An algorithmic study of the Maximum Flow problem: A comparative statistical analysis

In this paper, we show the results of an experimental study about the most important algorithms proposed to solve the Maximum Flow problem. The appropriate statistical analysis not only allows us to justify comparisons between the different procedures but also to obtain classifications of their practical efficiency. Furthermore, an empirical experiment allows us to identify the influence of several parameters that are not included in a theoretical study.     

A general model of parameterized OWA aggregation with given orness level

The paper proposes a general optimization model with separable strictly convex objective function to obtain the consistent OWA (ordered weighted averaging) operator family. The consistency means that the aggregation value of the operator monotonically changes with the given orness level. Some properties of the problem are discussed with its analytical solution. The model includes the two most commonly used maximum entropy OWA operator and minimum variance OWA operator determination methods as its special cases. The solution equivalence to the general minimax problem is proved. Then, with the conclusion that the RIM (regular increasing monotone quantifier) can be seen as the continuous case of OWA operator with infinite dimension, the paper further proposes a general RIM quantifier determination model, and analytically solves it with the optimal control technique. Some properties of the optimal solution and the solution equivalence to the minimax problem for RIM quantifier are also proved. Comparing with that of the OWA operator problem, the RIM quantifier solutions are usually more simple, intuitive, dimension free and can be connected to the linguistic terms in natural language. With the solutions of these general problems, we not only can use the OWA operator or RIM quantifier to obtain aggregation value that monotonically changes with the orness level for any aggregated set, but also can obtain the parameterized OWA or RIM quantifier families in some specific function forms, which can incorporate the background knowledge or the required characteristic of the aggregation problems.     

Complete rank theorem of advanced calculus and singularities of bounded linear operators

Let E and F be Banach spaces, f: U ⊂ E → F be a map of C ^r (r ⩾ 1), x ₀ ∈ U, and ft (x ₀) denote the FréLechet differential of f at x ₀. Suppose that f′(x ₀) is double split, Rank(f′(x ₀)) = ∞, dimN(f′(_{x 0})) > 0 and codimR(f′(x ₀)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x ₀) near x ₀. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x ₀ is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.      

Optimal control of heterogeneous systems: Basic theory

A general model of a heterogeneous control system is introduced in the form of a first order distributed system with nonlocal dynamics and exogenous side-conditions. The heterogeneity is represented by a parameter taking values in an abstract measurable space, so that both continuous and discrete heterogeneity, as well as probabilistic heterogeneity without density, are included. A distributed and a lumped controls are involved, the latter appearing also in the side conditions. An existence theorem is proved for the uncontrolled system, and the sensitivity of the solution with respect to the control variables is estimated. The main result is an optimality condition in the form of the Pontryagin local maximum principle. A global maximum principle holds for the distributed control under an additional condition that rules out discrete measurable heterogeneity spaces. A number of possible applications are outlined: age-structured systems, size-structured systems, (nonlocal) advection-reaction equations, static parametric heterogeneity in epidemiology, and two-stage control systems with uncertain switching time.     

模糊权数与方案排序

讨论了由Fuzzy权数导出方案排序中的问题，提出了评价Fuzzy权数取值大小及取值集中程度的两个指标，并据此给出了由Fuzzy权数导出方案排序的方法。     

Parameter estimation for stochastic equations with additive fractional Brownian sheet

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.      

Distance graphs with maximum chromatic number

The distance graph G(D) has the set of integers as vertices and two vertices are adjacent in G(D) if their difference is contained in the set D⊂Z. A conjecture of Zhu states that if the chromatic number of G(D) achieves its maximum value |D|+1 then the graph has a triangle. The conjecture is proven to be true if |D|?3. We prove that the chromatic number of a distance graph with D={a,b,c,d} is five only if either D={1,2,3,4k} or D={a,b,a+b,b-a}. This confirms a stronger version of Zhu's conjecture for |D|=4, namely, if the chromatic number achieves its maximum value then the graph contains K₄.     

Simple ingredients leading to very efficient heuristics for the maximum clique problem

Starting from an algorithm recently proposed by Pullan and Hoos, we formulate and analyze iterated local search algorithms for the maximum clique problem. The basic components of such algorithms are a fast neighbourhood search (not based on node evaluation but on completely random selection) and simple, yet very effective, diversification techniques and restart rules. A detailed computational study is performed in order to identify strengths and weaknesses of the proposed algorithms and the role of the different components on several classes of instances. The tested algorithms are very fast and reliable: most of the DIMACS benchmark instances are solved within very short CPU times. For one of the hardest tests, a new putative optimum was discovered by one of our algorithms. Very good performances were also shown on recently proposed and more difficult instances. It is important to remark that the heuristics tested in this paper are basically parameter free (the appropriate value for the unique parameter is easily identified and was, in fact, the same value for all problem instances used in this paper).     

Uniqueness of maximum planar five-distance sets

A subset X in the Euclidean plane is called a k-distance set if there are exactly k distances between two distinct points in X. We denote the largest possible cardinality of k-distance sets by g(k). Erdős and Fishburn proved that g(5)=12 and also conjectured that 12-point five-distance sets are unique up to similar transformations. We classify 8-point four-distance sets and prove the uniqueness of the 12-point five-distance sets given in their paper. We also introduce diameter graphs of planar sets and characterize these graphs.     

Optimal birth control of age-dependent competitive species III. Overtaking problem

A study is made of an overtaking optimal problem for a population system consisting of two competing species, which is controlled by fertilities. The existence of optimal policy is proved and a maximum principle is carefully derived under less restrictive conditions. Weak and strong turnpike properties of optimal trajectories are established.