denote them with set SmHi – Case 2:, for each rounded profit value, choose at most smallest items with profit round(k) and store in A(k) Guess a subset with high profit items for the bins in B 2, through enumeration. In total, only need to consider choices.

(3) Overall algorithm Compute approximate solution Modify structure of bins Compute the set SmHi and sets A(k) For each subset – Test whether A guess fits into B 2 ; if not, discard it – If yes, take a feasible placement of A guess into B 2 and set up a linear program to select the remaining items (similar to the instances with similar capacities) – Place the selected items into bins Take a solution among all feasible choices A guess with maximum total profit. Total running time

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