Estimation of sparse hessian matrices and graph coloring problems

Large scale optimization problems often require an approximation to the Hessian matrix. If the Hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem. The problem of estimating Hessian matrices by differences can be phrased as follows: Given the sparsity structure of a symmetric matrixA, obtain vectorsd ₁,d ₂, …d _p such thatAd ₁,Ad ₂, …Ad _p determineA uniquely withp as small as possible. We approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation. The complexity of the problem is analyzed and efficient practical heuristic procedures are developed. Numerical results illustrate the differences between the various approaches. Work supported in part by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.     

Approximating smooth and sparse functions by deep neural networks: Optimal approximation rates and saturation

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks with three hidden layers using a sigmoidal activation function to approximate smooth and sparse functions. Specifically, we prove that the constructed deep nets with controllable magnitude of free parameters can reach the optimal approximation rate in approximating both smooth and sparse functions. In particular, we prove that neural networks with three hidden layers can avoid the phenomenon of saturation, i.e., the phenomenon that for some neural network architectures, the approximation rate stops improving for functions of very high smoothness.     

On a generalization of the Gallai-Roy-Vitaver theorem to the bandwidth coloring problem

We generalize to the bandwidth coloring problem a classical theorem, discovered independently by Gallai, Roy and Vitaver, in the context of the graph coloring problem. Two proofs are given, a simple one and a more complex one that is based on a series of equivalent mathematical programming models.     

A short proof of a conjecture on the connectivity of graph coloring complexes

The -complexes were introduced by Lovász to study topological obstructions to graph colorings. It was conjectured by Babson and Kozlov, and proved by Cukic and Kozlov, that is -connected, where is the maximal degree of a vertex of , and the number of colors. We give a short proof of the conjecture.

     

The simplified partial digest problem: Approximation and a graph-theoretic model

The goal of the simplified partial digest problem (SPDP) is motivated by the reconstruction of the linear structure of a DNA chain with respect to a given nucleotide pattern, based on the multiset of distances between the adjacent patterns (interpoint distances) and the multiset of distances between each pattern and the two unlabeled endpoints of the DNA chain (end distances). We consider optimization versions of the problem, called SPDP-Min and SPDP-Max. The aim of SPDP-Min (SPDP-Max) is to find a DNA linear structure with the same multiset of end distances and the minimum (maximum) number of incorrect (correct) interpoint distances. Results are presented on the worst-case efficiency of approximation algorithms for these problems. We suggest a graph-theoretic model for SPDP-Min and SPDP-Max, which can be used to reduce the search space for an optimal solution in either of these problems. We also present heuristic polynomial time algorithms based on this model. In computational experiments with randomly generated and real-life input data, our best algorithm delivered an optimal solution in 100% of the instances for a number of restriction sites not greater than 50.     

On the circumference of a graph and its complement

Let G be a graph of order n and circumference c(G). Let be the complement of G. We prove that and show sharpness of this bound.     

The product of the restrained domination numbers of a graph and its complement

Let G=(V,E) be a graph.A set S■V is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and to a vertex in V-S.The restrained domination number of G,denoted γr(G),is the smallest cardinality of a restrained dominating set of G.In this paper,we show that if G is a graph of order n≥4,then γr(G)γr(G)≤2n.We also characterize the graphs achieving the upper bound.     

Decomposition theorems and approximation by a ``floating" system of exponentials

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.

     

A multi-round generalization of the traveling tournament problem and its application to Japanese baseball

In a double round-robin tournament involving n teams, every team plays 2(n − 1) games, with one home game and one away game against each of the other n − 1 teams. Given a symmetric n by n matrix representing the distances between each pair of home cities, the traveling tournament problem (TTP) seeks to construct an optimal schedule that minimizes the sum total of distances traveled by the n teams as they move from city to city, subject to several natural constraints to ensure balance and fairness. In the TTP, the number of rounds is set at r = 2. In this paper, we generalize the TTP to multiple rounds (r = 2k, for any k ? 1) and present an algorithm that converts the problem to finding the shortest path in a directed graph, enabling us to apply Dijkstra’s Algorithm to generate the optimal multi-round schedule. We apply our shortest-path algorithm to optimize the league schedules for Nippon Professional Baseball (NPB) in Japan, where two leagues of n = 6 teams play 40 sets of three intra-league games over r = 8 rounds. Our optimal schedules for the Pacific and Central Leagues achieve a 25% reduction in total traveling distance compared to the 2010 NPB schedule, implying the potential for considerable savings in terms of time, money, and greenhouse gas emissions.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈R ^n×n , a set of complex n-vectors {x _i } _i=1 ^m , a set of complex numbers {λ _i } _i=1 ^m , and an s-by-s real matrix C ₀, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C ₀, and {x _i } _i=1 ^m and {λ _i } _i=1 ^m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix $\tilde{C}$ , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.     

On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.This research supported in part by the Natural Sciences and Engineering Research Council, Canada.     

The study of a dynamic dial-a-ride problem under time-dependent and stochastic environments

This paper studies a dynamic dial-a-ride problem bearing complex constraints on a time-dependent network. A flexible scheduling scheme is proposed to dynamically cope with different stochastic events, such as the travelling time fluctuation, new requests, absences of customers, vehicle breakdowns, cancellations of requests, traffic jams and so on. A fast heuristic is proposed to re-optimize the schedule when a new event occurs. This heuristic consists of a properly organized local search strategy and uses a secondary objective function to drive the search out of local optima. Intensive computational simulations were carried out to evaluate the performance of this scheduling scheme and the influence of different stochastic factors. The simulation results of different scenarios with different percentage of dynamic requests reveal that this scheduling scheme can generate high quality schedules and is capable of coping with various stochastic events.     

The problem of split convex feasibility and its alternating approximation algorithms

This paper studies the problem of split convex feasibility and a strong convergent alternating algorithm is established. According to this algorithm, some strong convergent theorems are obtained and an affirmative answer to the question raised by Moudafi is given. At the same time, this paper also generalizes the problem of split convex feasibility.     

The behavior of a finite difference approximation to a singular initial boundary value problem

We develop difference approximations to a singular parabolic initial-boundary value problem and its corresponding steady-state problem. A critical value for the existence of nonnegative solutions to the discrete steady state system is established. Convergence of the computed critical values is obtained. The long time behavior for the approximated solution of the parabolic problem is investigated. It is shown that the behavior of the discrete system is consistent with that of the continuous one