A probabilistic analysis of a measure of combinatorial complexity for the central curve

We investigate certain combinatorial properties of the central curve associated with interior point methods for linear optimization. We define a measure of complexity for the curve in terms of the number of turns, or changes of direction, that it makes in a geometric sense, and then perform an average case analysis of this measure for P-matrix linear complementarity problems. We show that the expected number of nondegenerate turns taken by the central curve is bounded by n ²-n, where the expectation is taken with respect to a sign-invariant probability distribution on the problem data. As an alternative measure of complexity, we also consider the number of times the central curve intersects with a wide class of algebraic hypersurfaces, including such objects as spheres and boxes. As an example of the results obtained, we show that the primal and dual variables in each coordinate of the central curve cross each other at most once, on average. As a further example, we show that the central curve intersects any sphere centered at the origin at most twice, on average. Received May 28, 1998 / Revised version received October 12, 1999?Published online December 15, 1999     

On computation complexity problems concerning relation algebras

In this paper, we study the computation complexity of some algebraic combinatorial problems that are closely associated with the graph isomorphism problem. The key point of our considerations is a relation algebra which is a combinatorial analog of a cellular algebra. We present upper bounds on the complexity of central algorithms for relation algebras such as finding the standard basis of extensions and intersection of relation algebras. Also, an approach is described to the graph isomorphism problem involving Schurian relation algebras (these algebras arise from the centralizing rings of permutation groups). We also discuss a number of open problems and possible directions for further investigation. Bibliography: 18 titles. Translated by I. N.Ponomarenko. Translated fromZapiski Nauchnykh Seminarov POMI, Vol 202, 1992, pp. 116–134.     

What Is the Complexity of Solution-Restricted Operator Equations?

We study the worst case complexity of operator equations Lu = f where L: G → X is a bounded linear injection of normed linear spaces. Past work on the complexity of such problems has generally required the class F of problem elements f to be the unit ball of X. However, there are many problems for which this choice of F yields unsatisfactory results. Mixed elliptic—hyperbolic problems are one example. the difficulty being that our technical tools are nor strong enough to give good complexity bounds. Ill-posed problems are another example. because we know that the complexity of computing finite-error approximations is infinite if F is a ball in X. In this paper, we pursue another idea. Rather than directly restrict the class F of problem elements f, we will consider problems that are solution-restricted: i.e., we restrict the class U of solution elements u. In particular, we assume that U is the unit hall of a normed linear space W that is densely, continuously embedded in G. The main idea is that our problem can now be reduced to the standard approximation problem of approximating the embedding of W into G.This allows us to characterize optimal information and algorithms for our problem..We use this idea to study three problems: the Tricomi problem (a mixed hyperbolic— elliptic problem arising in the study of transonic flow), the inverse finite Laplace transform (an ill-posed problem arising. e.g.. in geomathematics), and the backwards heat equation. We determine the problem complexity and derive nearly optimal algorithms for each of these problems.     

On the Intrinsic Complexity of Elimination Theory

We consider the intrinsic complexity of selected algorithmic problems of classical elimination theory in algebraic geometry. The inputs and outputs of these problems are given by finite sets of polynomials which we represent alternatively in dense form or by straight line programs. We begin with an overview on the known upper bounds for the sequential and parallel time complexity of these problems and show then that in the most important cases these bounds are tight. Our lower bound results include both the relative and the absolute viewpoint of complexity theory. On one side we give reductions of fundamental questions of elimination theory to NP- and P^#-complete problems and on the other side we show that some of these questions may have exponential size outputs. In this way we confirm the intrinsically exponential character of algorithmic problems in elimination theory whatever the type of data structure may be.     

Minisum location problem with farthest Euclidean distances

The paper formulates an extended model of Weber problem in which the customers are represented by convex demand regions. The objective is to generate a site in R ² that minimizes the sum of weighted Euclidean distances between the new facility and the farthest points of demand regions. This location problem is decomposed into a polynomial number of subproblems: constrained Weber problems (CWPs). A projection contraction method is suggested to solve these CWPs. An algorithm and the complexity analysis are presented. Three techniques of bound test, greedy choice and choice of starting point are adopted to reduce the computational time. The restricted case of the facility is also considered. Preliminary computational results are reported, which shows that with the above three techniques adopted the algorithm is efficient.The authors were supported by the dissertation fund of Nari-Relays Corporation.     

An Order-theoretic Framework for the Greedy Algorithm with Applications to the Core and Weber Set of Cooperative Games

An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in R ⁿ . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory.As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model.     

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located. The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.     

On the Complexity of Computing Error Bounds

We consider the cost of estimating an error bound for the computed solution of a system of linear equations, i.e., estimating the norm of a matrix inverse. Under some technical assumptions we show that computing even a coarse error bound for the solution of a triangular system of equations costs at least as much as testing whether the product of two matrices is zero. The complexity of the latter problem is in turn conjectured to be the same as matrix multiplication, matrix inversion, etc. Since most error bounds in practical use have much lower complexity, this means they should sometimes exhibit large errors. In particular, it is shown that condition estimators that: (1) perform at least one operation on each matrix entry; and (2) are asymptotically faster than any zero tester, must sometimes over or underestimate the inverse norm by a factor of at least , where n is the dimension of the input matrix, k is the bitsize, and where either or grows faster than any polynomial in n . Our results hold for the RAM model with bit complexity, as well as computations over rational and algebraic numbers, but not real or complex numbers. Our results also extend to estimating error bounds or condition numbers for other linear algebra problems such as computing eigenvectors. September 10, 1999. Final version received: August 23, 2000.     

Approximating Rough Stochastic PDEs

We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.     

Index theory of geometric Fredholm operators on varieties with isolated singularities

It sometimes happens that geometric elliptic differential operators on a noncompact Riemannian manifold are Fredholm. The smooth parts of singular algebraic varieties provide examples of complete and incomplete manifolds where this can happen. The indices of such operators often provide topological or geometric information about the singular variety. This paper shows that the operators of the title represent K homology elements and solves the index problem for these operators by exhibiting equivalent K homology cycles in topological form.This material is based upon work supported by the National Science Foundation under Grant No. DMS-8501513.     

The theoretical and empirical rate of convergence for geometric branch-and-bound methods

Geometric branch-and-bound solution methods, in particular the big square small square technique and its many generalizations, are popular solution approaches for non-convex global optimization problems. Most of these approaches differ in the lower bounds they use which have been compared empirically in a few studies. The aim of this paper is to introduce a general convergence theory which allows theoretical results about the different bounds used. To this end we introduce the concept of a bounding operation and propose a new definition of the rate of convergence for geometric branch-and-bound methods. We discuss the rate of convergence for some well-known bounding operations as well as for a new general bounding operation with an arbitrary rate of convergence. This comparison is done from a theoretical point of view. The results we present are justified by some numerical experiments using the Weber problem on the plane with some negative weights.     

Binary Representations of Finite Fields and Their Application to Complexity Theory

Binary representations of finite fields are defined as an injective mapping from a finite field tol-tuples with components in {0, 1} where 0 and 1 are elements of the field itself. This permits one to study the algebraic complexity of a particular binary representation, i.e., the minimum number of additions and multiplications in the field needed to compute the binary representation. The two-way complexity of a binary representation is defined as the sum of the algebraic complexities of the binary representation and of its inverse mapping. Two particular binary representations are studied: the standard representation and the logarithmic representation. A method of surrogate computation is developed and used to deduce relationships between the algebraic complexities of certain functions. The standard representation of a finite field is shown to be among the two-way easiest representations of this field. In particular, the standard representation of a finite field with characteristicpis two-way easy wheneverp− 1 has only small prime factors. For any finite field having a two-way easy binary representation, the algebraic complexity in this field is shown to be essentially equivalent to Boolean circuit complexity. For any finite field, the Boolean circuit complexity of Zech's (or Jacobi's) logarithm is shown to be closely related to the Boolean circuit complexity of the discrete logarithm problem that is used in public-key cryptography.     

On Markovian Multi-Class,Multi-Server Queueing

Multi-class multi-server queueing problems are a generalisation of the well-known M/M/k queue to arrival processes with clients of N types that require exponentially distributed service with different average service times. In this paper, we give a procedure to construct exact solutions of the stationary state equations using the special structure of these equations. Essential in this procedure is the reduction of a part of the problem to a backward second order difference equation with constant coefficients. It follows that the exact solution can be found by eigenmode decomposition. In general eigenmodes do not have a simple product structure as one might expect intuitively. Further, using the exact solution, all kinds of interesting performance measures can be computed and compared with heuristic approximations (insofar available in the literature). We provide some new approximations based on special multiplicative eigenmodes, including the dominant mode in the heavy traffic limit. We illustrate our methods with numerical results. It turns out that our approximation method is better for higher moments than some other approximations known in the literature. Moreover, we demonstrate that our theory is useful to applications where correlation between items plays a role, such as spare parts management.     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Bottleneck shortest paths on a partially ordered scale

In bottleneck combinatorial problems, admissible solutions are compared with respect to their maximal elements. In such problems, one may work with an ordinal evaluation scale instead of a numerical scale. We consider here a generalization of this problem in which one only has a partially ordered scale (instead of a completely ordered scale). After the introduction of a mappimax comparison operator between sets of evaluations (which boils down to the classical operator when the order is complete), we establish computational complexity results for this variation of the shortest path problem. Finally, we formulate our problem as an algebraic shortest path problem and suggest adequate algorithms to solve it in the subsequent semiring.Received: June 2002, Revised: March 2003, AMS classification: 05C50, 16Y60, 90B06Olivier Spanjaard: Corresponding author.     

Computation Models for Parameterized Complexity

A parameterized computational problem is a set of pairs (x, k), where k is a distinguished item called “parameter”. FPT is the class of fixed-parameter tractable problems: for any fixed value of k, they are solvable in time bounded by a polynomial of degree α, where α is a constant not dependent on the parameter. In order to deal with parameterized intractability, Downey and Fellows have introduced a hierarchy of classes W[l] ? W[2] ? ? containing likely intractable parameterized problems, and they have shown that such classes have many natural, complete languages. In this paper we analyze several variations of the halting problem for nondeterministic Turing machines with parameterized time, and we show that its parameterized complexity strongly depends on some resources like the number of tapes, head and internal states, and on the size of the alphabet. Notice that classical polynomial-time complexity fails in distinguishing such features. As byproducts, we show that parameterized complexity is a useful tool for the study of the intrinsic power of some computational models, and we underline the different “computational powers” of some levels of the parameterized hierarchy.     

Duality for location problems with unbounded unit balls

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the “distances” are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications.     

Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$

How to find “best rational approximations” of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)? In this paper we specify this problem and make first steps in its study. It contains the classical problems of Diophantine and simultaneous approximation as particular subcases but in general is much wider. We prove estimates for n = 2 for both totally real and complex cases and give an algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.