In [11], a general packing algorithm is given for theGBP problem, with absolute approximation ratio 3, in the case where the graph which is to be packed is bipartite. Now we will present an improved algorithm with absolute approximation ratio 19/12 + 1 = 31/12 ≈ 2.5833, for this restricted case.
LetG(A, B, E)be the bipartite graph in question, whereV = A∪B is the set of points (also called items) of the bipartite graph, andE is the set of edges. Any pointv ∈V has a size, denoted by0 < s(v) ≤ 1. Moreover for anye(a, b)edge (a ∈ A, b∈ B) an integer lower bound will be denoted by d(a, b) ≥ 0. In this special case of the GBP problem, the goal is as follows: Let us pack the items into as small a number of bins as possible, in such a way that for any two items a ∈ A, b ∈ B, if they are packed into bins Bi andBj, respectively, the indices of their bins must satisfy|i−j| ≥ d(a, b), and the total size of items in any bin cannot exceed1. Let ddenote the maximum of the prescribed lower bounds, i.e. letd = max{d(a, b)|e(a, b)∈E}. (We suppose thatd≥1, otherwise there is no lower bound restriction.) In this section, letOP T be the number of bins in an optimum packing. Then it naturally follows thatOP T ≥LB1 =d+ 1.
Let OP TR denote the optimum value of the relaxed problem, where the lower bounds on the edges are neglected; i.e. we simply consider the packing problem of itemsA∪B. It trivially follows that OP T ≥ OP TR. Moreover if we apply ourBP P algorithm for the two sets of nodes after each other, Theorem 7.1.1 holds, henceF F D(A, B)≤ 1912 ·OP TR+ 2. Now we will introduce a very simple algorithm calledM asterwith the absolute approximation ratio of31/12. The number of bins created will also be denoted byM aster.
Algorithm Master
1. Pack items of A by bin packing algorithm F F D. The packing (and also the number of the bins used) will be be denoted by F F D(A). Now we pack items of B into new bins (independently of the packing of setA); let the packing of setB; and let the number of bins used be denoted byF F D(B). In this step, the lower bound constraints are totally neglected.
2. We leaved−1empty bins between the two packings; end.
Note that we are quite liberal when performing this Step 2. However, by applying Step 2, the packing of Master naturally will be feasible, so we do not need to deal with the lower bound restrictions. Roughly saying, Step 2 increases the absolute approximation ratio by1, as is shown in the following theorem.
Theorem 7.2.1 The absolute approximation ratio of the Master algorithm is at most31/12.
Proof. For any input, it follows that
M aster =F F D(A) +F F D(B) +d−1
=F F D(A, B) +d−1
≤19/12·OP TR+ 2 + (d−1)
= 19/12·OP TR+LB1 ≤31/12·OP T.
2
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