Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Every triangle-free planar graph has a planar upward drawing

A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices.Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.Supported in part by the Alexander von Humboldt Stiftung.Supported in part by the Deutsche Forschungsgemeinschaft.     

Using a facility location algorithm to solve large set covering problems

Erlenkotter has developed an efficient exact (guarantees optimality) algorithm to solve the uncapacitated facility location problem (UFLP). In this paper, we use his algorithm to solve large instances of an important subset of the UFLP; the set covering problem (SCP). In addition, we present further empirical evidence that a heuristic algorithm developed by Vasko and Wilson for the SCP is capable of quickly generating good solutions to large SCP's.     

An algorithm for large scale 0–1 integer programming with application to airline crew scheduling

We present an approximation algorithm for solving large 0–1 integer programming problems whereA is 0–1 and whereb is integer. The method can be viewed as a dual coordinate search for solving the LP-relaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems that have appeared in the literature.     

Simultaneous metric projection onto closed sets

We examine simultaneous metric projection by closed sets in a class of ordered normed spaces. First, we study simultaneous metric projection onto downward and upward sets and separation properties of these sets. The results obtained are used for examination of simultaneous metric projection by arbitrary closed sets, and we examine the minimization of the distance from a bounded set to an arbitrary closed set in a class of ordered normed spaces.     

I m -Quasi Upward Sets,with Their Best Approximations

The main purpose of this article is to introduce particular subsets of R ^I , which are not necessarily convex, and we call them I _m -quasi upward, or I _m -quasi downward. We show that these sets can be translated to downward or upward sets. We introduce the connection of these sets with downward and upward subsets of R ^I , and discuss the best approximation of these sets. Also we introduce embedded I _m -quasi upward and embedded I _m -quasi downward subsets of a normed space X.     

Computing tournament solutions using relation algebra and RelView

We describe a simple computing technique for the tournament choice problem. It rests upon relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView’s labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.     

Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a naive linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal–dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem. These results match the currently known best results for purely edge-weighted graphs.     

Maximally Disjoint Solutions of the Set Covering Problem

This paper is concerned with finding two solutions of a set covering problem that have a minimum number of variables in common. We show that this problem is NP-complete, even in the case where we are only interested in completely disjoint solutions. We describe three heuristic methods based on the standard greedy algorithm for set covering problems. Two of these algorithms find the solutions sequentially, while the third finds them simultaneously. A local search method for reducing the overlap of the two given solutions is then described. This method involves the solution of a reduced set covering problem. Finally, extensive computational tests are given demonstrating the nature of these algorithms. These tests are carried out both on randomly generated problems and on problems found in the literature.     

Note on: N.E. Aguilera, M.S. Escalante, G.L. Nasini, “The disjunctive procedure and blocker duality”

Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–13] give a generalization of a theorem of Lehman through an extension of the disjunctive procedure defined by Balas, Ceria and Cornuéjols. This generalization can be formulated as(A) For every clutter , the disjunctive index of its set covering polyhedron coincides with the disjunctive index of the set covering polyhedron of its blocker, .In Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–3], (A) is indeed a corollary of the stronger result(B) .Motivated by the work of Gerards et al. [Math. Oper. Res. 28 (2003) 884–885] we propose a simpler proof of (B) as well as an alternative proof of (A), independent of (B). Both of them are based on the relationship between the “disjunctive relaxations” obtained by and the set covering polyhedra associated with some particular minors of .     

An accelerated covering relaxation algorithm for solving 0–1 positive polynomial programs

The purpose of this note is to present an accelerated algorithm for solving 0–1 positive polynomial (PP) problems. Like our covering relaxation algorithm (Management Science 1979), the accelerated algorithm is a cutting plane method, which uses the linear set covering problem as a relaxation for PP. However, a unique and novel feature of the accelerated algorithm is that it attempts to generate cutting planes from heuristic solutions to the set covering problem whenever possible. Computational results reveal that this strategy of generating cutting planes has led to a significant reduction in the computational time required to solve a PP problem.This research was partially supported by National Sciences and Engineering Research Council Canada Grants 67-4181 and 67-3998, Office of Naval Research Contract N00014-76-C-0418, and National Science Foundation Grant ECS80-22027.     

Location coverage models with demand originating from nodes and paths: Application to cellular network design

Location covering problems, though well studied in the literature, typically consider only nodal (i.e. point) demand coverage. In contrast, we assume that demand occurs from both nodes and paths. We develop two separate models – one that handles the situation explicitly and one which handles it implicitly. The explicit model is formulated as a Quadratic Maximal Covering Location Problem – a greedy heuristic supported by simulated annealing (SA) that locates facilities in a paired fashion at each stage is developed for its solution. The implicit model focuses on systems with network structure – a heuristic algorithm based on geometrical concepts is developed. A set of computational experiments analyzes the performance of the algorithms, for both models. We show, through a case study for locating cellular base stations in Erie County, New York State, USA, how the model can be used for capturing demand from both stationary cell phone users as well as cell phone users who are in moving vehicles.     

基于改进遗传算法的集合覆盖问题

集合覆盖问题是组合优化中的典型问题,在日常生活中有着广泛的应用.提出了一种改进遗传算法来解决集合覆盖问题.算法对标准遗传算法的改进主要表现在:1)结合启发式算法和随机生成,设计了新的产生初始种群的方法;2)引入修补操作处理不可行解使其转换成可行解;3)对重复个体进行处理再利用;4)对多点交叉进行推广,提出了新的交叉算子;5)针对可行解和不可行解,采取两种自适应多位变异操作.数值实验结果表明该算法对于解决规模较大的集合覆盖问题是有效的.     

Dominance-based fuzzy rough set analysis of uncertain and possibilistic data tables

In this paper, we propose a dominance-based fuzzy rough set approach for the decision analysis of a preference-ordered uncertain or possibilistic data table, which is comprised of a finite set of objects described by a finite set of criteria. The domains of the criteria may have ordinal properties that express preference scales. In the proposed approach, we first compute the degree of dominance between any two objects based on their imprecise evaluations with respect to each criterion. This results in a valued dominance relation on the universe. Then, we define the degree of adherence to the dominance principle by every pair of objects and the degree of consistency of each object. The consistency degrees of all objects are aggregated to derive the quality of the classification, which we use to define the reducts of a data table. In addition, the upward and downward unions of decision classes are fuzzy subsets of the universe. Thus, the lower and upper approximations of the decision classes based on the valued dominance relation are fuzzy rough sets. By using the lower approximations of the decision classes, we can derive two types of decision rules that can be applied to new decision cases.     

On Representing Faces of a Cube by Subfaces

If each of a set A of l-dimensional faces of an n-dimensional cube is to be represented by a l–b dimensional subface, then it is possible to choose distinct representatives provided the cardinality of A does not exceed a certain bound. We give an algorithm for calculating this bound.     

Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.     

An interior point algorithm to solve computationally difficult set covering problems

We present an interior point approach to the zero–one integer programming feasibility problem based on the minimization of a nonconvex potential function. Given a polytope defined by a set of linear inequalities, this procedure generates a sequence of strict interior points of this polytope, such that each consecutive point reduces the value of the potential function. An integer solution (not necessarily feasible) is generated at each iteration by a rounding scheme. The direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid. We illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1-width of the incidence matrix of Steiner triple systems.     

Minimal periods of holomorphic maps on complex tori

We study the set of minimal periods of holomorphic self-maps of one- and two-dimensional complex tori. In particular, we characterize when the set of minimal periods of such maps is finite. In fact, we have an algorithm for doing this characterization for holomorphic self-maps of an arbitrary dimensional complex torus.     

Red-blue covering problems and the consecutive ones property

Set Cover problems are of core importance in many applications. In recent research, the “red-blue variants” where blue elements all need to be covered whereas red elements add further constraints on the optimality of a covering have received considerable interest. Application scenarios range from data mining to interference reduction in cellular networks. As a rule, these problem variants are computationally at least as hard as the original set cover problem. In this work we investigate whether and how the well-known consecutive ones property, restricting the structure of the input sets, makes the red-blue covering problems feasible. We explore a sharp border between polynomial-time solvability and NP-hardness for these problems.