Approximation algorithms for constrained generalized tree alignment problem

In generalized tree alignment problem, we are given a set S of k biologically related sequences and we are interested in a minimum cost evolutionary tree for S. In many instances of this problem partial phylogenetic tree for S is known. In such instances, we would like to make use of this knowledge to restrict the tree topologies that we consider and construct a biologically relevant minimum cost evolutionary tree. So, we propose the following natural generalization of the generalized tree alignment problem, a problem known to be MAX-SNP Hard, stated as follows:

Constrained Generalized Tree Alignment Problem [S. Divakaran, Algorithms and heuristics for constrained generalized alignment problem, DIMACS Technical Report 2007-21, 2007]: Given a set S of k related sequences and a phylogenetic forest comprising of node-disjoint phylogenetic trees that specify the topological constraints that an evolutionary tree of S needs to satisfy, construct a minimum cost evolutionary tree for S.

In this paper, we present constant approximation algorithms for the constrained generalized tree alignment problem. For the generalized tree alignment problem, a special case of this problem, our algorithms provide a guaranteed error bound of 2−2/k.     

Ill‐conditioning in the virtual element method: Stabilizations and bases

In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.     

Expected number of steps of a random optimization method

In this paper, we give an estimate of the expected number of steps of Matya's random optimization method applied to the constrained nonlinear minimization problem. It is also shown that, in a sense, this random optimization method can be optimized by the uniform distribution, in which case the exact value of the expected number of steps is computed.     

Minimizing the number of machines for scheduling jobs with equal processing times

In this paper, we consider a parallel machine environment when all jobs have the same processing time and arbitrary release dates and deadlines of the jobs are given. We suppose that the available number of machines, which can be used simultaneously, may vary over time. The aim is to construct a feasible schedule in such a way that the maximal number of simultaneously used machines is minimal. We give a polynomial algorithm for this problem.     

非光滑约束问题的既约次梯度法

１引言 对带约束的不可微的非线性规划问题,由于不能使用梯度,求极小点就比较困难．本文给出解决此问题的一种有效的算法． ２　非光滑约束问题的既约次梯度法 １）非线性规划问题的Ｌａｅｒａｎｅ对偶理论 考虑下面非线性规划问题其中ｇ（ｘ）＝（ｇ１（ｘ）,…,ｇｒ（ｘ））Ｔ,ｈ（ｘ））＝（ｈ１（ｘ）,…,ｈｍ（ｘ））Ｔ,ｆ（ｘ）＝　　　　　　Ｒｎ中是Ｌｉｓｐｓｃｈｉｔｚ连续的ｉ＝１,２,…,ｒ,ｊ＝１,２,…,ｍ相应的Ｌａｇｒａｎｇｅ对偶问题为其中　　（ｕ,　）＝ｉｎｆＬ（ｘ;ｕ,ｖ）＝ｉｎｆ（ｆ（ｘ）＋ｕＴ…     

Unifications,deunifications, and their complexity

The execution of a Prolog program can be viewed as a sequence of unifications and backtracks over unifications. We study the time requirement of executing a sequence of such operations (the unify-deunify problem). It is shown that the well-known set union problem is reducible to this problem, even in the case when no function symbols are allowed (the Datalog unify-deunify problem). As the set union problem requires nonlinear time on a large class of algorithms, the same holds for the unify-deunify problem. Thus the linearity of single unifications does not give a complete picture of the time complexity of Prolog primitives. We discuss the methods for executing sequences of Datalog unifications used in Prolog interpreters and show that some of them require even quadratic time in the worst case. Complementing these results, we show that if the number of variables occurring in one clause is bounded by a constant, then the Datalog unify-deunify problem can be solved in linear time.A preliminary version of this paper appeared in the Third International Conference on Logic Programming, London, July 1986. This work was supported by the Academy of Finland and by TEKES.     

On the best case performance of hit and run methods for detecting necessary constraints

The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collector's problem.     

Notes on Duality in Second Order and p -Order Cone Optimization

Recently, the so-called second order cone optimization problem has received much attention, because the problem has many applications and the problem can in theory be solved efficiently by interior-point methods. In this note we treat duality for second order cone optimization problems and in particular whether a nonzero duality gap can be obtained when casting a convex quadratically constrained optimization problem as a second order cone optimization problem. Furthermore, we also discuss the p -order cone optimization problem which is a natural generalization of the second order case. Specifically, we suggest a new self-concordant barrier for the p -order cone optimization problem.     

Pareto optimal matchings with lower quotas

We consider the problem of allocating applicants to courses, where each applicant has a capacity, possibly greater than 1, and a subset of acceptable courses that she ranks in a strict order of preference. Each course has a lower and an upper quota, indicating that if it is assigned some applicants then their number has to be between these two bounds. We further suppose that applicants extend their preferences over courses to preferences over bundles of courses lexicographically.In this setting we present several algorithmic results concerned with the computation of Pareto optimal matchings (POMs). Firstly, we extend the Serial Dictatorship with Project Closures mechanism to the case when an applicant can be assigned more than one course. We show that unlike in the one-to-many case no mechanism is strategy-proof against dropping manipulations and that this mechanism is strategy-proof against reordering strategies only for some picking sequences. We further show the intractability of the following problems: deciding about the Pareto optimality of a given matching, computation of a POM with maximum cardinality and computation of a POM in case of indifferences.     

An efficient co-swarm particle swarm optimization for non-linear constrained optimization

This paper proposes a new co-swarm PSO (CSHPSO) for constrained optimization problems, which is obtained by hybridizing the recently proposed shrinking hypersphere PSO (SHPSO) with the differential evolution (DE) approach. The total swarm is subdivided into two sub swarms in such a way that the first sub swarms uses SHPSO and second sub swarms uses DE. Experiments are performed on a state-of-the-art problems proposed in IEEE CEC 2006. The results of the CSHPSO is compared with SHPSO and DE in a variety of fashions. A statistical approach is applied to provide the significance of the numerical experiments. In order to further test the efficacy of the proposed CSHPSO, an economic dispatch (ED) problem with valve points effects for 40 generating units is solved. The results of the problem using CSHPSO is compared with SHPSO, DE and the existing solutions in the literature. It is concluded that CSHPSO is able to give the minimal cost for the ED problem in comparison with the other algorithms considered. Hence, CSHPSO is a promising new co-swarm PSO which can be used to solve any real constrained optimization problem.     

A provisioning problem with stochastic payments

We consider the problem of determining the minimal requirement one must establish in order to meet a series of future random payments. It is shown in a very general setting that this problem can be recast as a chance constrained model and how the technique of Sample Average Approximation can be employed to find solutions. We also use comonotonic theory to analyze analytical approximations in a restricted Gaussian setting. Our numerical illustrations demonstrate that the Sample Average Approximation is a viable and efficient way to solve the stated problem generally and outperforms the analytical approximations. In passing we present a result that is related to Stein’s famous lemma (Stein, 1981) and is of interest in itself.     

Asymptotic behavior of central order statistics from stationary processes

In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations.     

A numerical solution of the constrained weighted energy problem

A numerical algorithm is presented to solve the constrained weighted energy problem from potential theory. As one of the possible applications of this algorithm, we study the convergence properties of the rational Lanczos iteration method for the symmetric eigenvalue problem. The constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the Ritz values. The region depends on the distribution of the eigenvalues, on the distribution of the poles, and on the ratio between the size of the matrix and the number of iterations. Our algorithm gives the possibility of finding the boundary of this region in an effective way.We give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational Lanczos algorithm.     

Novel evolutionary models and applications to sequence alignment problems

In this paper, we present a novel graph-theoretical approach for representing a wide variety of sequence analysis problems within a single model. The model allows incorporation of the operations “insertion”, “deletion”, and “substitution”, and various parameters such as relative distances and weights. Conceptually, we refer the problem as the minimum weight common mutated sequence (MWCMS) problem. The MWCMS model has many applications including multiple sequence alignment problem, the phylogenetic analysis, the DNA sequencing problem, and sequence comparison problem, which encompass a core set of very difficult problems in computational biology. Thus the model presented in this paper lays out a mathematical modeling framework that allows one to investigate theoretical and computational issues, and to forge new advances for these distinct, but related problems. Through the introduction of supernodes, and the multi-layer supergraph, we proved that MWCMS is -complete. Furthermore, it was shown that a conflict graph derived from the multi-layer supergraph has the property that a solution to the associated node-packing problem of the conflict graph corresponds to a solution of the MWCMS problem. In this case, we proved that when the number of input sequences is a constant, MWCMS is polynomial-time solvable. We also demonstrated that some well-known combinatorial problems can be viewed as special cases of the MWCMS problem. In particular, we presented theoretical results implied by the MWCMS theory for the minimum weight supersequence problem, the minimum weight superstring problem, and the longest common subsequence problem. Two integer programming formulations were presented and a simple yet elegant decomposition heuristic was introduced. The integer programming instances have proven to be computationally intensive. Consequently, research involving simultaneous column and row generation and parallel computing will be explored. The heuristic algorithm, introduced herein for multiple sequence alignment, overcomes the order-dependent drawbacks of many of the existing algorithms, and is capable of returning good sequence alignments within reasonable computational time. It is able to return the optimal alignment for multiple sequences of length less than 1500 base pairs within 30 minutes. Its algorithmic decomposition nature lends itself naturally for parallel distributed computing, and we continue to explore its flexibility and scalability in a massive parallel environment.     

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.     

An algorithm for solving linearly constrained minimax problems

In this study, we propose an algorithm for solving a minimax problem over a polyhedral set defined in terms of a system of linear inequalities. At each iteration a direction is found by solving a quadratic programming problem and then a suitable step size along that direction is taken through an extension of Armijo's approximate line search technique. We show that each accumulation point is a Kuhn-Tucker solution and give a condition that guarantees convergence of the whole sequence of iterations. Through the use of an exact penalty function, the algorithm can be used for solving constrained nonlinear programming. In this case, our algorithm resembles that of Han, but differs from it both in the direction-finding and the line search steps.     

The knapsack problem with neighbour constraints

We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected.We give approximation algorithms and hardness results when the vertices have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.     

A dual approach to multiple exercise option problems under constraints

This paper considers the pricing of multiple exercise options in discrete time. This type of option can be exercised up to a finite number of times over the lifetime of the contract. We allow multiple exercise of the option at each time point up to a constraint, a feature relevant for pricing swing options in energy markets. It is shown that, in the case where an option can be exercised an equal number of times at each time point, the problem can be reduced to the case of a single exercise possibility at each time. In the general case there is not a solution of this type. We develop a dual representation for the problem and give an algorithm for calculating both lower and upper bounds for the prices of such multiple exercise options.     

Regularized collocation method for Fredholm integral equations of the first kind

In this paper we consider a collocation method for solving Fredholm integral equations of the first kind, which is known to be an ill-posed problem. An “unregularized” use of this method can give reliable results in the case when the rate at which smallest singular values of the collocation matrices decrease is known a priori. In this case the number of collocation points plays the role of a regularization parameter. If the a priori information mentioned above is not available, then a combination of collocation with Tikhonov regularization can be the method of choice. We analyze such regularized collocation in a rather general setting, when a solution smoothness is given as a source condition with an operator monotone index function. This setting covers all types of smoothness studied so far in the theory of Tikhonov regularization. One more issue discussed in this paper is an a posteriori choice of the regularization parameter, which allows us to reach an optimal order of accuracy for deterministic noise model without any knowledge of solution smoothness.     

Distribution of the Number of Successes in Success Runs of Length at Least <Emphasis Type="Italic">k</Emphasis> in Higher-Order Markovian Sequences

We consider the distribution of the number of successes in success runs of length at least k in a binary sequence. One important application of this statistic is in the detection of tandem repeats among DNA sequence segments. In the literature, its distribution has been computed for independent sequences and Markovian sequences of order one. We extend these results to Markovian sequences of a general order. We also show that the statistic can be represented as a function of the number of overlapping success runs of lengths k and k + 1 in the sequence, and give immediate consequences of this representation. AMS 2000 Subject Classification 60E05, 60J05