An exact algorithm for the maximum stable set problem

We describe a new branch-and-bound algorithm for the exact solution of the maximum cardinality stable set problem. The bounding phase is based on a variation of the standard greedy algorithm for finding a colouring of a graph. Two different node-fixing heuristics are also described. Computational tests on random and structured graphs and very large graphs corresponding to real-life problems show that the algorithm is competitive with the fastest algorithms known so far.This work has been supported by Agenzia Spaziale Italiana.     

An efficient algorithm for the complete set partitioning problem

The complete set partitioning problem is the well known set partitioning problem with all possible nonzero binary columns in the constraint matrix. A highly specialized enumerative algorithm, which never requires the explicit maintenance of the constraint matrix, is presented. Computational results, with data reflecting a particular corporate tax payment scenario that can be modelled as a complete set partitioning problem, is also given.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

An exact algorithm for graph partitioning

An exact algorithm is presented for solving edge weighted graph partitioning problems. The algorithm is based on a branch and bound method applied to a continuous quadratic programming formulation of the problem. Lower bounds are obtained by decomposing the objective function into convex and concave parts and replacing the concave part by an affine underestimate. It is shown that the best affine underestimate can be expressed in terms of the center and the radius of the smallest sphere containing the feasible set. The concave term is obtained either by a constant diagonal shift associated with the smallest eigenvalue of the objective function Hessian, or by a diagonal shift obtained by solving a semidefinite programming problem. Numerical results show that the proposed algorithm is competitive with state-of-the-art graph partitioning codes.     

An exact algorithm for the bi-objective timing problem

The timing problem in the bi-objective just-in-time single-machine job-shop scheduling problem (JiT-JSP) is the task to schedule N jobs whose order is fixed, with each job incurring a linear earliness penalty for finishing ahead of its due date and a linear tardiness penalty for finishing after its due date. The goal is to minimize the earliness and tardiness simultaneously. We propose an exact greedy algorithm that finds the entire Pareto front in \(O(N^2)\) time. This algorithm is asymptotically optimal.     

An exact algorithm for the pallet loading problem

The two-dimensional packing problem of finding optimal layouts for identical rectangular boxes on a rectangular pallet has interested OR practitioners for many years. The problem is NP-complete and solution methods to date tend to be heuristic. This paper discusses the development of an exact tree search algorithm based on a graph-theoretic model of the problem.     

An exact algorithm for solving a capacitated location-routing problem

In location-routing problems, the objective is to locate one or many depots within a set of sites (representing customer locations or cities) and to construct delivery routes from the selected depot or depots to the remaining sites at least system cost. The objective function is the sum of depot operating costs, vehicle acquisition costs and routing costs. This paper considers one such problem in which a weight is assigned to each site and where sites are to be visited by vehicles having a given capacity. The solution must be such that the sum of the weights of sites visited on any given route does not exceed the capacity of the visiting vehicle. The formulation of an integer linear program for this problem involves degree constraints, generalized subtour elimination constraints, and chain barring constraints. An exact algorithm, using initial relaxation of most of the problem constraints, is presented which is capable of solving problems with up to twenty sites within a reasonable number of iterations.     

An exact algorithm for the network pricing problem

This work focuses on an improved exact algorithm for addressing an NP-hard network pricing problem. The method involves an efficient and partial generation of candidate solutions, a recursive scheme for generating improved upper bounds, and a column generation procedure for solving the network-structured subproblems. Its efficiency is assessed against both randomly generated instances involving three distinct topologies as well as instances based on real life situations in telecommunication and freight transportation.     

An exact algorithm for the fuzzy p-median problem

In this paper we propose a fuzzy version of the classical p-median problem. We consider a fuzzy set of constraints so that the decision-maker will be able to take into account solutions which provide significantly lower costs by leaving a part of the demand uncovered. We propose an algorithm for solving the problem which is based on Hakimi's works and we compare the crisp and the fuzzy approach by means of an example.     

An approximating polynomial algorithm for a sequence partitioning problem

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.     

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.     

An exact algorithm for the petrol station replenishment problem

In the petrol station replenishment problem (PSRP), the aim is to deliver petroleum products to petrol stations by means of an unlimited heterogeneous fleet of compartmented tank trucks. The problem consists of jointly determining quantities to deliver within a given interval, of allocating products to tank truck compartments and of designing delivery routes to stations. This article describes an exact algorithm which decomposes the PSRP into a truck loading problem and a routing problem. An algorithm which makes use of assignment, optimality tests and possibly standard ILP algorithm is proposed to solve the loading problem. The routing problem is handled using two different strategies, based either on a matching approach or on a column generation scheme. This algorithm was extensively tested on randomly generated data and on a real-life case arising in Eastern Quebec.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

An exact algorithm for the general quadratic assignment problem

We develop an algorithm that is based on the linearization and decomposition of a general Quadratic Assignment Problem of size n into n² Linear Assignment Problems of size (n − 1). The solutions to these subproblems are used to calculate a lower bound for the original problem, and this bound is then used in an exact branch and bound procedure. These subproblems are similar to the ‘minors’ defined by Lawler [16], but permit us to calculate tighter bounds. Computational experience is given for solution to optimality of general quadratic assignment problems is sizes up to n = 10.     

An exact algorithm for the fixed-charge multiple knapsack problem

We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528–541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.     

An exact depth-first algorithm for the pallet loading problem

This paper proposes a fast exact algorithm to solve the Pallet Loading Problem (PLP) using depth-first strategy. A new concept called Maximal Breadth Filling Sequence (MBFS) is introduced to bring down the size of the search tree. The algorithm makes use of two pruning rules — lower-bound pruning and state-dominance pruning. Although depth-first search, by itself, requires very little memory, the dominance pruning rule makes effective utilization of the available memory. For large problems, more the memory available, more effective is the dominance pruning. The algorithm has been tested on standard problem sets. It has been found to be quite fast in outputting optimal solutions. Empirical findings are given in detail.     

An exact algorithm for solving the vertex separator problem

Given G = (V, E) a connected undirected graph and a positive integer β(|V|), the vertex separator problem is to find a partition of V into no-empty three classes A, B, C such that there is no edge between A and B, max{|A|, |B|} ≤ β(|V|) and |C| is minimum. In this paper we consider the vertex separator problem from a polyhedral point of view. We introduce new classes of valid inequalities for the associated polyhedron. Using a natural lower bound for the optimal solution, we present successful computational experiments.     

An efficient memetic algorithm for the graph partitioning problem

We consider the component testing problem of a device that is designed to perform a mission consisting of a random sequence of phases with random durations. Testing is done at the component level to attain desired levels of mission reliability at minimum cost. The components fail exponentially where the failure rate depends on the phase of the mission. The reliability structure of the device involves a series connection of nonidentical components with different failure characteristics. The optimal component testing problem is formulated as a semi-infinite linear program. We present an algorithmic procedure to compute optimal test times based on the column generation technique, and illustrate it with numerical examples.     

Penalty computations for the set partitioning problem

The computation of penalties associated with the continuous relaxation of integer programming problems can be useful to derive conditional and relational tests which allow to fix some variables at their optimal value or to generate new constraints (cuts). We study in this paper the computation and the use of penalties as a tool to improve the efficiency of algorithms for solving set partitioning problems. This leads to a preprocessing scheme which can be embedded within any exact or approximate algorithm. The strength of these penalties is illustrated through computational results on some real-world set partitioning problems.This work was sponsored by FINEP (research contract number 4.3.86.0689-00), CNPq (research contract numbers 11.1592-84, 30.2281-85 and 40.2002-86.5), IBM Brazil and NSERC (grant # GP0036426).On leave from the Catholic University of Rio de Janeiro, Department of Electrical Engineering, Caixa Postal 38063, Gávea, Rio de Janeiro 22452, Brazil.