Solving polynomial least squares problems via semidefinite programming relaxations

A polynomial optimization problem whose objective function is represented as a sum of positive and even powers of polynomials, called a polynomial least squares problem, is considered. Methods to transform a polynomial least square problem to polynomial semidefinite programs to reduce degrees of the polynomials are discussed. Computational efficiency of solving the original polynomial least squares problem and the transformed polynomial semidefinite programs is compared. Numerical results on selected polynomial least square problems show better computational performance of a transformed polynomial semidefinite program, especially when degrees of the polynomials are larger.     

Construction of polynomial solutions to some boundary value problems for Poisson’s equation

A polynomial solution of the inhomogeneous Dirichlet problem for Poisson’s equation with a polynomial right-hand side is found. An explicit representation of the harmonic functions in the Almansi formula is used. The solvability of a generalized third boundary value problem for Poisson’s equation is studied in the case when the value of a polynomial in normal derivatives is given on the boundary. A polynomial solution of the third boundary value problem for Poisson’s equation with polynomial data is found.     

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.     

Reid-Zhi符号数值混合消元方法的一个应用

将Reid和Zhi提出的符号数值混合消元方法应用于求解多项式优化问题,将多项式优化问题转化为矩阵最小特征值求解问题,并在Maple软件中实现了算法.     

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].     

A Linear System Arising from a Polynomial Problem and Its Applications

A linear system arising from a polynomial problem in the approximation theory is studied, and the necessary and sufficient conditions for existence and uniqueness of its solutions are presented. Together with a class of determinant identities, the resulting theory is used to determine the unique solution to the polynomial problem. Some homogeneous polynomial identities as well as results on the structure of related polynomial ideals are just by-products.     

Parameter choices on Guruswami–Sudan algorithm for polynomial reconstruction

Guruswami–Sudan algorithm for polynomial reconstruction problem plays an important role in the study of error-correcting codes. In this paper, we study new better parameter choices in Guruswami–Sudan algorithm for the polynomial reconstruction problem. As a consequence, our result gives a better upper bound for the number of solutions for the polynomial reconstruction problem comparing with the original algorithm.     

The complexity of approximating a nonlinear program

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

A polynomially solvable case of the pooling problem

Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview of known complexity results and remaining open problems to further characterize the border between (strongly) NP-hard and polynomially solvable cases of the pooling problem.     

The external constraint 4 nonempty part sandwich problem

List partitions generalize list colourings. Sandwich problems generalize recognition problems. The polynomial dichotomy (NP-complete versus polynomial) of list partition problems is solved for 4-dimensional partitions with the exception of one problem (the list stubborn problem) for which the complexity is known to be quasipolynomial. Every partition problem for 4 nonempty parts and only external constraints is known to be polynomial with the exception of one problem (the 2K₂-partition problem) for which the complexity of the corresponding list problem is known to be NP-complete. The present paper considers external constraint 4 nonempty part sandwich problems. We extend the tools developed for polynomial solutions of recognition problems obtaining polynomial solutions for most corresponding sandwich versions. We extend the tools developed for NP-complete reductions of sandwich partition problems obtaining the classification into NP-complete for some external constraint 4 nonempty part sandwich problems. On the other hand and additionally, we propose a general strategy for defining polynomial reductions from the 2K₂-partition problem to several external constraint 4 nonempty part sandwich problems, defining a class of 2K₂-hard problems. Finally, we discuss the complexity of the Skew Partition Sandwich Problem.     

A strongly polynomial FPTAS for the symmetric quadratic knapsack problem

The symmetric quadratic knapsack problem (SQKP), which has several applications in machine scheduling, is NP-hard. An approximation scheme for this problem is known to achieve an approximation ratio of (1 + ?) for any ? > 0. To ensure a polynomial time complexity, this approximation scheme needs an input of a lower bound and an upper bound on the optimal objective value, and requires the ratio of the bounds to be bounded by a polynomial in the size of the problem instance. However, such bounds are not mentioned in any previous literature. In this paper, we present the first such bounds and develop a polynomial time algorithm to compute them. The bounds are applied, so that we have obtained for problem (SQKP) a fully polynomial time approximation scheme (FPTAS) that is also strongly polynomial time, in the sense that the running time is bounded by a polynomial only in the number of integers in the problem instance.     

On the computational complexity of a merge recognition problem

A problem arising from the work of C.A.R. Hoare on parallel programming is that of deciding whether a given string ? is a “merge” of two other given strings σ and τ. We describe a polynomial time algorithm for this problem. This algorithm can easily be extended to check, in polynomial time, whether ? is a merge of any fixed number of strings. The problem for an arbitrary number of strings is shown to be NP-complete and so is unlikely to have a polynomial time algorithm.     

Cauchy problem for a class of integro-differential equations

We consider the Cauchy problem for integro-differential equations containing Wiener-Hopf operators. We define the characteristic polynomial of the problem. We show that if all roots of the characteristic polynomial have negative imaginary parts and the corresponding strong moment problem is solvable, then the problem is equivalent to a Wiener-Hopf integral equation. We consider an example of an application of the result to a problem related to nonlocal wave interaction.     

Construction of polynomial solutions to the Dirichlet problem for the polyharmonic equation in a ball

An algorithm is proposed for the analytical construction of a polynomial solution to Dirichlet problem for an inhomogeneous polyharmonic equation with a polynomial right-hand side and polynomial boundary data in the unit ball.     

Polynomial-time algorithms for regular set-covering and threshold synthesis

A set-covering problem is called regular if a cover always remains a cover when any column in it is replaced by an earlier column. From the input of the problem - the coefficient matrix of the set-covering inequalities - it is possible to check in polynomial time whether the problem is regular or can be made regular by permuting the columns. If it is, then all the minimal covers are generated in polynomial time, and one of them is an optimal solution. The algorithm also yields an explicit bound for the number of minimal covers. These results can be used to check in polynomial time whether a given set-covering problem is equivalent to some knapsack problem without additional variables, or equivalently to recognize positive threshold functions in polynomial time. However, the problem of recognizing when an arbitrary Boolean function is threshold is NP-complete. It is also shown that the list of maximal non-covers is essentially the most compact input possible, even if it is known in advance that the problem is regular.     

The weighted euclidean 1-center problem

We present a polynomial algorithm for the weighted 1-center problem (indeed minimization of the ratio of convex quadratic and an affine function over a polyhedral set). In the location problem, the complexity is polynomial in the dimension of the space.     

Generating Functions for Computing the Myerson Value

The complexity of a computational problem is the order of computational resources which are necessary and sufficient to solve the problem. The algorithm complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce polynomial time algorithms based in generating functions for computing the Myerson value in weighted voting games restricted by a tree. Moreover, we apply the new generating algorithm for computing the Myerson value in the Council of Ministers of the European Union restricted by a communication structure.     

工件加工时间增加的排序问题(1‖Cmax)

讨论了工件加工时间随工件开工时间线性增加的排序问题,考虑的目标函数是最大完工时间,证明了加工时间是简单线性增加情况下最大完工时间问题是多项式时间可解的,对于加工时间是一般线性增加情况,研究了最优排序的性质,同时证明了两种特殊情况下最大完工时间问题也是多项式时间可解的。     

A note on the complexity of a phaseless polynomial interpolation

In this paper we revisit the classical problem of polynomial interpolation, with a slight twist; namely, polynomial evaluations are available up to a group action of the unit circle on the complex plane. It turns out that this new setting allows for a phaseless recovery of a polynomial in a polynomial time.