Two Variations of the Minimum Steiner Problem

Given a set S of starting vertices and a set T of terminating vertices in a graph G = (V,E) with non-negative weights on edges, the minimum Steiner network problem is to find a subgraph of G with the minimum total edge weight. In such a subgraph, we require that for each vertex s S and t T, there is a path from s to a terminating vertex as well as a path from a starting vertex to t. This problem can easily be proven NP-hard. For solving the minimum Steiner network problem, we first present an algorithm that runs in time and space that both are polynomial in n with constant degrees, but exponential in |S|+|T|, where n is the number of vertices in G. Then we present an algorithm that uses space that is quadratic in n and runs in time that is polynomial in n with a degree O(max {max {|S|,|T|}–2,min {|S|,|T|}–1}). In spite of this degree, we prove that the number of Steiner vertices in our solution can be as large as |S|+|T|–2. Our algorithm can enumerate all possible optimal solutions. The input graph G can either be undirected or directed acyclic. We also give a linear time algorithm for the special case when min {|S|,|T|} = 1 and max {|S|,|T|} = 2.The minimum union paths problem is similar to the minimum Steiner network problem except that we are given a set H of hitting vertices in G in addition to the sets of starting and terminating vertices. We want to find a subgraph of G with the minimum total edge weight such that the conditions required by the minimum Steiner network problem are satisfied as well as the condition that every hitting vertex is on a path from a starting vertex to a terminating vertex. Furthermore, G must be directed acyclic. For solving the minimum union paths problem, we also present algorithms that have a time and space tradeoff similar to algorithms for the minimum Steiner network problem. We also give a linear time algorithm for the special case when |S| = 1, |T| = 1 and |H| = 2.An extended abstract of part of this paper appears in Hsu et al. (1996).Supported in part by the National Science Foundation under Grants CCR-9309743 and INT-9207212, and by the Office of Naval Research under Grant No. N00014-93-1-0272.Supported in part by the National Science Council, Taiwan, ROC, under Grant No. NSC-83-0408-E-001-021.     

Broadcast Routing with Minimum Wavelength Conversion in WDM Optical Networks

We consider dynamic routing of broadcast connections in WDM optical networks. Given the current network state, we want to find a minimum set of network nodes S such that a broadcast routing using only the nodes in S as wavelength conversion nodes can be found. This ensures that the average conversion delay from the source to all destinations is minimized. We refer to the problem as Broadcast Conversion Node Selection (BCNS) problem. We prove that BCNS has no polynomial-time approximation with performance ratio ln n for < 1 unless NPDTIME(n^{O(log log n)}) where n is the number of vertices in the input graph. We present a greedy approximation algorithm for BCNS that achieves approximation ratio 2+ln n. Simulation results show that the algorithm performs very well in practice, obtaining optimal solutions in most of the cases.     

A Tighter Extra-Resource Analysis of Online Deadline Scheduling

This paper is concerned with online algorithms for scheduling jobs with deadlines on a single processor. It has been known for long that unless the system is underloaded, no online scheduling algorithm can be 1-competitive, i.e., matching the performance of the optimal offline algorithm. Nevertheless, recent work has revealed that some online algorithms using a moderately faster processor (or extra processors) can guarantee very competitive performance Kalyanasundaram and Pruhs, 2000 or even be 1-competitive Koo et al., 2002; Lam and To, 2001. This paper takes a further step to investigate online scheduling algorithms with an even higher performance guarantee (i.e., better than 1-competitive algorithms) and in particular, presents an extra-resource analysis of the earliest-deadline-first strategy (EDF) with respect to such a higher performance guarantee.A preliminary version of this paper has been accepted by The Australian Theory Symposium on Computing, 2004.This research was supported in part by Hong Kong RGC Grant HKU-7024/01E.     

Feature Selection for Consistent Biclustering via Fractional 0–1 Programming

Biclustering consists in simultaneous partitioning of the set of samples and the set of their attributes (features) into subsets (classes). Samples and features classified together are supposed to have a high relevance to each other which can be observed by intensity of their expressions. We define the notion of consistency for biclustering using interrelation between centroids of sample and feature classes. We prove that consistent biclustering implies separability of the classes by convex cones. While previous works on biclustering concentrated on unsupervised learning and did not consider employing a training set, whose classification is given, we propose a model for supervised biclustering, whose consistency is achieved by feature selection. The developed model involves solution of a fractional 0–1 programming problem. Preliminary computational results on microarray data mining problems are reported.This research work was partially supported by NSF, NIH and AirForce grants.     

The Center Location Improvement Problem Under the Hamming Distance

In this paper, we consider the center location improvement problems under the sum-type and bottleneck-type Hamming distance. For the sum-type problem, we show that achieving an algorithm with a worst-case ratio of O(log |V|) is NP-hard, and for the bottleneck-type problem, we present a strongly polynomial algorithm.     

Clustering of Microarray data via Clique Partitioning

Microarrays are repositories of gene expression data that hold tremendous potential for new understanding, leading to advances in functional genomics and molecular biology. Cluster analysis (CA) is an early step in the exploration of such data that is useful for purposes of data reduction, exposing hidden patterns, and the generation of hypotheses regarding the relationship between genes and phenotypes. In this paper we present a new model for the clique partitioning problem and illustrate how it can be used to perform cluster analysis in this setting.     

The surprising effectiveness of earliest operation due-date sequencing

In make-to order production, schedule reliability is very important but still not sufficiently accomplished in industrial practice by the vast majority of companies. It has been known for long that processing the orders at a workstation in the order of their operation due-dates can compensate for lateness in the arrival at the workstation within certain boundaries. The paper analyses the effectiveness of earliest operation due date (EODD) sequencing by comparing it to an optimistic theoretical boundary. The surprising result is that EODD can nearly fully exploit the theoretical potential. It should therefore be used in practice whenever schedule reliability is important, with only few exceptions. Its effectiveness though is increasing with the workstation’s WIP level and thus is in conflict with the objective to reduce WIP levels and throughput times. A simple forecasting model allows to assess the extent to which lateness can be compensated by EODD sequencing and which schedule compliance can be achieved.     

Decision model for product rollover in manufacturing operations

The purpose of this study is to investigate critical decisions when planning for product rollover at a manufacturing company, and how to organise these decisions. A literature review and a case study are used to develop a decision model. The findings indicate that product rollover can be organised in a five-phase decision model. The research quality is strengthened by a structured literature review, but it can be argued that more empirical research is needed for validation. For researchers, this paper contributes with the identification of critical decisions and a model for product rollover. For practitioners, the study highlights the need to acknowledge the importance of product rollover and its role as a competitive weapon. In earlier studies, product rollover has mainly been investigated from a market perspective, so that this study contributes by investigating the issue from a manufacturing perspective.