Fencol C. C. Yung

Online optimization of busy time on parallel machines

We consider the following online scheduling problem in which the input consists of n jobs to be scheduled on identical machines of bounded capacity g (the maximum number of jobs that can be processed simultaneously on a single machine). Each job is associated with a release time and a completion time between which it is supposed to be processed. When a job is released, the online algorithm has to make decision without changing it afterwards. We consider two versions of the problem. In the minimization version, the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version, the goal is to maximize the number of jobs that are scheduled under a budget constraint given in terms of busy time. This is the first study on online algorithms for these problems. We show a rather large lower bound on the competitive ratio for general instances. This motivates us to consider special families of input instances for which we show constant competitive algorithms. Our study has applications in power aware scheduling, cloud computing and optimizing switching cost of optical networks.

An 8/3 Lower Bound for Online Dynamic Bin Packing

We study the dynamic bin packing problem introduced by Coffman, Garey and Johnson. This problem is a generalization of the bin packing problem in which items may arrive and depart dynamically. The objective is to minimize the maximum number of bins used over all time. The main result is a lower bound of 8/3 ~2.666 on the achievable competitive ratio, improving the best known 2.5 lower bound. The previous lower bounds were 2.388, 2.428, and 2.5. This moves a big step forward to close the gap between the lower bound and the upper bound, which currently stands at 2.788. The gap is reduced by about 60% from 0.288 to 0.122. The improvement stems from an adversarial sequence that forces an online algorithm \({\mathcal{A}}\) to open 2s bins with items having a total size of s only and this can be adapted appropriately regardless of the current load of other bins that have already been opened by \({\mathcal{A}}\). Comparing with the previous 2.5 lower bound, this basic step gives a better way to derive the complete adversary and a better use of items of slightly different sizes leading to a tighter lower bound. Furthermore, we show that the 2.5-lower bound can be obtained using this basic step in a much simpler way without case analysis.

Competitive multi-dimensional dynamic bin packing via l-shape bin packing

We study d-dimensional dynamic bin packing for general d-dimensional 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 3d-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 ×⋯×sd 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.

Approximating border length for DNA microarray synthesis

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 log2 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(log2 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.

Online Multi-dimensional Dynamic Bin Packing of Unit-Fraction Items

We study the 2-D and 3-D dynamic bin packing problem, in which items arrive and depart at arbitrary times. The 1-D problem was first studied by Coffman, Garey, and Johnson motivated by the dynamic storage problem. Bar-Noy et al. have studied packing of unit fraction items (i.e., items with length 1/k for some integer k ≥ 1), motivated by the window scheduling problem. In this paper, we extend the study of 2-D and 3-D dynamic bin packing problem to unit fractions items. The objective is to pack the items into unit-sized bins such that the maximum number of bins ever used over all time is minimized. We give a scheme that divides the items into classes and show that applying the First-Fit algorithm to each class is 6.7850- and 21.6108-competitive for 2-D and 3-D, respectively, unit fraction items. This is in contrast to the 7.4842 and 22.4842 competitive ratios for 2-D and 3-D, respectively, that would be obtained using only existing results for unit fraction items.

Hardness and approximation of the asynchronous border minimization problem

Abstract We study a combinatorial problem arising from the microarray synthesis. The objective of the Border Minimization Problem (BMP) is to place a set of sequences in the array and to find an embedding of these sequences into a common supersequence such that the sum of the “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. An exponential time algorithm has been proposed for the problem but it is unknown whether the problem is NP-hard or not. In this paper, we give a comprehensive study of different variations of BMP by presenting NP-hardness proofs and approximation algorithms. We show that BMP, P-BMP, and 1D-BMP are all NP-hard and 1D-BMP is polynomial time solvable. The interesting implications include (i) the BMP is NP-hard regardless of the dimension (1D or 2D) of the array; (ii) the array dimension differentiates the complexity of the P-BMP; and (iii) for 1D array, whether placement is given differentiates the complexity of the BMP. Another contribution of the paper is devising approximation algorithms, and in particular, we present a randomized approximation algorithm for BMP with approximation ratio O ( n 1 ∕ 4 log 2 n ) , where n is the total number of sequences.