Scheduling precedence graphs of bounded height

The existence of a schedule for a partially ordered set of unit length tasks on m identical processors is known to be NP-complete (J. D. Ullman, NP-complete scheduling problems, J. Comput. System Sci., 10 (1975), 384–393). The problem remains NP-complete even if we restrict the precedence graph to be of height bounded by a constant. (J. K. Lenkstra and A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res., 26 (1978), 22–35; D. Dolev and M. K. Warmuth, “Scheduling Flat Graphs,” IBM Research Report RJ 3398, 1982). In these NP-completeness proofs the upper bound on the number of available processors varies with the problem instance. We present a polynomial algorithm for the case where the upper bound on the number of available processors and the height of the precedence graph are both constants.     

Boundary penalty finite element methods for blending surfaces,III. Superconvergence and stability and examples

This paper is Part III of the study on blending surfaces by partial differential equations (PDEs). The blending surfaces in three dimensions (3D) are taken into account by three parametric functions, x(r,t),y(r,t) and z(r,t). The boundary penalty techniques are well suited to the complicated tangent (i.e., normal derivative) boundary conditions in engineering blending. By following the previous papers, Parts I and II in Li (J. Comput. Math. 16 (1998) 457–480; J. Comput. Appl. Math. 110 (1999) 155–176) the corresponding theoretical analysis is made to discover that when the penalty power σ=2, σ=3 (or 3.5) and 0<σ⩽1.5 in the boundary penalty finite element methods (BP-FEMs), optimal convergence rates, superconvergence and optimal numerical stability can be achieved, respectively. Several interesting samples of 3D blending surfaces are provided, to display the remarkable advantages of the proposed approaches in this paper: unique solutions of blending surfaces, optimal blending surfaces in minimum energy, ease in handling the complicated boundary constraint conditions, and less CPU time and computer storage needed. This paper and Li (J. Comput. Math. 16 (1998) 457–480; J. Comput. Appl. Math.) provide a foundation of blending surfaces by PDE solutions, a new trend of computer geometric design.     

An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

A filter-line-search method for unconstrained optimization

A new filter-line-search algorithm for unconstrained nonlinear optimization problems is proposed. Based on the filter technique introduced by Fletcher and Leyffer (Math. Program. 91:239–269, 2002) it extends an existing technique of Wächter and Biegler (SIAM J. Comput. 16:1–31, 2005) for nonlinear equality constrained problem to the fully general unconstrained optimization problem. The proposed method, which differs from their approach, does not depend on any external restoration procedure. Global and local quadratic convergence is established under some reasonable conditions. The results of numerical experiments indicate that it is very competitive with the classical line search algorithm.     

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.     

Steady state analysis of balanced‐allocation routing

We compare the long‐term, steady‐state performance of a variant of the standard Dynamic Alternative Routing (DAR) technique commonly used in telephone and ATM networks, to the performance of a path‐selection algorithm based on the “balanced‐allocation” principle [Y. Azer, A. Z. Broder, A. R. Karlin, and E. Upfal, SIAM J Comput 29(1) (2000), 180–200; M. Mitzenmacher, Ph.D. Thesis, University of California, Berkeley, August 1996]; we refer to this new algorithm as the Balanced Dynamic Alternative Routing (BDAR) algorithm. While DAR checks alternative routes sequentially until available bandwidth is found, the BDAR algorithm compares and chooses the best among a small number of alternatives. We show that, at the expense of a minor increase in routing overhead, the BDAR algorithm gives a substantial improvement in network performance, in terms both of network congestion and of bandwidth requirement. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Blossom V: a new implementation of a minimum cost perfect matching algorithm

We describe a new implementation of the Edmonds’s algorithm for computing a perfect matching of minimum cost, to which we refer as Blossom V. A key feature of our implementation is a combination of two ideas that were shown to be effective for this problem: the “variable dual updates” approach of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and the use of priority queues. We achieve this by maintaining an auxiliary graph whose nodes correspond to alternating trees in the Edmonds’s algorithm. While our use of priority queues does not improve the worst-case complexity, it appears to lead to an efficient technique. In the majority of our tests Blossom V outperformed previous implementations of Cook and Rohe (INFORMS J Comput 11(2):138–148, 1999) and Mehlhorn and Schäfer (J Algorithmics Exp (JEA) 7:4, 2002), sometimes by an order of magnitude. We also show that for large VLSI instances it is beneficial to update duals by solving a linear program, contrary to a conjecture by Cook and Rohe.     

General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a local as well as a semilocal convergence analysis for Newton’s method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. Our hypotheses involve m-Fréchet-differentiable operators and general Lipschitz-type hypotheses, where m≥2 is a positive integer. The new convergence analysis unifies earlier results; it is more flexible and provides a finer convergence analysis than in earlier studies such as Argyros in J. Comput. Appl. Math. 131:149–159, 2001, Argyros and Hilout in J. Appl. Math. Comput. 29:391–400, 2009, Argyros and Hilout in J. Complex. 28:364–387, 2012, Argyros et al. Numerical Methods for Equations and Its Applications, CRC Press/Taylor & Francis, New York, 2012, Gutiérrez in J. Comput. Appl. Math. 79:131–145, 1997, Ren and Argyros in Appl. Math. Comput. 217:612–621, 2010, Traub and Wozniakowski in J. Assoc. Comput. Mech. 26:250–258, 1979. Numerical examples are presented further validating the theoretical results.     

Comment on: a two-stage fourth-order “almost” P-stable method for oscillatory problems

In Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118) a fourth-order “almost” P-stable method for y″=f(x,y) is proposed. We claim that it is possible to retrieve this combination of multistep methods by means of the theory of parameterized Runge-Kutta-Nyström (RKN) methods and moreover to generalize the method discussed by Chawla and Al-Zanaidi (J. Comput. Appl. Math. 89 (1997) 115–118).     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

Estimates in the Hardy-Sobolev space of the annulus and stability result

The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space H ^k,∞; k ∈ ?* of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L.Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S.Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces H ^k,∞, C. R., Math., Acad. Sci. Paris 347 (2009), 1001–1006. As an application, we prove a logarithmic stability result for the inverse problem of identifying a Robin parameter on a part of the boundary of an annular domain starting from its behavior on the complementary boundary part.     

Global Bifurcation Diagrams of Steady States of Systems of PDEs via Rigorous Numerics: a 3-Component Reaction-Diffusion System

In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol. 53(4):617–641, 2006] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comput. 79(271):1565–1584, 2010], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.     

A note on domination, girth and minimum degree

Let G be a graph of order n, minimum degree δ?2, girth g?5 and domination number γ. In 1990 Brigham and Dutton [Bounds on the domination number of a graph, Q. J. Math., Oxf. II. Ser. 41 (1990) 269-275] proved that γ?⌈n/2-g/6⌉. This result was recently improved by Volkmann [Upper bounds on the domination number of a graph in terms of diameter and girth, J. Combin. Math. Combin. Comput. 52 (2005) 131-141; An upper bound for the domination number of a graph in terms of order and girth, J. Combin. Math. Combin. Comput. 54 (2005) 195-212] who for i∈{1,2} determined a finite set of graphs G_i such that γ?⌈n/2-g/6-(3i+3)/6⌉ unless G is a cycle or G∈G_i.Our main result is that for every i∈N there is a finite set of graphs G_i such that γ?n/2-g/6-i unless G is a cycle or G∈G_i. Furthermore, we conjecture another improvement of Brigham and Dutton's bound and prove a weakened version of this conjecture.     

Complementary Lidstone interpolation on scattered data sets

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell’Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691–1707, 2012) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell’Accio, Appl Numer Math 52:339–361, 2005). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77–90, 2005) and drawn on by Agarwal et al. (2009) and Agarwal and Wong (J Comput Appl Math 234:2543–2561, 2010). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided.     

Periodic multisequences with large error linear complexity

Generalizing the theory of k-error linear complexity for single sequences over a finite field, Meidl et al. (J. Complexity 23(2), 169–192 (2007)) introduced three possibilities of defining error linear complexity measures for multisequences. A good keystream sequence must possess a large linear complexity and a large k-error linear complexity simultaneously for suitable values of k. In this direction several results on the existence, and lower bounds on the number, of single sequences with large k-error linear complexity were proved in Meidl and Niederreiter (Appl. Algebra Eng. Commun. Comput. 14(4), 273–286 (2003)), Niederreiter (IEEE Trans. Inform. Theory 49(2), 501–505 (2003)) and Niederreiter and Shparlinski (In: Paterson (ed.) 9th IMA International Conference on Cryptography and Coding (2003)). In this paper we discuss analogous results for the case of multisequences. We also present improved bounds on the error linear complexity and on the number of sequences satisfying such bounds for the case of single sequences.     

Better upper bounds on the QOBDD size of integer multiplication

We show that the middle bit of the multiplication of two n-bit integers can be computed by an ordered binary decision diagram (OBDD) of size less than 2.8·2^6n/5. This improves the previously known upper bound of by Woelfel (New Bounds on the OBDD-size of integer multiplication via Universal Hashing, J. Comput. System Sci. 71(4) (2005) 520-534). The experimental results suggest that our exponent of 6n/5 is optimal or at least very close to optimal. A general upper bound of O(2^3n/2) on the OBDD size of each output bit of the multiplication is also presented.     

An optimal Steffensen-type family for solving nonlinear equations

In this paper, a general family of Steffensen-type methods with optimal order of convergence for solving nonlinear equations is constructed by using Newton’s iteration for the direct Newtonian interpolation. It satisfies the conjecture proposed by Kung and Traub [H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651] that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 2^m−1. Its error equations and asymptotic convergence constants are obtained. Finally, it is compared with the related methods for solving nonlinear equations in the numerical examples.     

A combinatorial characterization of smooth LTCs and applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet, we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate‐limit of LTCs and whether asymptotically good linear LTCs with constant query complexity exist. In this paper, we provide a combinatorial characterization of smooth locally testable codes, which are locally testable codes whose associated tester queries every bit of the tested word with equal probability. Our main contribution is a combinatorial property defined on the Tanner graph associated with the code tester (“well‐structured tester”). We show that a family of codes is smoothly locally testable if and only if it has a well‐structured tester. As a case study we show that the standard tester for the Hadamard code is “well‐structured,” giving an alternative proof of the local testability of the Hadamard code, originally proved by (Blum, Luby, Rubinfeld, J. Comput. Syst. Sci. 47 (1993) 549–595) (STOC 1990). Additional connections to the works of (Ben‐Sasson, Harsha, Raskhodnikova, SIAM J. Comput 35 (2005) 1–21) (SICOMP 2005) and of (Lachish, Newman and Shapira, Comput Complex 17 (2008) 70–93) are also discussed. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 280–307, 2016     

Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h???h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), pp. 135–162], the two-level error estimator from [M. Maischak, P. Mund, and E. Stephan, Adaptive multilevel BEM for acoustic scattering, 585 Comput. Methods Appl. Mech. Eng. 150 (1997), pp. 351–367], and the averaging error estimator from [C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comput. 29 (2007), pp. 782–810]. All of these a posteriori error estimators are simple in the following sense: first, the numerical analysis can be done within the same mathematical framework, namely localization techniques for the energy norm. Second, there is almost no implementational overhead for the realization.     

Pareto Optimizing and Kuhn–Tucker Stationary Sequences

We study the relation between weakly Pareto minimizing and Kuhn–Tucker stationary nonfeasible sequences for vector optimization under constraints, where the weakly Pareto (efficient) set may be empty. The work is placed in a context of Banach spaces and the constraints are described by a functional taking values in a cone. We characterize the asymptotic feasibility in terms of the constraint map and the asymptotic efficiency via a Kuhn–Tucker system completely approximate, distinguishing the classical bounded case from the nontrivial unbounded one. The latter requires Auslender–Crouzeix type conditions and Ekeland's variational principle for constrained vector problems.