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Theorem 3. For the Grid Scheduling Problem, CRFFD ≥ 2 − Θ log M M . Proof. Let k be the largest integer such that 2k+1 ≤ M , let M = 2k , and let n be a large integer. Consider the following instance of the problem: Items: For 0 ≤ i ≤ k, 2i n items of size xi = ( 12 + 2−i )M . Bins: For 0 ≤ i ≤ k − 1, 2i n bins of size bi = (1 + 2−i )M , then n bins of size 3 2 M , and ﬁnally (M − 2)n bins of size M + 1. FFD packs the items of size xi in the bins of size bi , 0 ≤ i ≤ k − 1. The last 2k n = M n items are packed, two by two, in the n bins of size 32 M and, one by one, in the (M − 2)n bins of size M + 1.

Our algorithm for the large requests computes k + 1 solutions and picks the cheapest solution among these. The ﬁrst solution is to pack all the large requests with a minimum number of unit capacity colors (using First-Fit on the sorted list of large requests). For each j = 1, 2, . . , k we deﬁne aj = 12 + jε and our (j + 1)-th solution is constructed as follows. We partition the large requests into two classes: the ﬁrst class consists of all large requests with bandwidth at most aj , and the second class consists of all the remaining large requests.

Since the largest possible capacity of a color is 1, no two overlapping intervals can receive the same color, and therefore the algorithm is forced to use 3k −2 colors, whereas an optimal oﬄine algorithm can use at most k colors, each of capacity 12 + ε. The second construction will use intervals of bandwidth 12 + jε for some 2 ≤ j ≤ P . In this construction as well the algorithm is forced to use 3k − 2 colors of capacity at least 12 + jε, whereas the construction is k-colorable. An optimal oﬄine algorithm uses k colors of capacity 12 + jε each, and these colors are used 36 L.