A Gröbner Free Alternative for Polynomial System Solving

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton's iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation of the system. We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We show that the theoretical complexity of our algorithm is well reflected in practice and we exhibit some cases for which our program is more efficient than the other available software.     

Theoretical Complexity of Grid Cover Problems Used in Radar Applications

Modern radars are highly flexible systems, relying on digital antennas to dynamically control the radar beam shape and position through electronic circuits. Radar surveillance is performed by sequential emission of different radar beams. Optimization of radar surveillance requires finding a minimal subset of radar beams, which covers and ensures detection over the surveillance space, among a collection of available radar beams with different shapes and positions, thus minimizing the required scanning time. Optimal radar surveillance can be modelled by grid covering, a specific geometric case of set covering where the universe set is laid out on a grid, representing the radar surveillance space, which must be covered using available subsets, representing the radar beams detection areas. While the set cover problem is generally difficult to solve optimally, certain geometric cases can be optimized in polynomial time. This paper studies the theoretical complexity of grid cover problems used for modelling radar surveillance, proving that unidimensional grids can be covered by strongly polynomial algorithms based on dynamic programming, whereas optimal covering of bidimensional grids is generally non-deterministic polynomially hard.     

Time complexity of single machine scheduling with stochastic precedence constraints

This paper deals with single machine scheduling problems with stochastic precedence relations (so calledGERT networks). Until now most investigations on such problems, dealt with algorithms running in polynomial time. On the other hand, for scheduling problems with deterministic precedence relations exist a lot of results about time complexity. Therefore, the object of this paper is to consider time complexity of scheduling problems with stochastic precedence constraints and to describe the boundary between theNP-hard problems and those which can be solved in polynomial time.     

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.     

The Computational Complexity of the Chow Form

We present a bounded probability algorithm for the computation of the Chowforms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L. M. Pardo, and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic overdetermined polynomial equation system.     

Exponential approximation schemata for some network design problems

We study approximation of some well-known network design problems such as the traveling salesman problem (for both minimization and maximization versions) and the min steiner tree problem by moderately exponential algorithms. The general goal of the issue of moderately exponential approximation is to catch up on polynomial inapproximability by designing superpolynomial algorithms achieving approximation ratios unachievable in polynomial time. Worst-case running times of such algorithms are significantly smaller than those needed for optimal solutions of the problems handled.     

Complexity of solving parametric polynomial systems

In this paper, we present three algorithms: the first one solves zero-dimensional parametric homogeneous polynomial systems within single exponential time in the number n of unknowns; it decomposes the parameter space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factirizes absolutely multivariate parametic polynomials within single exponential time in n and in the upper bound d on the degree of the factorized polynomials. The third algorithm decomposes algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly in the values of the parameters. The complexity bound for this algorithm is double exponential in n. On the other hand, the lower bound on the complexity of the problem of resolution of parametric polynomial systems is double exponential in n. Bibliography: 72 titles.     

Fast Linear Homotopy to Find Approximate Zeros of Polynomial Systems

We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.     

PotLLL: a polynomial time version of LLL with deep insertions

Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in practice but the running time seems to explode. Weaker variants of DeepLLL, where the insertions are restricted to blocks, behave nicely in practice concerning the running time. However no proof of polynomial running time is known. In this paper PotLLL, a new variant of DeepLLL with provably polynomial running time, is presented. We compare the practical behavior of the new algorithm to classical LLL, BKZ as well as blockwise variants of DeepLLL regarding both the output quality and running time.     

A deterministic view of random sampling and its use in geometry

The combination of divide-and-conquer and random sampling has proven very effective in the design of fast geometric algorithms. A flurry of efficient probabilistic algorithms have been recently discovered, based on this happy marriage. We show that all those algorithms can be derandomized with only polynomial overhead. In the process we establish results of independent interest concerning the covering of hypergraphs and we improve on various probabilistic bounds in geometric complexity. For example, givenn hyperplanes ind-space and any integerr large enough, we show how to compute, in polynomial time, a simplicial packing of sizeO(r ^d ) which coversd-space, each of whose simplices intersectsO(n/r) hyperplanes.Bernard Chazelle wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8700917. Joel Friedman wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and this Office of Naval Research under Grant N00014-87-K-0467.A preliminary version of this work has appeared in the proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science (1988). 539–549.     

A strongly polynomial simplex method for the linear fractional assignment problem

In this paper we show that the complexity of the simplex method for the linear fractional assignment problem (LFAP) is strongly polynomial. Although LFAP can be solved in polynomial time using various algorithms such as Newton’s method or binary search, no polynomial time bound for the simplex method for LFAP is known.     

On scheduling with the non-idling constraint

In this paper, we give an overview of the main results obtained on the complexity of scheduling under the non-idling constraint, i.e, when the jobs assigned to each machine must be processed with no intermediate delay. That constraint is met in practice when the cost of intermediate idle time is too high due to the idle time itself and/or the machine restarting. The non idling constraint is a strong constraint that often needs a new solving approach and most results about classical scheduling problems do not easily extend to the non-idling variant of the problem. In this survey, we mainly consider the non-idling variants of the basic scheduling problems. So, we first present basic properties, complexity results and some algorithms concerning the one-machine non-idling scheduling problem. Then we consider the $m$ -machine non idling scheduling problem. We show that a few basic problems may be solved by rather easy extensions of the algorithm solving their classical counterpart. However, the complexity status of the non idling version of quite easy polynomial basic problems remains an open question. We finally consider a more constrained version of non-idling, called the “homogeneously non idling” constraint, where for any subset of machines, the union of their busy intervals must make an interval and we present the structural property that leads to a polynomial algorithm for unit time jobs and a weak precedence. We conclude by giving some research directions that seem quite interesting to study both for theoretical and practical issues.     

Polar Varieties,Real Equation Solving,and Data Structures: The Hypersurface Case

In this paper we apply for the first time a new method for multivariate equation solving which was developed for complex root determination to therealcase. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem-adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in nonuniform sequential time that ispolynomialin the length of the input (given in straight-line program representation) and an adequately definedgeometric degree of the equation system.Replacing the notion of geometric degree of the given polynomial equation system by a suitably definedreal (or complex) degreeof certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.     

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.     

Some parallel methods for polynomial root-finding

Parallelizations of various different methods for determining the roots of a polynomial are discussed. These include methods which locate a single root only as well as those which find all roots. Some techniques for parallelizing such methods are identified and some examples are given. Further places in polynomial root-finding algorithms where parallel behaviour can be introduced are described. Results are presented for a range of programs written to test the effectiveness of methods presented here.     

Efficient Algorithms for the Discrete Gabor Transform with a Long Fir Window

The Discrete Gabor Transform (DGT) is the most commonly used signal transform for signal analysis and synthesis using a linear frequency scale. The development of the Linear Time-Frequency Analysis Toolbox (LTFAT) has been based on a detailed study of many variants of the relevant algorithms. As a side result of these systematic developments of the subject, two new methods are presented here. Comparisons are made with respect to the computational complexity, and the running time of optimised implementations in the C programming language. The new algorithms have the lowest known computational complexity and running time when a long FIR window is used. The implementations are freely available for download. By summarizing general background information on the state of the art, this article can also be seen as a research survey, sharing with the readers experience in the numerical work in Gabor analysis.     

On the class of restricted linear information systems

In the paper the class of restricted linear information systems is described completely. For decision tables over each such information system there exist low upper bounds on minimal complexity of decision trees and polynomial algorithms of decision tree optimization for various complexity measures. A corollary connected with combinatorial geometry is considered.     

A Theory of Complexity for Continuous Time Systems

We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P.     

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require n^δ (δ>0) colors even for bounded chromatic (k-colorable for fixed k) n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19 , 204–234 (1995)] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work as long as p≥n^−α(k)+ϵ where α(k)=(2k)/((k−1)(k+2)) and ϵ is a positive constant. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13, 125–158 (1998)     

The computation of multiple roots of a polynomial

This paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise, and whose exact form contains multiple roots. The addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots, but the algorithms presented in this paper preserve the multiplicities of the roots. In particular, even though the given polynomial is corrupted by noise, and all computations are performed on these inexact coefficients, the algorithms ‘sew’ together the simple roots that originate from the same multiple root, thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial. The algorithms described in this paper do not require that the noise level imposed on the coefficients be known, and all parameters are calculated from the given inexact coefficients. Examples that demonstrate the theory are presented.