A Global Optimization RLT-based Approach for Solving the Hard Clustering Problem

The field of cluster analysis is primarily concerned with the sorting of data points into different clusters so as to optimize a certain criterion. Rapid advances in technology have made it possible to address clustering problems via optimization theory. In this paper, we present a global optimization algorithm to solve the hard clustering problem, where each data point is to be assigned to exactly one cluster. The hard clustering problem is formulated as a nonlinear program, for which a tight linear programming relaxation is constructed via the Reformulation-Linearization Technique (RLT) in concert with additional valid inequalities that serve to defeat the inherent symmetry in the problem. This construct is embedded within a specialized branch-and-bound algorithm to solve the problem to global optimality. Pertinent implementation issues that can enhance the efficiency of the branch-and-bound algorithm are also discussed. Computational experience is reported using several standard data sets found in the literature as well as using synthetically generated larger problem instances. The results validate the robustness of the proposed algorithmic procedure and exhibit its dominance over the popular k-means clustering technique. Finally, a heuristic procedure to obtain a good quality solution at a relative ease of computational effort is also described.     

Spaces With Compact-Countable k-Systems

In this paper the relations among k-covers, cs ^*-covers and k-systems are discussed. The following question is partially answered: Does every separable k'-space with a point-countable k-system have a countable k-system?     

Linear bilevel multi-follower programming with independent followers

This paper considers a particular case of linear bilevel programming problems with one leader and multiple followers. In this model, the followers are independent, meaning that the objective function and the set of constraints of each follower only include the leader’s variables and his own variables. We prove that this problem can be reformulated into a linear bilevel problem with one leader and one follower by defining an adequate second level objective function and constraint region. In the second part of the paper we show that the results on the optimality of the linear bilevel problem with multiple independent followers presented in Shi et al. [The kth-best approach for linear bilevel multi-follower programming, J. Global Optim. 33, 563–578 (2005)] are based on a misconstruction of the inducible region.     

k-systems, k-networks and k-covers

The concepts of k-systems, k-networks and k-covers were defined by A. Arhangel’skii in 1964, P. O’Meara in 1971 and R. McCoy, I. Ntantu in 1985, respectively. In this paper the relationships among k-systems, k-networks and k-covers are further discussed and are established by mk-systems. As applications, some new characterizations of quotients or closed images of locally compact metric spaces are given by means of mk-systems.     

On k-primitivity of two classes of digraphs

In this paper, we study two classes of primitive digraphs, and show that there are k-colorings that are k-primitive.     

Dense trees: a new look at degenerate graphs

Dense trees are undirected graphs defined as natural extensions of trees. They are already known in the realm of graph coloring under the name of k-degenerate graphs. For a given integer k1, a k-dense cycle is a connected graph, where the degree of each vertex is greater than k. A k-dense forest F=(V,E) is a graph without k-dense cycles as subgraphs. If F is connected, then is a k-dense tree. 1-dense trees are standard trees. We have |E|k|V|−k(k+1)/2. If equality holds F is connected and is called a maximal k-dense tree. k-trees (a subfamily of triangulated graphs) are special cases of maximal k-dense trees.We review the basic theory of dense trees in the family of graphs and show their relation with k-trees. Vertex and edge connectivity is thoroughly investigated, and the role of maximal k-dense trees as “reinforced” spanning trees of arbitrary graphs is presented. Then it is shown how a k-dense forest or tree can be decomposed into a set of standard spanning trees connected through a common “root” of k vertices. All sections include efficient construction algorithms. Applications of k-dense trees in the fields of distributed systems and data structures are finally indicated.     

The Width-Volume Inequality

We prove that a bounded open set U in has k-width less than C(n) Volume(U) ^k/n . Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in . In particular, we estimate the smallest (n – 1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n – 1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor. Received: January 2006, Revision: May 2006, Accepted: June 2006     

The <Emphasis Type="Italic">k</Emphasis>-domatic number of a graph

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d _k (G). In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d ₁(G).      

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

Map Preserving Local Properties of a Digital Image

In order to study digital topological properties of a k-surface in Z ⁿ , we generalize the topological number in Bertrand (Pattern Recogn. Lett. 15:1003–1011, 1994). Furthermore, we show that a local (k ₀,k ₁)-isomorphism preserves some digital-topological properties, such as a generalized topological number and a simple k ₀-point, and prove that a local (k ₀,k ₁)-isomorphism takes a simple k ₀-surface in into a simple k ₁-surface in .      

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Orderings on Graphs and Game Coloring Number

Many graph theoretic algorithms rely on an initial ordering of the vertices of the graph which has some special properties. We discuss new ways to measure the quality of such orders, give methods for constructing high quality orders, and provide applications for these orders. While our main motivation is the study of game chromatic number, there have been other applications of these ideas and we expect there will be more. This revised version was published online in June 2006 with corrections to the Cover Date.     

Piece-wise affine identification in dialysis

Piece-wise affine identification of hybrid dynamical systems, a quite general tool to analyse dynamical data as time series, is briefly reviewed. It is then shown to be a useful tool in order to manage a quite diffuse health impairment, namely the forecasting of the correct length of an individual session of dialysis, with the final outcome of approximating to a switch the transition from the faster initial blood dialysis to the slower inner compartment depletion.     

Applying mod-k-cuts for solving linear ordering problems

The linear ordering problem consists of finding an ordering of the nodes of the weighted complete digraph on n nodes such that the sum of the weights of the arcs compatible with the ordering is maximized. In this paper, we report about the usefulness of mod-k cuts in a branch-and-cut algorithm for solving linear ordering problems to optimality.      

Some comments on a paper onk-HNBUE life distributions

Summary In a recent paper by A. P. Basu and N. Ebrahimi (1984, Ann. Inst. Statist. Math., A, 36, 87–100) a new class of life distributions calledk-HNBUE (with dualk-HNWUE) is introduced. Closure properties and bounds on the moments and on the survival function to ak-HNBUE (k-HNWUE) life distribution are presented. However, some of the results presented are incorrect. This research was supported by Swedish Natural Science Research Council Post Doctoral Fellowship F-PD 1564-100.     

On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

New algorithms for lineark-matroid intersection and matroidk-parity problems

We present algorithms for thek-Matroid Intersection Problem and for the Matroidk-Parity Problem when the matroids are represented over the field of rational numbers andk > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of thek-Dimensional Assignment Problem and of thek-Dimensional Matching Problem. The algorithms use some new identities in multilinear algebra including the generalized Binet—Cauchy formula and its analogue for the Pfaffian. These techniques extend known methods developed earlier fork = 2.A preliminary version of this paper appeared in the Proceedings of the Second IPCO Conference [2].Supported by the Mittag-Leffler Institute and KTH, Stockholm.     

The construction of infinite families of any κ-tight optimal and singular κ-tight optimal directed double loop networks

The double loop network (DLN) is a circulant digraph with n nodes and outdegree 2. It is an important topological structure of computer interconnection networks and has been widely used in the designing of local area networks and distributed systems. Given the number n of nodes, how to construct a DLN which has minimum diameter? This problem has attracted great attention. A related and longtime unsolved problem is for any given non-negative integer k, is there an infinite family of k-tight optimal DLN? In this paper, two main results are obtained (1) for any k ≥ 0, the infinite families of k-tight optimal DLN can be constructed, where the number n(k,e,c) of their nodes is a polynomial of degree 2 in e with integral coefficients containing a parameter c. (2) for any k ≥ 0,an infinite family of singular k-tight optimal DLN can be constructed.     

Norm of pre-Schwarzian derivative for the class of k-uniformly convex and k-starlike functions

A quasiconformal extension for the class of k-uniformly convex functions, denoted , and for the class of k-starlike functions, denoted is provided. Also, estimation of the norm of pre-Schwarzian derivative in is given.     

Estimation of parameters in the discrete distributions of order k

This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one.