Covering partially directed graphs with directed paths

We consider graphs which contain both directed and undirected edges (partially directed graphs). We show that the problem of covering the edges of such graphs with a minimum number of edge-disjoint directed paths respecting the orientations of the directed edges is polynomially solvable. We exhibit a good characterization for this problem in the form of a min-max theorem. We introduce a more general problem including weights on possible orientations of the undirected edges. We show that this more general weighted formulation is equivalent to the weighted bipartite b-factor problem. This implies the existence of a strongly polynomial algorithm for this weighted generalization of Euler's problem to partially directed graphs (compare this with the negative results for the mixed Chinese postman problem). We also provide a compact linear programming formulation for the weighted generalization that we propose.     

A preference-based approach to spanning trees and shortest paths problems****

Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems).     

Direct newton method for a linear problem of semidefinite programming

We consider a linear problem of semidefinite programming. To solve this problem, we propose a direct Newton method, which is a generalization of the direct barrier-Newton method for problems of linear programming. We study properties of the method and prove its local convergence.     

Local search heuristics for multi-index assignment problems with decomposable costs

The multi-index assignment problem (MIAP) with decomposable costs is a natural generalization of the well-known assignment problem. Applications of the MIAP arise, for instance, in the field of multi-target multi-sensor tracking. We describe an (exponentially sized) neighbourhood for a solution of the MIAP with decomposable costs, and show that one can find a best solution in this neighbourhood in polynomial time. Based on this neighbourhood, we propose a local search algorithm. We empirically test the performance of published constructive heuristics and the local search algorithm on random instances; a straightforward iterated local search algorithm is also tested. Finally, we compute lower bounds to our problem, which enable us to assess the quality of the solutions found.     

Effectiveness-based Voronoi partition: a new tool for solving a class of location optimization problems

In this article we provide a framework for optimal placement or deployment of facilities in a region of interest. We present a generalization of Voronoi partition, where functions modeling the effectiveness of facilities are used in the place of the usual distance measure used in the standard Voronoi partition and its variations. We illustrate the usefulness of the generalization in designing strategies for optimal deployment of multiple vehicles equipped with sensors, optimal placement of base stations in a cellular network design problem, and locational optimization of power plants.     

Surfaces of revolution in the Heisenberg group and the spectral generalization of the Willmore functional

We study the generalization of the Willmore functional for surfaces in the three-dimensional Heisenberg group. Its construction is based on the spectral theory of the Dirac operator entering into theWeierstrass representation of surfaces in this group. Using the surfaces of revolution we demonstrate that the generalization resembles the Willmore functional for the surfaces in the Euclidean space in many geometrical aspects. We also observe the relation of these functionals to the isoperimetric problem.     

Model Selection for Regularized Least-Squares Algorithm in Learning Theory

We investigate the problem of model selection for learning algorithms depending on a continuous parameter. We propose a model selection procedure based on a worst-case analysis and on a data-independent choice of the parameter. For the regularized least-squares algorithm we bound the generalization error of the solution by a quantity depending on a few known constants and we show that the corresponding model selection procedure reduces to solving a bias-variance problem. Under suitable smoothness conditions on the regression function, we estimate the optimal parameter as a function of the number of data and we prove that this choice ensures consistency of the algorithm.     

Convergence Results for an Analytical Trial Free-Boundary Method

We consider a free-boundary problem, in two dimensions, involvingan ideal fluid flow with constant pressure on the free boundary.(A three-dimensional generalization is also considered.) Forthis problem, we give a novel definition for the iterationsof an analytical trial free-boundary method for approximatingthe solution. We then show that the successive approximationsdo converge to the solution, and we derive an error estimate.     

Approximation of the Quadratic Set Covering problem

We study in this article the polynomial approximation properties of the Quadratic Set Covering problem. This problem, which arises in many applications, is a natural generalization of the usual Set Covering problem. We show that this problem is very hard to approximate in the general case, and even in classical subcases (when the size of each set or when the frequency of each element is bounded by a constant). Then we focus on the convex case and give both positive and negative approximation results. Finally, we tackle the unweighted version of this problem.     

Recovering differential pencils on compact graphs

We study boundary value problems on compact graphs without circles (i.e. on trees) for second-order ordinary differential equations with nonlinear dependence on the spectral parameter. We establish properties of the spectral characteristics and investigate the inverse spectral problem of recovering the coefficients of the differential equation from the so-called Weyl vector which is a generalization of the Weyl function (m-function) for the classical Sturm-Liouville operator. For this inverse problem we prove the uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mappings.     

The vehicle routing problem with flexible time windows and traveling times

We generalize the standard vehicle routing problem by allowing soft time window and soft traveling time constraints, where both constraints are treated as cost functions. With the proposed generalization, the problem becomes very general. In our algorithm, we use local search to determine the routes of vehicles. After fixing the route of each vehicle, we must determine the optimal start times of services at visited customers. We show that this subproblem is NP-hard when cost functions are general, but can be efficiently solved with dynamic programming when traveling time cost functions are convex even if time window cost functions are non-convex. We deal with the latter situation in the developed iterated local search algorithm. Finally we report computational results on benchmark instances, and confirm the benefits of the proposed generalization.     

On the first eigenvalue of the Dirichlet-to-Neumann operator on forms

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Dirichlet-to-Neumann (or Steklov) problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of these estimates will be sharp, and for some of them we characterize equality. We also relate these new eigenvalues with those of other operators, like the Hodge Laplacian or the biharmonic Steklov operator.     

Buyback problem with discrete concave valuation functions

We discuss an online discrete optimization problem called the buyback problem. In the literature of the buyback problem, the valuation function representing the total value of selected elements is given by a linear function. In this paper, we consider a generalization of the buyback problem using nonlinear valuation functions. We propose an online algorithm for the problem with discrete concave valuation functions, and show that it achieves the tight competitive ratio, i.e., the competitive ratio of the proposed algorithm is equal to the known lower bound for the problem.     

A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.     

A new neighborhood and tabu search for the Blocking Job Shop

The Blocking Job Shop is a version of the job shop scheduling problem with no intermediate buffers, where a job has to wait on a machine until being processed on the next machine. We study a generalization of this problem which takes into account transfer operations between machines and sequence-dependent setup times. After formulating the problem in a generalized disjunctive graph, we develop a neighborhood for local search. In contrast to the classical job shop, there is no easy mechanism for generating feasible neighbor solutions. We establish two structural properties of the underlying disjunctive graph, the concept of closures and a key result on short cycles, which enable us to construct feasible neighbors by exchanging critical arcs together with some other arcs. Based on this neighborhood, we devise a tabu search algorithm and report on extensive computational experience, showing that our solutions improve most of the benchmark results found in the literature.     

The Red–Blue transportation problem

This paper considers the Red–Blue Transportation Problem (Red–Blue TP), a generalization of the transportation problem where supply nodes are partitioned into two sets and so-called exclusionary constraints are imposed. We encountered a special case of this problem in a hospital context, where patients need to be assigned to rooms. We establish the problem’s complexity, and we compare two integer programming formulations. Furthermore, a maximization variant of Red–Blue TP is presented, for which we propose a constant-factor approximation algorithm. We conclude with a computational study on the performance of the integer programming formulations and the approximation algorithms, by varying the problem size, the partitioning of the supply nodes, and the density of the problem.     

A k-Sample Test for Functional Data Based on Generalized Maximum Mean Discrepancy

Lithuanian Mathematical Journal - In this paper, we deal with the problem of testing the equality of k probability distributions. We introduce a generalization of the maximum mean discrepancy that...     

A generalization of König-Egervary graphs and heuristics for the maximum independent set problem with improved approximation ratios

We first study a generalization of the König-Egervary graphs, the class of the κ-KE graphs, and propose an exact polynomial time algorithm solving maximum independent set problem in this class. Next, we show how this result can be efficiently used to devise polynomial time approximation algorithms with improved approximation ratios for the maximum independent set problem in general graphs.     

Beyond Canonical DC-Optimization: The Single Reverse Polar Problem

We propose a novel generalization of the Canonical Difference of Convex problem (CDC), and we study the convergence of outer approximation algorithms for its solution, which use an approximated oracle for checking the global optimality conditions. Although the approximated optimality conditions are similar to those of CDC, this new class of problems is shown to significantly differ from its special case. Indeed, outer approximation approaches for CDC need be substantially modified in order to cope with the more general problem, bringing to new algorithms. We develop a hierarchy of conditions that guarantee global convergence, and we build three different cutting plane algorithms relying on them.     

Second-Order Conditions for Efficiency in Nonsmooth Multiobjective Optimization Problems

In this paper, we are concerned with a nonsmooth multiobjective optimization problem with inequality constraints. We introduce a second-order constraint qualification, which is a generalization of the Abadie constraint qualification and derive second-order Kuhn-Tucker type necessary conditions for efficiency under the constraint qualification. Moreover, we give some conditions which ensure the constraint qualification holds.