A Lagrangean-based heuristics for the target covering problem in wireless sensor network

We study the target coverage problem in wireless sensor networks. The problem consists in maximizing the network lifetime by grouping the sensors in disjoint set covers of the targets. A binary integer programing model is formulated to maximize the network lifetime. Since the problem is NP-complete, we provide an iterative approximation based on Lagrangean relaxation and subgradient optimization.     

On the independent queens covering problem

The minimum number of queens which can be placed on ann × n chessboard so that all other squares are dominated by at least one queen but no queen covers another, is shown to be less than 0.705n + 2.305.     

Fuzzy versions of the covering circle problem

In this paper we consider two fuzzy versions of the well-known problem of determining the smallest circle (center and radius) that would cover a given finite set of points in the plane when the locations of points are not precise but fuzzy. The first is modeled as a possibility-constrained mathematical program while the second is modeled as a necessity-constrained one. Polynomial algorithms are presented for both the versions. Also, a numerical example is included for one case. These models and solutions are of interest in both theoretical and practical contexts.     

The variable radius covering problem

In this paper we propose a covering problem where the covering radius of a facility is controlled by the decision-maker; the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance (i.e., it costs more to establish a facility with a greater covering radius). The problem is to cover all demand points at a minimum cost by finding optimal number, locations and coverage radii for the facilities. Both, the planar and discrete versions of the model are considered. Heuristic approaches are suggested for solving large problems in the plane. These methods were tested on a set of planar problems. Mathematical programming formulations are proposed for the discrete problem, and a solution approach is suggested and tested.     

An extension of the edge covering problem

In this paper we define a weightedr-covering problem, and show that there exists an optimum solution of ther-covering problem which can be decomposed into the sum of a rounded down solution of its linear relaxation and an optimal solution of a weighted edge covering problem on a reduced graph. Vertexr-packing problem can also be reduced to ther-covering problem.     

On a cone covering problem

Given a set of polyhedral cones C₁,…,Ck⊂Rd, and a convex set D, does the union of these cones cover the set D? In this paper we consider the computational complexity of this problem for various cases such as whether the cones are defined by extreme rays or facets, and whether D is the entire Rd or a given linear subspace Rt. As a consequence, we show that it is coNP-complete to decide if the union of a given set of convex polytopes is convex, thus answering a question of Bemporad, Fukuda and Torrisi.     

The hub covering flow problem

The hub covering flow problem (HCFP) seeks to find the minimal cost hub-and-spoke network by optimally locating hub nodes and assigning non-hub nodes to the hub nodes subject to a coverage constraint. The cost of establishing such a hub network is based on a fixed cost of opening hubs and the cost of transporting demand flow through the network. We also present an extension called the multi-aircraft HCFP. The results from computational experiments are presented and discussed.     

Equivalence between the minimum covering problem and the maximum matching problem

The minimum covering problem in weighted graphs with n vertices is transformed in time O(n²) to the maximum matching problem with n or n + 1 vertices, and conversely.     

A covering problem for plane lattices

Let be a plane lattice and {v ₁, v ₂} a Minkowski reduced base of . In this note we prove that if a convex body K has minimal width w(K) |v ₂| sin +|v ₁|( ), where is the acute angle between v ₁ and v ₂, then K is a covering set for .This work was partially supported by MURST (40–60%).     

An algorithm for set covering problem

In this paper we present an algorithm for the set covering problem that combines problem reduction tests with dual ascent, subgradient optimisation and linear programming. Computational results are presented for problems involving up to 400 rows and 4000 columns.     

MIP reformulations of the probabilistic set covering problem

In this paper, we address the following probabilistic version (PSC) of the set covering problem: where A is a 0-1 matrix, is a random 0-1 vector and is the threshold probability level. We introduce the concepts of p-inefficiency and polarity cuts. While the former is aimed at deriving an equivalent MIP reformulation of (PSC), the latter is used as a strengthening device to obtain a stronger formulation. Simplifications of the MIP model which result when one of the following conditions hold are briefly discussed: A is a balanced matrix, A has the circular ones property, the components of are pairwise independent, the distribution function of is a stationary distribution or has the disjunctive shattering property. We corroborate our theoretical findings by an extensive computational experiment on a test-bed consisting of almost 10,000 probabilistic instances. This test-bed was created using deterministic instances from the literature and consists of probabilistic variants of the set covering model and capacitated versions of facility location, warehouse location and k-median models. Our computational results show that our procedure is orders of magnitude faster than any of the existing approaches to solve (PSC), and in many cases can reduce hours of computing time to a fraction of a second. Anureet Saxena’s research was supported by the National Science Foundation through grant #DMI-0352885 and by the Office of Naval Research through contract N00014-03-1-0133. Vineet Goyal’s research was supported in part by NSF grant CCF-0430751 and ITR grant CCR-0122581.     

Surrogate constraint normalization for the set covering problem

The set covering problem (SCP) is central in a wide variety of practical applications for which finding good feasible solutions quickly (often in real-time) is crucial. Surrogate constraint normalization is a classical technique used to derive appropriate weights for surrogate constraint relaxations in mathematical programming. This framework remains the core of the most effective constructive heuristics for the solution of the SCP chiefly represented by the widely-used Chvátal method. This paper introduces a number of normalization rules and demonstrates their superiority to the classical Chvátal rule, especially when solving large scale and real-world instances. Directions for new advances on the creation of more elaborate normalization rules for surrogate heuristics are also provided.     

A probabilistic analysis of the set covering problem

A probabilistic analysis of the minimum cardinality set covering problem (SCP) is developed, considering a stochastic model of the (SCP), withn variables andm constraints, in which the entries of the corresponding (m, n) incidence matrix are independent Bernoulli distributed random variables, each with constant probabilityp of success. The behaviour of the optimal solution of the (SCP) is then investigated as bothm andn grow asymptotically large, assuming either an incremental model for the evolution of the matrix (for each size, the matrixA is obtained bordering a matrix of smaller size by new columns and rows) or an independent one (for each size, an entirely new set of entries forA are considered). Two functions ofm are identified, which represent a lower and an upper bound onn in order the (SCP) to be a.e. feasible and not trivial. Then, forn lying within these bounds, an asymptotic formula for the optimum value of the (SCP) is derived and shown to hold a.e.The performance of two simple randomized algorithms is then analyzed. It is shown that one of them produces a solution value whose ratio to the optimum value asymptotically approaches 1 a.e. in the incremental model, but not in the independent one, in which case the ratio is proved to be tightly bounded by 2 a.e. Thus, in order to improve the above result, a second randomized algorithm is proposed, for which it is proved that the ratio between the approximate solution value and the optimum approaches 1 a.e. also in the independent model.     

A-priori upper bounds for the set covering problem

A matrix M∈R^n×n is said to be a column sufficient matrix if the solution set of LCP(M,q) is convex for every q∈Rⁿ. In a recent article, Qin et al. (Optim. Lett. 3:265–276, 2009) studied the concept of column sufficiency property in Euclidean Jordan algebras. In this paper, we make a further study of this concept and prove numerous results relating column sufficiency with the Z and Lypaunov-like properties. We also study this property for some special linear transformations.     

Fuzzy minimum weight edge covering problem

Minimum weight edge covering problem, known as a classic problem in graph theory, is employed in many scientific and engineering applications. In the applications, the weight may denote cost, time, or opponent’s payoff, which can be vague in practice. This paper considers the edge covering problem under fuzzy environment, and formulates three models which are expected minimum weight edge cover model, α-minimum weight edge cover model, and the most minimum weight edge cover model. As an extension for the models, we respectively introduce the crisp equivalent of each model in the case that the weights are independent trapezoidal fuzzy variables. Due to the complexity of the problem, a hybrid intelligent algorithm is employed to solve the models, which can deal with the problem with any type of fuzzy weights. At last, some numerical experiments are given to show the application of the models and the robustness of the algorithm.     

Two new heuristics for the location set covering problem

Summary In this paper we present two new greedy-type heuristics for solving the location set covering problem. We compare our new pair of algorithms with the pair GH1 and GH2 [Vasko and Wilson (1986)] and show that they perform better for a selected set of test problems.     

A Lagrangean heuristic for the maximal covering location problem

We develop a Lagrangean heuristic for the maximal covering location problem. Upper bounds are given by a vertex addition and substitution heuristic and lower bounds are produced through a subgradient optimization algorithm. The procedure was tested in networks of up to 150 vertices. A duality gap was generally present at the end of the heuristic for the larger problems. The test problems were run in an IBM 3090-600J ‘super-computer’; the maximum computing time was kept below three minutes of CPU.