The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

Planar Posets,Dimension, Breadth and the Number of Minimal Elements

In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar poset in terms of the number of minimal elements, where the starting point is the 1977 theorem of Trotter and Moore asserting that the dimension of a planar poset with a single minimal element is at most 3. By carefully analyzing and then refining the details of this argument, we are able to show that the dimension of a planar poset with t minimal elements is at most 2t + 1. This bound is tight for t = 1 and t = 2. But for t ≥ 3, we are only able to show that there exist planar posets with t minimal elements having dimension t + 3. Our lower bound construction can be modified in ways that have immediate connections to the following challenging conjecture: For every d ≥ 2, there is an integer f(d) so that if P is a planar poset with dim(P) ≥ f(d), then P contains a standard example of dimension d. To date, the best known examples only showed that the function f, if it exists, satisfies f(d) ≥ d + 2. Here, we show that lim _d→∞ f(d)/d ≥ 2.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

Conditions for the existence of a classical solution of a cauchy type problem for the diffusion equation with a Riemann-Liouville partial derivative

We study a Cauchy type problem for a differential equation containing a fractional Riemann-Liouville partial derivative of order α, 0 < α < 2. Conditions under which the solution of the problem tends to zero as |x| → ∞ are obtained. We prove an existence theorem for a classical solution of the Cauchy type problem and show that the solution has a singularity as t → 0 of order 1 ? α if 0 < α ≤ 1 and of order 2 ? α if 1 < α < 2.     

The Transient Behaviour of a Cyclic Queue

This paper studies a cyclic queueing problem arising in civil engineering earthmoving projects. There are m excavators and n trucks which queue to be loaded by the excavators. The problem is therefore equivalent to a machine interference problem with m operators and n machines. The size of the haul fleet must be optimized, so as to minimize the cost per unit volume of earth moved. Graphs which show regions of optimal n in the parameter space are presented for the steady-state D/D/1, M/D/1 and E _k /M/1 models. The effect of operating the M/M/m system in short work shifts, so that the initial conditions and transient behaviour are important, is then analysed and quantified as a correction to the optimal number of trucks in the haul fleet from the steady-state solution.     

Solvability of the problem of taking the discrete logarithm

It is proved that consideration of the solvability problem for taking the discrete logarithm in a residue ring modulo composite M can be reduced to consideration of a similar problem in residue rings modulo pq, where p and q are prime divisors of M. For moduli of form pq, necessary and sufficient conditions for solvability checking are obtained in some cases. In addition, the problem of raising a solution of an exponential comparison in a residue ring modulo p ^α is solved.     

The pos/neg-weighted 2-medians in balanced trees with subtree-shaped customers

This paper deals with the pos/neg-weighted p-median problem on tree graphs where all customers are modeled as subtrees. We present a polynomial algorithm for the 2-median problem on an arbitrary tree. Then we improve the time complexity to O(n log n) for the problem on a balanced tree, where n is the number of the vertices in the tree.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

A Minimum Principle for Potentials with Application to Chebyshev Constants

For “Riesz-like” kernels K(x,y) = f(|x?y|) on A×A, where A is a compact d-regular set \(A\subset \mathbb {R}^{p}\), we prove a minimum principle for potentials \(U_{K}^{\mu }=\int K(x,y)\textup {d}\mu (x)\), where μ is a Borel measure supported on A. Setting \(P_{K}(\mu )=\inf _{y\in A}U^{\mu }(y)\), the K-polarization of μ, the principle is used to show that if {ν _N } is a sequence of measures on A that converges in the weak-star sense to the measure ν, then P _K (ν _N )→P _K (ν) as \(N\to \infty \). The continuous Chebyshev (polarization) problem concerns maximizing P _K (μ) over all probability measures μ supported on A, while the N-point discrete Chebyshev problem maximizes P _K (μ) only over normalized counting measures for N-point multisets on A. We prove for such kernels and sets A, that if {ν _N } is a sequence of N-point measures solving the discrete problem, then every weak-star limit measure of ν _N as \(N \to \infty \) is a solution to the continuous problem.     

On DP-coloring of graphs and multigraphs

While solving a question on the list coloring of planar graphs, Dvo?ák and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph G reduces the problem of finding a coloring of G from a given list L to the problem of finding a “large” independent set in the auxiliary graph H(G,L) with vertex set {(v, c): v ∈ V (G) and c ∈ L(v)}. It is similar to the old reduction by Plesnevi? and Vizing of the k-coloring problem to the problem of finding an independent set of size |V(G)| in the Cartesian product G□K _k, but DP-coloring seems more promising and useful than the Plesnevi?–Vizing reduction. Some properties of the DP-chromatic number χ _DP (G) resemble the properties of the list chromatic number χ _l (G) but some differ quite a lot. It is always the case that χ _DP (G) ≥ χ _l (G). The goal of this note is to introduce DP-colorings for multigraphs and to prove for them an analog of the result of Borodin and Erd?s–Rubin–Taylor characterizing the multigraphs that do not admit DP-colorings from some DP-degree-lists. This characterization yields an analog of Gallai’s Theorem on the minimum number of edges in n-vertex graphs critical with respect to DP-coloring.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Pairwise sums in colourings of the reals

Suppose that we have a finite colouring of \(\mathbb R\). What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums monochromatic. We also show that there is such a colouring such that there is no infinite set X with \(X+X\) (the pairwise sums from X, allowing repetition) monochromatic. These results assume CH. In the other direction, we show that if each colour class is measurable, or each colour class is Baire, then there is an infinite set X (and even an uncountable X, of size the reals) with \(X+X\) monochromatic. We also give versions for all of these results for k-wise sums in place of pairwise sums.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered in a rectangular domain in x and t; the perturbation parameter ? multiplying the highest derivative takes arbitrary values in the half-open interval (0,1]. For the boundary value problem, we construct a scheme based on the method of lines in x passing through N ₀+1 points of the mesh with respect to t. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width δ), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter ? (as well as the values of N ₀, δ) influence the convergence rate of the decomposition scheme (as N ₀ → ∞), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection-diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.     

New heuristics for flow shop problem to minimize makespan

This paper investigates the flow shop problem with the objective to minimizemakespan. New algorithms are designed: one is off-line heuristic, Single JobFirst, for problemF _m ∥C _max ;and the other is on-line heuristic, Dynamic Single Job First (DSJF), for problemF _m |r _i |C _max and its on-line version. It is proved that the two heuristics are asymptoticallyoptimal when the size of the problem is large enough. In addition, theasymptotical optimality of First-Come, First-Served manner is obtained as abyproduct of the asymptotical analysis of DSJF heuristic. At the end of thepaper, a new lower bound with performance guarantee is provided for problemF _m ∥C _max , andcomputational experiments show the effectiveness of these heuristicalgorithms.     

Expanding operators for the independent set problem

The notion is introduced of an expanding operator for the independent set problem. This notion is a useful tool for the constructive formation of new cases with the efficient solvability of this problem in the family of hereditary classes of graphs and is applied to hereditary parts of the set Free({P ₅, C ₅}). It is proved that if for a connected graph G the problem is polynomial-time solvable in the class Free({P ₅, C ₅,G}) then it remains so in the class Free({P ₅, C ₅,G ○ \(\bar K_2 \), G ⊕ K _1,p ) for every p. We also find two new hereditary subsets of Free({P ₅, C ₅}) with polynomially solvable independent set problem that are not a corollary of applying the revealed operators.     

Optimal Linear Bernoulli Factories for Small Mean Problems

Suppose a coin with unknown probability p of heads can be flipped as often as desired. A Bernoulli factory for a function f is an algorithm that uses flips of the coin together with auxiliary randomness to flip a single coin with probability f(p) of heads. Applications include perfect sampling from the stationary distribution of certain regenerative processes. When f is analytic, the problem can be reduced to a Bernoulli factory of the form f(p) = C p for constant C. Presented here is a new algorithm that for small values of C p, requires roughly only C coin flips. From information theoretic considerations, this is also conjectured to be (to first order) the minimum number of flips needed by any such algorithm. For large values of C p, the new algorithm can also be used to build a new Bernoulli factory that uses only 80 % of the expected coin flips of the older method. In addition, the new method also applies to the more general problem of a linear multivariate Bernoulli factory, where there are k coins, the kth coin has unknown probability p _k of heads, and the goal is to simulate a coin flip with probability C ₁ p ₁+? + C _k p _k of heads.     

Some minimax problems in lexicographic order

In this paper, minimax theorems and saddle points for a class of vector-valued mappings f(x, y) = u(x)+β(x)v(y) are first investigated in the sense of lexicographic order, where u, v are two general vector-valued mappings and β is a non-negative real-valued function. Then, by applying the existence theorem of lexicographic saddle point, we investigate a lexicographic equilibrium problem and establish an equivalent relationship between the lexicographic saddle point theorem and existence theorem of a lexicographic equilibrium problem for vector-valued mappings.