A greedy heuristic for the capacitated minimum spanning tree problem

This paper develops a greedy heuristic for the capacitated minimum spanning tree problem (CMSTP), based on the two widely known methods of Prim and of Esau–Williams. The proposed algorithm intertwines two-stages: an enhanced combination of the Prim and Esau–Williams approaches via augmented and synthetic node selection criteria, and an increase of the feasible solution space by perturbing the input data using the law of cosines. Computational tests on benchmark problems show that the new heuristic provides extremely good performance results for the CMSTP, justifying its effectiveness and robustness. Furthermore, excluding the feasible space expansion, we show that we can still obtain good quality solutions in very short computational times.     

A tabu search heuristic for the generalized minimum spanning tree problem

This paper describes an attribute based tabu search heuristic for the generalized minimum spanning tree problem (GMSTP) known to be NP-hard. Given a graph whose vertex set is partitioned into clusters, the GMSTP consists of designing a minimum cost tree spanning all clusters. An attribute based tabu search heuristic employing new neighborhoods is proposed. An extended set of TSPLIB test instances for the GMSTP is generated and the heuristic is compared with recently proposed genetic algorithms. The proposed heuristic yields the best results for all instances. Moreover, an adaptation of the tabu search algorithm is proposed for a variation of the GMSTP in which each cluster must be spanned at least once.     

A hybrid heuristic for the diameter constrained minimum spanning tree problem

We propose a hybrid GRASP and ILS based heuristic for the diameter constrained minimum spanning tree problem. The latter typically models network design applications where, under a given quality requirement, all vertices must be connected at minimum cost. An adaptation of the one time tree heuristic is used to build feasible diameter constrained spanning trees. Solutions thus obtained are then attempted to be improved through local search. Four different neighborhoods are investigated, in a scheme similar to VND. Upper bounds within 2% of optimality were obtained for problems in two test sets from the literature. Additionally, upper bounds stronger than those previously obtained in the literature are reported for OR-Library instances.     

A heuristic method for the multi-story layout problem

In this paper we present a heuristic procedure designed expressly for solving a large layout problem in a multi-story setting, where the objective is to minimize total fixed and interaction costs. This is achieved by decomposing the original facilities layout problem into several similar but smaller problems, thus enabling solution of problems with as many as 150 facilities in reasonable time. Some of the novel features of the procedure described are the use of a heuristic K-median subroutine to obtain groupings of facilities, and a simple and fast exchange-improvement method. Computational results for randomly generated problems compare the effectiveness of this method with the space planning heuristic method of Liggett and Mitchell.     

A heuristic method for the open vehicle routing problem

The open vehicle routing problem (OVRP) differs from the classic vehicle routing problem (VRP) because the vehicles either are not required to return to the depot, or they have to return by revisiting the customers assigned to them in the reverse order. Therefore, the vehicle routes are not closed paths but open ones. A heuristic method for solving this new problem, based on a minimum spanning tree with penalties procedure, is presented. Computational results are provided.     

Solving the minimum toll revenue problem in real transportation networks

As a means to relieve traffic congestion, toll pricing has recently received significant attention by transportation planners. Inappropriate use of transportation networks is one of the major causes of network congestion. Toll pricing is a method of traffic management in which traffic flow is guided to proper time and path in order to reduce the total delay in the network. This article investigates a method for solving the minimum toll revenue problem in real and large-scale transportation networks. The objective of this problem is to find link tolls that simultaneously cause users to efficiently use the transportation network and to minimize the total toll revenues to be collected. Although this model is linear, excessive number of variables and constraints make it very difficult to solve for large-scale networks. In this paper, a path-generation algorithm is proposed for solving the model. Implementation of this algorithm for different networks indicates that this method can achieve the optimal solution after a few iterations and a proper CPU time.     

A simple enhancement of the Esau–Williams heuristic for the capacitated minimum spanning tree problem

The Esau–Williams algorithm is one of the best known heuristics for the capacitated minimum spanning tree problem. This paper describes a simple enhancement of this heuristic. On benchmark test problems, the modified method does not result in much larger computation times and almost always produces lower solution values than does the Esau–Williams algorithm. The improvements are often significant.     

Worst-case behavior of the MVCA heuristic for the minimum labeling spanning tree problem

In this paper, we review recent work on the minimum labeling spanning tree problem and obtain a new worst-case ratio for the MVCA heuristic. We also present a family of graphs in which the worst-case ratio can be attained. This implies that the new ratio cannot be improved any further.     

A gamma heuristic for the p-median problem

Heuristic concentration (HC) is a two-stage metaheuristic that can be applied to a wide variety of combinatorial problems. It is particularly suited to location problems in which the number of facilities is given in advance. In such settings, the first stage of HC repeatedly applies some random-start interchange (or other) heuristic to produce a number of alternative facility configurations. A subset of the best of these alternatives is collected and the union of the facility sites in them is called a concentration set (CS). Among the component elements of the CS are likely to be included those sites which are members of the optimal solution. In earlier studies the second stage of HC has consisted of an exact procedure to extract the best possible solution from the CS. In this paper we demonstrate, for the p-median problem, the use of two sequentially active heuristics in the second stage of HC. That is, we offer two additional layers of heuristics to improve solutions which are found in the first stage of HC. Thus we are describing a variant of the HC metaheuristic consisting of three layers of heuristics which are utilized in sequence. We propose for this procedure the name of Gamma Heuristic.     

An average case analysis of the minimum spanning tree heuristic for the power assignment problem

We present an average case analysis of the minimum spanning tree heuristic for the power assignment problem. The worst‐case approximation ratio of this heuristic is 2. We show that in Euclidean d‐dimensional space, when the vertex set consists of a set of i.i.d. uniform random independent, identically distributed random variables in [0,1]^d, and the distance power gradient equals the dimension d, the minimum spanning tree‐based power assignment converges completely to a constant depending only on d.     

A hybrid heuristic for the maximum clique problem

In this paper we present a heuristic based steady-state genetic algorithm for the maximum clique problem. The steady-state genetic algorithm generates cliques, which are then extended into maximal cliques by the heuristic. We compare our algorithm with three best evolutionary approaches and the overall best approach, which is non-evolutionary, for the maximum clique problem and find that our algorithm outperforms all the three evolutionary approaches in terms of best and average clique sizes found on majority of DIMACS benchmark instances. However, the obtained results are much inferior to those obtained with the best approach for the maximum clique problem.     

A heuristic for the Steiner problem in graphs

In this paper, we present a heuristic for the Steiner problem in graphs (SPG) along with some experimental results. The heuristic is based on an approach similar to Prim's algorithm for the minimum spanning tree. However, in this approach, arcs are associated with preference weights which are used to break ties among alternative choices of shortest paths occurring during the course of the algorithm. The preference weights are calculated according to a global view which takes into consideration the effect of all the regular nodes, nodes to be connected, on determining the choice of an arc in the solution tree.     

A heuristic procedure for the Capacitated m-Ring-Star problem

In this paper we propose a heuristic method to solve the Capacitated m-Ring-Star Problem which has many practical applications in communication networks. The problem consists of finding m rings (simple cycles) visiting a central depot, a subset of customers and a subset of potential (Steiner) nodes, while customers not belonging to any ring must be “allocated” to a visited (customer or Steiner) node. Moreover, the rings must be node-disjoint and the number of customers allocated or visited in a ring cannot be greater than the capacity Q given as an input parameter. The objective is to minimize the total visiting and allocation costs. The problem is a generalization of the Traveling Salesman Problem, hence it is NP-hard. In the proposed heuristic, after the construction phase, a series of different local search procedures are applied iteratively. This method incorporates some random aspects by perturbing the current solution through a “shaking” procedure which is applied whenever the algorithm remains in a local optimum for a given number of iterations. Computational experiments on the benchmark instances of the literature show that the proposed heuristic is able to obtain, within a short computing time, most of the optimal solutions and can improve some of the best known results.     

A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.     

A dynamic programming heuristic for the P-median problem

A new heuristic algorithm is proposed for the P-median problem. The heuristic restricts the size of the state space of a dynamic programming algorithm. The approach may be viewed as an extension of the myopic or greedy adding algorithm for the P-median model. The approach allows planners to identify a large number of solutions all of which perform well with respect to the P-median objective of minimizing the demand weighted average distance between customer locations and the nearest of the P selected facilities. In addition, the results indicate regions in which it is desirable to locate facilities. Computational results from three test problems are discussed.     

A heuristic for the three-dimensional strip packing problem

The contribution presents a heuristic for the three-dimensional strip packing problem (3D-SPP) with rectangular pieces (boxes). The considered 3D-SPP can be formulated as follows: for a given set of boxes and a given longitudinal open container, determine an arrangement of all boxes within the container so that the required container length is minimized.     

A hybrid heuristic for the maximum dispersion problem

In this paper we propose a hybrid heuristic for the Maximum Dispersion Problem of finding a balanced partition of a set of objects such that the shortest intra-part distance is maximized. In contrast to clustering problems, dispersion problems aim for a large spread of objects in the same group. They arise in many practical applications such as waste collection and the formation of study groups. The heuristic alternates between finding a balanced solution, and increasing the dispersion. Balancing is achieved by a combination of a minimum cost flow algorithm to find promising pairs of parts and a branch-and-bound algorithm that searches for an optimal balance, and the dispersion is increased by a local search followed by an ejection chain method for escaping local minima. We also propose new upper bounds for the problem. In computational experiments we show that the heuristic is able to find solutions significantly faster than previous approaches. Solutions are close to optimal and in many cases provably optimal.     

A heuristic algorithm for the strip packing problem

The two-dimensional strip packing problem is to pack a given set of rectangles into a strip with a given width and infinite height so as to minimize the required height of the packing. From the computational point of view, the strip packing problem is an NP-hard problem. With the B*-tree representation, this paper first presents a heuristic packing strategy which evaluates the positions used by the rectangles. Then an effective local search method is introduced to improve the results and a heuristic algorithm (HA) is further developed to find a desirable solution. Computational results on randomly generated instances and popular test instances show that the proposed method is efficient for the strip packing problem.