For a positive integer k ≥ 1 , a k -star ( k + -star, k − -star, respectively) is a connected graph containing a degree- ℓ vertex and ℓ degree-1 vertices, where ℓ = k ( ℓ ≥ k , 1 ≤ ℓ ≤ k , respectively). The k + -star packing problem is to cover as many vertices of an input graph G as possible using vertex-disjoint k + -stars in G ; and given k > t ≥ 1 , the k − / t -star packing problem is to cover as many vertices of G as possible using vertex-disjoint k − -stars but no t -stars in G . Both problems are NP-hard for any fixed k ≥ 2 . We present a ( 1 + k 2 2 k + 1 ) - and a 3 2 -approximation algorithms for the k + -star packing problem when k ≥ 3 and k = 2 , respectively, and a ( 1 + 1 t + 1 + 1 / k ) -approximation algorithm for the k − / t -star packing problem when k > t ≥ 2 . They are all local search algorithms and they improve the best known approximation algorithms for the problems, respectively.

This paper studies MPCv5+, which is to cover as many vertices as possible in a given graph G=(V,E) by vertex-disjoint 5+-paths (i.e., paths each with at least five vertices). MPCv5+ is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of [Formula presented] and runs in O(|V|6) time. In this paper, we present a new approximation algorithm for MPCv5+ which achieves a ratio of 2.511 and runs in O(|V|2.5|E|2) time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion. © 2025 The Author(s)

Scheduling within a limited budget is closely related to several much studied scheduling with compression and rescheduling problems, and is also a variant of the more recent model of scheduling with testing. In this problem, one seeks to minimize the makespan for a set of jobs that are to be processed on a number of parallel identical machines. Each job J j is given an upper bound u j on its actual processing time p j , and a testing fee c j . In the offline case, the processing time p j is known to the scheduler; while in the oblivious case, p j is revealed to the scheduler only if the testing fee c j is paid. So the scheduler can choose to execute J j on any one of the machines non-preemptively for u j time or pay for the testing and then execute the job for p j time. The scheduler is given a budget B to pay for testing and seeks to minimize the makespan within that budget. The offline problem is denoted as P ∣ u j , p j , c j , B ∣ C max , and the oblivious problem is denoted as P ∣ u j , − , c j , B ∣ C max , where P stands for multiple parallel identical machines with the number of machines being part of the input. We contribute a polynomial-time approximation scheme (PTAS) for the offline problem P ∣ u j , p j , c j , B ∣ C max , which leads to an almost tight ( 2 + ϵ ) -competitive algorithm for the oblivious problem P ∣ u j , − , c j , B ∣ C max .

In a controllable scheduling environment, the processing time of a job can be shortened by allocating extra resource at a cost, or the job can be declined for processing by paying a penalty. We investigate the single machine controllable scheduling to minimize the sum of the total resource consumption cost, the total job rejection cost, and the makespan of the accepted jobs, where the makespan is upper bounded and the job processing time is a decreasing linear function in the amount of allocated resource. We first show that the studied problem is polynomial solvable if the makespan is unbounded, but otherwise is NP-hard, and characterize important structural properties for the optimal solution; we then take advantage of the structural properties to design several algorithms for the problem, including a pseudo-polynomial time dynamic programming exact algorithm, an O ( n 2 )-time n -approximation algorithm where n is the number of jobs, and building on top of the dynamic programming exact algorithm, the n -approximation algorithm and the bound improvement procedure, two fully polynomial time approximation schemes.

We study the fair allocation of indivisible items to n agents to maximize the utilitarian social welfare, where the fairness criterion is envy-free up to one item and there are only two different utility functions shared by the agents. We present a 2-approximation algorithm when the two utility functions are normalized, improving the previous best ratio of 16 √n shown for general normalized utility functions; thus this constant ratio approximation algorithm confirms the APX-completeness in this special case previously shown APX-hard. When there are only three agents, i.e., n = 3, the previous best ratio is 3 shown for general utility functions, and we present an improved and tight 5/3-approximation algorithm when the two utility functions are normalized, and a best possible and tight 2-approximation algorithm when the two utility functions are unnormalized.

The problem studied in this paper is to find a collection of vertex-disjoint paths in a given graph G = ( V , E ) such that each path has length at least k , called a long path, and the total number of edges on these paths is maximized. The problem is NP-hard for any fixed k or when k is part of the input, by a reduction from the Hamiltonian path problem. Berman and Karpinski presented a 7/6-approximation algorithm for k = 1 , but for a general k ≥ 2 , there is no approximation algorithm directly for the problem. We present the first local search ( 0.4394 k + O ( 1 ) ) -approximation algorithm for any fixed k ≥ 1 , and a 1.4254-approximation algorithm for k = 2 built on top of a maximum triangle-free path-cycle cover.

We study a variant of the well-known Path Cover problem where the candidate paths in a solution have orders up to a fixed integer k . In Path Cover, one finds a minimum number of vertex-disjoint paths in an input graph to cover all the vertices; in our variant, not all paths but only those short ones, i.e., containing up to k vertices, can be used as candidates. The problem is NP-hard when k ≥ 3 ; in the literature, there exist quite a number of approximation algorithms, especially for small k 's. We present an improved k 3 -approximation algorithm for k ∈ { 6 , 7 , 8 } , an improved 55 31 -approximation algorithm for k = 5 , and an improved 8 5 -approximation algorithm for k = 4 . The novelty inside these improved algorithms is observing a close connection between an optimal path cover and a certain polynomial-time computed edge set.

The problem of computing the longest common subsequence of two sequences ( LCS for short) is a classical and fundamental problem in computer science. In this article, we study four variants of LCS : the Repetition-Bounded Longest Common Subsequence problem ( RBLCS ), the Multiset-Restricted Common Subsequence problem ( MRCS ), the Two-Side-Filled Longest Common Subsequence problem ( 2FLCS ), and the One-Side-Filled Longest Common Subsequence problem ( 1FLCS ). Although the original LCS can be solved in polynomial time, all these four variants are known to be NP-hard. Recently, an exact, O ( 1 . 4422 5 n ) -time, dynamic programming (DP) based algorithm for RBLCS was proposed, where the two input sequences have lengths n and p o l y ( n ) . Here, we first establish that each of MRCS , 1FLCS , and 2FLCS is polynomially equivalent to RBLCS . Then, we design a refined DP-based algorithm for RBLCS that runs in O ( 1 . 4142 2 n ) time, which implies that MRCS , 1FLCS , and 2FLCS can also be solved in O ( 1 . 4142 2 n ) time. Finally, we give a polynomial-time 2-approximation algorithm for 2FLCS .

Given a digraph G=(V,E), the k-path partition problem aims to find a minimum collection of vertex-disjoint directed paths, each of order at most k, to cover all the vertices of V. The problem has various applications in facility location, network monitoring, transportation networks and others. Its special case on undirected graphs is NP-hard when k≥3, and has received much study recently from the approximation algorithm perspective. However, the general problem on digraphs is seemingly untouched in the literature. We fill the gap with the first k/2-approximation algorithm, for any k≥3, based on a novel concept of enlarging walk to minimize the number of singletons in the k-path partition. Secondly, for k=3, we define a second novel kind of enlarging walks to greedily reduce the number of 2-paths in the 3-path partition and propose an improved 13/9-approximation algorithm. Lastly, for any k≥7, we present an improved (k+2)/3-approximation algorithm built on the maximum path-cycle cover followed by a careful 2-cycle elimination process.

In this paper, we study a type of incremental optimization variant of the Maximum Independent Set problem (MaxIS), called Bounded-Deletion Maximum Independent Set problem (BD-MaxIS): Given an unweighted graph $$G = (V, E)$$ , an initial feasible solution (i.e., an independent set) $$S^0\subseteq V$$ , and a non-negative integer k, the objective of BD-MaxIS is to find an independent set $$S\subseteq V$$ such that $$|S^0\setminus S|\le k$$ and |S| is maximized. The original MaxIS is generally NP-hard, but, it can be solved in polynomial time for perfect graphs (and therefore, comparability, co-comparability, bipartite, chordal, and interval graphs). In this paper, we show that BD-MaxIS is NP-hard even if the input is restricted to bipartite graphs, and hence to comparability graphs. On the other hand, fortunately, BD-MaxIS on co-comparability, interval, convex bipartite, and chordal graphs can be solved in polynomial time. Finally, we study the computational complexity on very similar variants of the Minimum Vertex Cover and the Maximum Clique problems for graph subclasses.

Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seemingly escapes from the literature. A path containing at least $k$ vertices is considered long. When $k \le 3$, the problem is polynomial time solvable; when $k$ is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed $k \ge 4$, the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a $k$-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when $k = 4$, the problem admits a $4$-approximation algorithm which was presented recently. We propose the first $(0.4394 k + O(1))$-approximation algorithm for the general problem and an improved $2$-approximation algorithm when $k = 4$. Both algorithms are based on local improvement, and their theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies.

In this paper, we study exact, exponential-time algorithms for a variant of the classic Longest Common Subsequence problem called the Repetition-Bounded Longest Common Subsequence problem (or RBLCS , for short): Let an alphabet S be a finite set of symbols and an occurrence constraint C o c c be a function C o c c : S → N , assigning an upper bound on the number of occurrences of each symbol in S . Given two sequences X and Y over the alphabet S and an occurrence constraint C o c c , the goal of RBLCS is to find a longest common subsequence of X and Y such that each symbol s ∈ S appears at most C o c c ( s ) times in the obtained subsequence. The special case where C o c c ( s ) = 1 for every symbol s ∈ S is known as the Repetition-Free Longest Common Subsequence problem ( RFLCS ) and has been studied previously; e.g., in [1] , Adi et al. presented a simple (exponential-time) exact algorithm for RFLCS . However, they did not analyze its time complexity in detail, and to the best of our knowledge, there are no previous results on the running times of any exact algorithms for this problem. Without loss of generality, we will assume that | X | ≤ | Y | and | X | = n . In this paper, we first propose a simpler algorithm for RFLCS based on the strategy used in [1] and show explicitly that its running time is O ( 1.44225 n ) . Next, we provide a dynamic programming (DP) based algorithm for RBLCS and prove that its running time is O ( 1.44225 n ) for any occurrence constraint C o c c , and even less in certain special cases. In particular, for RFLCS , our DP-based algorithm runs in O ( 1.41422 n ) time, which is faster than the previous one. Furthermore, we prove NP-hardness and APX-hardness results for RBLCS on restricted instances.

We study the scheduling problem with calibrations. In 2013, Bender et al. (SPAA '13) proposed a theoretical framework for the problem. Jobs of unit processing time with release times and deadlines are to be scheduled on parallel identical machines. The machines need to be calibrated to run jobs while a single calibration remains valid on a machine only for a time period of length T. The objective is to find a schedule that completes all jobs within their timing constraints and minimizes the total number of calibrations. In this paper, we aim to design an approximation algorithm to solve the problem. We propose a dynamic programming algorithm with polynomial running time when the number of machines is constant. In addition, we give a PTAS when the number of machines is input.

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition (MCSP) problem, both of which have applications in many fields including text compression and bioinformatics. k-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most k times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree \Delta \le 6(k-1). In particular, 2-Max-Duo can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the MIS problem. 2-Max-Duo was proved APX-hard and very recently a (1.6 + \epsilon)-approximation was claimed, for any \epsilon > 0. In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-Max-Duo can be approximated arbitrarily close to 1.4.

We consider the single machine scheduling problem with job-dependent machine deterioration. In the problem, we are given a single machine with an initial non-negative maintenance level, and a set of jobs each with a non-preemptive processing time and a machine deterioration. Such a machine deterioration quantifies the decrement in the machine maintenance level after processing the job. To avoid machine breakdown, one should guarantee a non-negative maintenance level at any time point; and whenever necessary, a maintenance activity must be allocated for restoring the machine maintenance level. The goal of the problem is to schedule the jobs and the maintenance activities such that the total completion time of jobs is minimized. There are two variants of maintenance activities: in the partial maintenance case each activity can be allocated to increase the machine maintenance level to any level not exceeding the maximum; in the full maintenance case every activity must be allocated to increase the machine maintenance level to the maximum. In a recent work, the problem in the full maintenance case has been proven NP-hard; several special cases of the problem in the partial maintenance case were shown solvable in polynomial time, but the complexity of the general problem is left open. In this paper we first prove that the problem in the partial maintenance case is NP-hard, thus settling the open problem; we then design a 2-approximation algorithm.

We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous (i.e., each individual at level-1 can mate with exactly one partner) then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research.

BACKGROUND: Despite ongoing reductions in the cost of sequencing technologies, whole genome SNP genotype imputation is often used as an alternative for obtaining abundant SNP genotypes for genome wide association studies. Several existing genotype imputation methods can be efficient for this purpose, while achieving various levels of imputation accuracy. Recent empirical results have shown that the two-step imputation may improve accuracy by imputing the low density genotyped study animals to a medium density array first and then to the target density. We are interested in building a series of staircase arrays that lead the low density array to the high density array or even the whole genome, such that genotype imputation along these staircases can achieve the highest accuracy. RESULTS: For genotype imputation from a lower density to a higher density, we first show how to select untyped SNPs to construct a medium density array. Subsequently, we determine for each selected SNP those untyped SNPs to be imputed in the add-one two-step imputation, and lastly how the clusters of imputed genotype are pieced together as the final imputation result. We design extensive empirical experiments using several hundred sequenced and genotyped animals to demonstrate that our novel two-step piecemeal imputation always achieves an improvement compared to the one-step imputation by the state-of-the-art methods Beagle and FImpute. Using the two-step piecemeal imputation, we present some preliminary success on whole genome SNP genotype imputation for genotyped animals via a series of staircase arrays. CONCLUSIONS: From a low SNP density to the whole genome, intermediate pseudo-arrays can be computationally constructed by selecting the most informative SNPs for untyped SNP genotype imputation. Such pseudo-array staircases are able to impute more accurately than the classic one-step imputation.

This paper describes a novel and fast placement algorithm for FPGA design space (e.g., area, power or reliability) exploration. The proposed algorithm generates the placement based on the topological similarity between two configurations (netlists) in the design space. Thus, it utilizes the sharing of reusable information during the design space exploration and avoids the time-consuming placement computation like VPR. Tested on logic-level and algorithm-level design space exploration cases, our similarity-based placement accurately depicts the “shape” of a design space and pinpoints the designs which are of most interest to IC designers. Moreover, a turbo version of circuit similarity-based placement performs an average of 30x (up to 100x) faster than VPR's while still achieving comparable placement results.

Modern biological applications usually involve the similarity comparison between two objects, which is often computationally very expensive, such as whole genome pairwise alignment and protein 3D structure alignment. Nevertheless, being able to quickly identify the closest neighboring objects from very large databases for a newly obtained sequence or structure can provide timely hints to its functions and more. This paper presents a substantial speedup technique for the well-studied k-nearest neighbor (k-nn) search, based on novel concepts of virtual pivots and partial pivots, such that a significant number of the expensive distance computations can be avoided. The new method is able to dynamically locate virtual pivots, according to the query, with increasing pruning ability. Using the same or less amount of database preprocessing effort, the new method outperformed the second best method by using no more than 40 percent distance computations per query, on a database of 10,000 gene sequences, compared to several best known k-nn search methods including M-Tree, OMNI, SA-Tree, and LAESA. We demonstrated the use of this method on two biological sequence data sets, one of which is for HIV-1 viral strain computational genotyping.

BACKGROUND: The "common disease--common variant" hypothesis and genome-wide association studies have achieved numerous successes in the last three years, particularly in genetic mapping in human diseases. Nevertheless, the power of the association study methods are still low, in particular on quantitative traits, and the description of the full allelic spectrum is deemed still far from reach. Given increasing density of single nucleotide polymorphisms available and suggested by the block-like structure of the human genome, a popular and prosperous strategy is to use haplotypes to try to capture the correlation structure of SNPs in regions of little recombination. The key to the success of this strategy is thus the ability to unambiguously determine the haplotype allele sharing status among the members. The association studies based on haplotype sharing status would have significantly reduced degrees of freedom and be able to capture the combined effects of tightly linked causal variants. RESULTS: For pedigree genotype datasets of medium density of SNPs, we present two methods for haplotype allele sharing status determination among the pedigree members. Extensive simulation study showed that both methods performed nearly perfectly on breakpoint discovery, mutation haplotype allele discovery, and shared chromosomal region discovery. CONCLUSION: For pedigree genotype datasets, the haplotype allele sharing status among the members can be deterministically, efficiently, and accurately determined, even for very small pedigrees. Given their excellent performance, the presented haplotype allele sharing status determination programs can be useful in many downstream applications including haplotype based association studies.

BACKGROUND: With the increasing availability of whole genome sequences, it is becoming more and more important to use complete genome sequences for inferring species phylogenies. We developed a new tool ComPhy, 'Composite Distance Phylogeny', based on a composite distance matrix calculated from the comparison of complete gene sets between genome pairs to produce a prokaryotic phylogeny. RESULTS: The composite distance between two genomes is defined by three components: Gene Dispersion Distance (GDD), Genome Breakpoint Distance (GBD) and Gene Content Distance (GCD). GDD quantifies the dispersion of orthologous genes along the genomic coordinates from one genome to another; GBD measures the shared breakpoints between two genomes; GCD measures the level of shared orthologs between two genomes. The phylogenetic tree is constructed from the composite distance matrix using a neighbor joining method. We tested our method on 9 datasets from 398 completely sequenced prokaryotic genomes. We have achieved above 90% agreement in quartet topologies between the tree created by our method and the tree from the Bergey's taxonomy. In comparison to several other phylogenetic analysis methods, our method showed consistently better performance. CONCLUSION: ComPhy is a fast and robust tool for genome-wide inference of evolutionary relationship among genomes. It can be downloaded from http://digbio.missouri.edu/ComPhy.

MOTIVATION: With the knowledge of large number of SNPs in human genome and the fast development in high-throughput genotyping technologies, identification of linked regions in linkage analysis through allele sharing status determination will play an ever important role, while consideration of recombination fractions becomes unnecessary. RESULTS: In this study, we have developed a rule-based program that identifies linked regions for underlined diseases using allele sharing information among family members. Our program uses high-density SNP genotype data and works in the face of genotyping errors. It works on nuclear family structures with two or more siblings. The program graphically displays allele sharing status for all members in a pedigree and identifies regions that are potentially linked to the underlined diseases according to user-specified inheritance mode and penetrance. Extensive simulations based on the chi(2) model for recombination show that our program identifies linked regions with high sensitivity and accuracy. Graphical display of allele sharing status helps to detect misspecification of inheritance mode and penetrance, as well as mislabeling or misdiagnosis. Allele sharing determination may represent the future direction of linkage analysis due to its better adaptation to high-density SNP genotyping data. AVAILABILITY: http://paed.hku.hk/uploadarea/yangwl/html/index.html

BACKGROUND: Serotypes of the Foot-and-Mouth disease viruses (FMDVs) were generally determined by biological experiments. The computational genotyping is not well studied even with the availability of whole viral genomes, due to uneven evolution among genes as well as frequent genetic recombination. Naively using sequence comparison for genotyping is only able to achieve a limited extent of success. RESULTS: We used 129 FMDV strains with known serotype as training strains to select as many as 140 most serotype-specific nucleotide strings. We then constructed a linear-kernel Support Vector Machine classifier using these 140 strings. Under the leave-one-out cross validation scheme, this classifier was able to assign correct serotype to 127 of these 129 strains, achieving 98.45% accuracy. It also assigned serotype correctly to an independent test set of 83 other FMDV strains downloaded separately from NCBI GenBank. CONCLUSION: Computational genotyping is much faster and much cheaper than the wet-lab based biological experiments, upon the availability of the detailed molecular sequences. The high accuracy of our proposed method suggests the potential of utilizing a few signature nucleotide strings instead of whole genomes to determine the serotypes of novel FMDV strains.

BACKGROUND: Contact order is a topological descriptor that has been shown to be correlated with several interesting protein properties such as protein folding rates and protein transition state placements. Contact order has also been used to select for viable protein folds from ab initio protein structure prediction programs. For proteins of known three-dimensional structure, their contact order can be calculated directly. However, for proteins with unknown three-dimensional structure, there is no effective prediction method currently available. RESULTS: In this paper, we propose several simple yet very effective methods to predict contact order from the amino acid sequence only. One set of methods is based on a weighted linear combination of predicted secondary structure content and amino acid composition. Depending on the number of components used in these equations it is possible to achieve a correlation coefficient of 0.857-0.870 between the observed and predicted contact order. A second method, based on sequence similarity to known three-dimensional structures, is able to achieve a correlation coefficient of 0.977. We have also developed a much more robust implementation for calculating contact order directly from PDB coordinates that works for > 99% PDB files. All of these contact order predictors and calculators have been implemented as a web server (see Availability and requirements section for URL). CONCLUSION: Protein contact order can be effectively predicted from the primary sequence, at the absence of three-dimensional structure. Three factors, percentage of residues in alpha helices, percentage of residues in beta strands, and sequence length, appear to be strongly correlated with the absolute contact order.

MOTIVATION: The availability of the whole genomic sequences of HIV-1 viruses provides an excellent resource for studying the HIV-1 phylogenies using all the genetic materials. However, such huge volumes of data create computational challenges in both memory consumption and CPU usage. RESULTS: We propose the complete composition vector representation for an HIV-1 strain, and a string scoring method to extract the nucleotide composition strings that contain the richest evolutionary information for phylogenetic analysis. In this way, a large-scale whole genome phylogenetic analysis for thousands of strains can be done both efficiently and effectively. By using 42 carefully curated strains as references, we apply our method to subtype 1156 HIV-1 strains (10.5 million nucleotides in total), which include 825 pure subtype strains and 331 recombinants. Our results show that our nucleotide composition string selection scheme is computationally efficient, and is able to define both pure subtypes and recombinant forms for HIV-1 strains using the 5000 top ranked nucleotide strings. AVAILABILITY: The Java executable and the HIV-1 datasets are accessible through 'http://www.cs.ualberta.ca/~ghlin/src/WebTools/hiv.php. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: Gene expression microarray is a powerful technology for genetic profiling diseases and their associated treatments. Such a process involves a key step of biomarker identification, which are expected to be closely related to the disease. A most important task of these identified genes is that they can be used to construct a classifier which can effectively diagnose disease and even recognize the disease subtypes. Binary classification, for example, diseased or healthy, in microarray data analysis has been successful, while multi-class classification, such as cancer subtyping, remains challenging. RESULTS: We target on the challenging multi-class classification in microarray data analysis, especially on the cancer subtyping using gene expression microarray. We present a novel class discrimination strength vector to represent individual genes and introduce a new measurement to quantify the class discrimination strength difference between two genes. Such a new distance measure is employed in gene clustering, and subsequently the gene cluster information is exploited to select a set of genes which can be used to construct a sample classifier. We tested our method on four real cancer microarray datasets each contains multiple subtypes of cancer patients. The experimental results show that the constructed classifiers all achieved a higher classification accuracy than the previously best classification results obtained on these four datasets. Additional tests show that the selected genes by our method are less correlated and they all contribute statistically significantly to the more accurate cancer subtyping. CONCLUSION: The proposed novel class discrimination strength vector is a better representation than the gene expression vector, in the sense that it can be used to effectively eliminate highly correlated but redundant genes for classifier construction. Such a method can build a classifier to achieve a higher classification accuracy, which is demonstrated via cancer subtyping.

A nearly complete sequential resonance assignment is a key factor leading to successful protein structure determination via NMR spectroscopy. Assuming the availability of a set of NMR spectral peak lists, most of the existing assignment algorithms first use the differences between chemical shift values for common nuclei across multiple spectra to provide the evidence that some pairs of peaks should be assigned to sequentially adjacent amino acid residues in the target protein. They then use these connectivities as constraints to produce a sequential assignment. At various levels of success, these algorithms typically generate a large number of potential connectivity constraints, and it grows exponentially as the quality of spectral data decreases. A key observation used in our sequential assignment program, CISA, is that chemical shift residual signature information can be used to improve the connectivity determination, and thus to dramatically decrease the number of predicted connectivity constraints. Fewer connectivity constraints lead to less ambiguities in the sequential assignment. Extensive simulation studies on several large test datasets demonstrated that CISA is efficient and effective, compared to three most recently proposed sequential resonance assignment programs RANDOM, PACES, and MARS.

In this paper, a new representation is presented for the Maximum Quartet Consistency (MQC) problem, where solving the MQC problem becomes searching for an ultrametric matrix that satisfies a maximum number of given quartet topologies. A number of structural properties of the MQC problem in this new representation are characterized through formulating into answer set programming, a recent powerful logic programming tool for modeling and solving search problems. Using these properties, a number of optimization techniques are proposed to speed up the search process. The experimental results on a number of simulated data sets suggest that the new representation, combined with answer set programming, presents a unique perspective to the MQC problem.

A single DNA microarray measures thousands to tens of thousands of gene expression levels, but experimental datasets normally consist of much fewer such arrays, typically in tens to hundreds, taken over a selection of tissue samples. The biological interpretation of these data relies on identifying subsets of induced or repressed genes that can be used to discriminate various categories of tissue, to provide experimental evidence for connections between a subset of genes and the tissue pathology. A variety of methods can be used to identify discriminatory gene subsets, which can be ranked by classification accuracy. But the high dimensionality of the gene expression space, coupled with relatively fewer tissue samples, creates the dimensionality problem: gene subsets that are too large to provide convincing evidence for any plausible causal connection between that gene subset and the tissue pathology. We propose a new gene selection method, clustered gene selection (CGS) which, when coupled with existing methods, can identify gene subsets that overcome the dimensionality problem and improve classification accuracy. Experiments on eight real datasets showed that CGS can identify many more cancer related genes and clearly improve classification accuracy, compared with three other non-CGS based gene selection methods

Ribonucleic acid (RNA) molecules contain the genetic information that regulates the functions of organisms. Given two different molecules, a preserved function corresponds to a preserved secondary RNA structure. Hence, RNA secondary-structure comparison is essential in predicting the functions of a newly discovered molecule. In this paper, we discuss our SPRC method for RNA structure comparison. In this work, we developed, a novel tree representation of RNA that reflects both its primary and secondary structure and a tree-alignment algorithm, which, given the tree representations of two RNA molecules, produces a sequence of mutations that could transform one RNA molecule to the other. Our SPRC algorithm extends the Zhang-Shasha tree-edit distance calculation algorithm in two ways: first, in addition to the distance, it reports all editing sequences with the same minimum edit cost, and second, it uses a biologically-inspired affine cost function. Furthermore, the SPRC method proposes set of heuristics designed to filter the produced solution set to recommend the simplest editing sequence, as corresponding to the most biologically correct alignment. Experiments on three 5S rRNA families: archaea, eubacteria, and eukaryota, show that SPRC is very effective in producing biologically meaningful RNA secondary structure alignments

Microarray data analysis is notoriously challenging as it involves a huge number of genes compared to only a limited number of samples. Gene selection, to detect the most significantly differentially expressed genes under different categories of conditions, is both computationally and biologically interesting, and has become a central research focus in all studies that use gene expression microarray technology. Despite many existing efforts, better gene selection methods that can effectively identify biologically significant biomarkers, yet computationally efficient, are still in need. In this paper, a model-free greedy (MFG) gene selection method is proposed, which implements several intuitive heuristics but doesn't assume any statistical distribution on the expression data. The experimental results on three real microarray datasets showed that the MFG method combined with a support vector machine (SVM) classifier or a k-nearest neighbor (KNN) classifier is efficient and robust in identifying discriminatory genes

BACKGROUND: Microarray data analysis is notorious for involving a huge number of genes compared to a relatively small number of samples. Gene selection is to detect the most significantly differentially expressed genes under different conditions, and it has been a central research focus. In general, a better gene selection method can improve the performance of classification significantly. One of the difficulties in gene selection is that the numbers of samples under different conditions vary a lot. RESULTS: Two novel gene selection methods are proposed in this paper, which are not affected by the unbalanced sample class sizes and do not assume any explicit statistical model on the gene expression values. They were evaluated on eight publicly available microarray datasets, using leave-one-out cross-validation and 5-fold cross-validation. The performance is measured by the classification accuracies using the top ranked genes based on the training datasets. CONCLUSION: The experimental results showed that the proposed gene selection methods are efficient, effective, and robust in identifying differentially expressed genes. Adopting the existing SVM-based and KNN-based classifiers, the selected genes by our proposed methods in general give more accurate classification results, typically when the sample class sizes in the training dataset are unbalanced.

Microarray data analysis is notorious for involving a huge number of genes compared to a relatively small number of samples. Detecting the most significantly differentially expressed genes under different conditions, or gene selection, has been a central focus for researchers. The gene selection problem becomes more difficult when the numbers of samples under different conditions vary significantly, or are unbalanced. A novel model-free and stable gene selection method is proposed in this paper, i.e., the method does not assume any statistical model on the gene expression data and it is not affected by the unbalanced samples. The method has been evaluated on two publicly available datasets, the leukemia dataset and the small round blue cell tumor dataset, where the experimental results showed that the proposed method is efficient and robust in identifying differentially expressed genes.

Evolution is an important subarea of study in biological science, where given a set of species, the goal is to reconstruct their evolutionary history, or phylogeny. Many kinds of data associated with the species can be deployed for this task and many reconstruction methods have been proposed and examined in the literature. One very recent approach is to build a local phylogeny for every subset of 4 species, which is called a quartet for these 4 species, and then to assemble a phylogeny for the whole set of species satisfying these predicted quartets. In general, those predicted quartets might not always agree each other; and thus the objective function becomes to satisfy a maximum number of predicted quartets. This is the well-known maximum quartet consistency (MQC) problem, which is studied by a lot of researchers in the last two decades. We present a new equivalent representation for the MQC problem, that is, to search for an ultrametric matrix to satisfy the maximum number of those predicted quartets. We examine a few number of structural properties of the MQC problem in this new representation, through formulating it into answer set programming (ASP), a recent powerful logic programming tool for modeling and solving searching problems. The efficiency and usefulness of our approach are confirmed by our computational experiments on the artificial data as well as two real datasets.

Given a set of species and their similarity data, an important problem in evolutionary biology is how to reconstruct a phylogeny (also called evolutionary tree) so that species are close in the phylogeny if and only if they have high similarity. Assume that the similarity data are represented as a graph G = (V, E), where each vertex represents a species and two vertices are adjacent if they represent species of high similarity. The phylogeny reconstruction problem can then be abstracted as the problem of finding a (phylogenetic) tree T from the given graph G such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3) $(u, v) \in E$ if and only if $d_T(u, v) \le k$ for some fixed threshold k, where d T (u,v) denotes the distance between u and v in tree T. This is called the phylogenetic kth root problem (PRk), and such a tree T, if it exists, is called a phylogenetic kth root of graph G. The computational complexity of PRk} is open, except for $k \le 4$. In this paper, we investigate PRk under a natural restriction that the maximum degree of the phylogenetic root is bounded from above by a constant. Our main contribution is a linear-time algorithm that determines if G has such a phylogenetic kth root, and if so, demonstrates one. On the other hand, because in practice the collected similarity data are usually not perfect and may contain errors, we propose to study a generalized version of PRk where the output phylogeny is required only to be an approximate root of the input graph. We show that this and other related problems are computationally intractable.

We consider the problem of routing the lightpaths of a logical topology of a WDM network on an arbitrary physical topology, such that the logical topology remains,connected even after the failure of a physical link. We focus our attention on the ring interconnection as the logical topology because it is widely used in many protection schemes. We first establish the necessary and sufficient condition for a ring logical topology to withstand failure of a single physical link. Next we show that the testing of this necessary and sufficient condition is an NP-complete problem. Finally, we give an algorithm for testing the necessary and sufficient condition and demonstrate the execution of the algorithm with the help of an example.

Arc-annotated sequences are useful in representiug the structural information of RNA sequences. Typically, RNA secondary and tertiary structures could be represented by a set of nested arcs and a set of crossing arcs, respectively. As the specified RNA functions are determined by the specified molecular confirmation and therefore the specified secondary and tertiary structures, the comparison between RNA secondary and tertiary structures have received much attention recently. In this paper, we propose the notion of edit distance to measure the similarity between two RNA secondary and tertiary structures, by incorporating the various edit operations performing on both bases and arcs (base-pairs). Several algorithms are presented to compute the edit distance two RNA sequences with various arc structures and under various score schemes, either exactly or approximately. Preliminary experimental tests confirm that our definition of edit distance and the computation model are among the most reasonable ones ever studied in the literature.

In this paper, we present a linear time algorithm for computing the hexagonal Steiner minimum tree for a set of points on the boundary of a regular hexagon. Computational results on randomly generated test problems show that our algorithm can find the optimal solutions on a 200 MHz Pentium within 18 seconds for n as large as 20000. It is expected that techniques of this paper may be generalized to the case where the points are on the boundary of a polygon.

With the advent of deep sub-micron technology, gate delays are much smaller than wire delays. As a result, hexagonal Steiner minimum trees have received extensive study recently because of their applications in VLSI physical design. Just like its rectilinear and Euclidean counterparts, the hexagonal Steiner minimum tree problem can be shown to be NP-hard. Therefore polynomial time approximation algorithms are of great interest. In a recent paper, Lin, Xue and Zhou (1999) proposed a quadratic time algorithm to compute an optimal layout of a hexagonal minimum spanning tree with attractive computational results. In this paper, we present an improved linear time algorithm for computing an optimal layout of a hexagonal minimum spanning tree.

We studied the Steiner tree problem in four uniform orientations where any line, half-line, or line segment must be on a line which makes an angle of (iπ)/4 with the positive x-axis, for some i ∈ {0, 1, 2, 3}, and the distance between two points is measured as the length of the shortest polygonal path connecting them. We show that for any set P of n terminal points there exists a Steiner minimum tree interconnecting P such that all Steiner points are in 𝒢⌈2n/3⌉ − 1(P), the (⌈(2n)/3⌉ − 1)st-generation grid points of P. Our result improves the previous best result which guarantees that for any set P of n terminal points there is a Steiner minimum tree in which all Steiner points are in 𝒢n − 2(P). © 2000 John Wiley & Sons, Inc.

We study algorithms for approximating a Steiner minimal tree interconnecting n points under hexagonal routing. We prove that: (1) every minimum spanning tree is separable; (2) a minimum spanning tree with maximum node degree no more than 5 can be computed in O (n log n) time; (3) an optimal L-shaped layout of a given minimum spanning tree can be computed in O(n) time; (4) an optimal stair-shaped layout of a given minimum spanning tree can be computed in O(n/sup 2/) time. Computational results on standard benchmarks show that our algorithm compares favorably to the current best algorithms.

Chung and Ross [SIAM J. Comput., 20 (1991), pp. 726--736] conjectured that the multirate three-stage Clos network C(n,2n-1,r) is rearrangeable in the general discrete bandwidth case; i.e., each connection has a weight chosen from a given finite set {p 1 , p 2 ,. . .,p k } where $1 \geq p_1 > p_2 > \cdots > p_k > 0$ and p i is an integer multiple of p i , denoted by $p_k \mid p_i$, for $1 \leq i \leq k-1$. In this paper, we prove that multirate three-stage Clos network C(n,2n-1,r) is rearrangeable when each connection has a weight chosen from a given finite set {p 1 , p 2 ,. . .,p k } where $1 \geq p_1 > p_2 > \cdots > p_{h} > 1/2 \geq p_{h+1} > \cdots > p_k > 0$ and p h+2 | p h+1 ,p h+3 |p h+2 ,. . . ,p k | p h+1 . We also prove that C(n,2n-1,r) is two-rate rearrangeable and $C(n, \lceil \frac{7n}{3} \rceil, r)$ is three-rate rearrangeable.