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Abstract. We study the border minimization problem (BMP), which arises in microarray synthesis to place and embed probes in the array. The synthesis is based on a light-directed chemical process in which unintended illumination may contaminate the quality of the experiments. Border length is a measure of the amount of unintended illumination and the objective of BMP is to find a placement and embedding of probes such that the border length is minimized. The problem is believed to be NP-hard. In this paper we show that BMP admits an O( √ n log 2 n)-approximation, where n is the number of probes to be synthesized. In the case where the placement is given in advance, we show that the problem is O(log 2 n)-approximable. We also study a related problem called agreement maximization problem (AMP). In contrast to BMP, we show that AMP admits a constant approximation even when placement is not given in advance.

Abstract. We study d-dimensional dynamic bin packing for general ddimensional boxes, for d ≥ 2. This problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. Our main result is a 3 d -competitive online algorithm. We further study the 2-and 3-dimensional problem closely and improve the competitive ratios. Technically speaking, our d-dimensional result is due to a space efficient offline single bin packing algorithm, which is a variant of d-dimensional NFDH. We introduce an interesting notion of d-dimensional L-shape bin and show that effective offline packing into L-shape bin leads to effective online dynamic packing into unit-sized bins. We also investigate the resource augmentation version of the problem where the online algorithm can use d-dimensional bins of size s1 × s2 × · · ·×s d for si ≥ 1 while the optimal offline algorithm uses unit-sized bins. We give conditions for the online algorithm to match the performance of the optimal offline algorithm, i.e., 1-competitive.

We study a combinatorial problem arising from microarrays synthesis. The synthesis is done by a light-directed chemical process. The objective is to minimize unintended illumination that may contaminate the quality of experiments. Unintended illumination is measured by a notion called border length and the problem is called Border Minimization Problem (BMP). The objective of the BMP is to place a set of probe sequences in the array and find an embedding (deposition of nucleotides/residues to the array cells) such that the sum of border length is minimized. A variant of the problem, called P-BMP, is that the placement is given and the concern is simply to find the embedding.Approximation algorithms have been proposed for the problem [22] but it is unknown whether the problem is NP-hard or not. In this paper, we give a thorough study of different variations of BMP by giving NP-hardness proofs and improved approximation algorithms. We show that P-BMP, 1D-BMP, and BMP are all NP-hard. Contrast with the result in [22] that 1D-P-BMP is polynomial time solvable, the interesting implications include (i) the array dimension (1D or 2D) differentiates the complexity of P-BMP; (ii) for 1D array, whether placement is given differentiates the complexity of BMP; (iii) BMP is NP-hard regardless of the dimension of the array. Another contribution of the paper is improving the approximation for BMP from O(n 1/2 log 2 n) to O(n 1/4 log 2 n), where n is the total number of sequences.