Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.     

Branch-and-bound algorithm for the maximum triangle packing problem

This work addresses the problem of finding the maximum number of unweighted vertex-disjoint triangles in an undirected graph G. It is a challenging NP-hard combinatorial problem and it is well-known to be APX-hard. A branch-and-bound algorithm which uses a lower bound based on neighborhood degree is presented. A naive upper bound is proposed as well as another one based on a surrogate relaxation of the related integer linear program which is analogous to a multidimensional knapsack problem. Further, a Greedy Search algorithm and a genetic algorithm are described to improve the lower bound. A computational comparison of lower bounds, branch-and-bound algorithm and CPLEX solver is provided using randomly generated benchmarks and well-known DIMACS implementation challenges. The empirical study shows that the branch-and-bound finds the optimal triangle packing solution for small randomly generated MTP instances (up to 100 vertices and 200 triangles) and some DIMACS graphs. For some larger instances and DIMACS challenges graphs, we remark that our lower bound outperforms CPLEX solver regarding the triangle packing solution and the computation time.     

Exact algorithms for a scheduling problem with unrelated parallel machines and sequence and machine-dependent setup times

A scheduling problem with unrelated parallel machines, sequence and machine-dependent setup times, due dates and weighted jobs is considered in this work. A branch-and-bound algorithm (B&B) is developed and a solution provided by the metaheuristic GRASP is used as an upper bound. We also propose a set of instances for this type of problem. The results are compared to the solutions provided by two mixed integer programming models (MIP) with the solver CPLEX 9.0. We carry out computational experiments and the algorithm performs extremely well on instances with up to 30 jobs.     

Minimum stabbing rectangular partitions of rectilinear polygons

We study integer programming (ip) models for the problem of finding a rectangular partition of a rectilinear polygon with minimum stabbing number. Strong valid inequalities are introduced for an existing formulation and a new model is proposed. We compare the dual bounds yielded by the relaxations of the two models and prove that the new one is stronger than the old one. Computational experiments with the problem are reported for the first time in which polygons with thousands of vertices are solved to optimality. The (ip) branch-and-bound algorithm based on the new model is faster and more robust than those relying on the previous formulation.     

A new formulation and approach for the black and white traveling salesman problem

This paper proposes a new formulation and a column generation approach for the black and white traveling salesman problem. This problem is an extension of the traveling salesman problem in which the vertex set is divided into black vertices and white vertices. The number of white vertices visited and the length of the path between two consecutive black vertices are constrained. The objective of this problem is to find the shortest Hamiltonian cycle that covers all vertices satisfying the cardinality and the length constraints. We present a new formulation for the undirected version of this problem, which is amenable to the Dantzig–Wolfe decomposition. The decomposed problem which is defined on a multigraph becomes the traveling salesman problem with an extra constraint set in which the variable set is the feasible paths between pairs of black vertices. In this paper, a column generation algorithm is designed to solve the linear programming relaxation of this problem. The resulting pricing subproblem is an elementary shortest path problem with resource constraints, and we employ acceleration strategies to solve this subproblem effectively. The linear programming relaxation bound is strengthened by a cutting plane procedure, and then column generation is embedded within a branch-and-bound algorithm to compute optimal integer solutions. The proposed algorithm is used to solve randomly generated instances with up to 80 vertices.     

Randomized heuristics for the family traveling salesperson problem

This paper introduces the family traveling salesperson problem (FTSP), a variant of the generalized traveling salesman problem. In the FTSP, a subset of nodes must be visited for each node cluster in the graph. The objective is to minimize the distance traveled. We describe an integer programming formulation for the FTSP and show that the commercial grade integer programming solver CPLEX 11 can only solve small instances of the problem in reasonable running time. We propose two randomized heuristics for finding optimal and near‐optimal solutions of this problem. These heuristics are a biased random‐key genetic algorithm and a GRASP with evolutionary path‐relinking. Computational results comparing both heuristics are presented in this study.     

The fixed set search applied to the power dominating set problem

In this article, we focus on solving the power dominating set problem and its connected version. These problems are frequently used for finding optimal placements of phasor measurement units in power systems. We present an improved integer linear program (ILP) for both problems. In addition, a greedy constructive algorithm and a local search are developed. A greedy randomised adaptive search procedure (GRASP) algorithm is created to find near optimal solutions for large scale problem instances. The performance of the GRASP is further enhanced by extending it to the novel fixed set search (FSS) metaheuristic. Our computational results show that the proposed ILP has a significantly lower computational cost than existing ILPs for both versions of the problem. The proposed FSS algorithm manages to find all the optimal solutions that have been acquired using the ILP. In the last group of tests, it is shown that the FSS can significantly outperform the GRASP in both solution quality and computational cost.     

Solving diameter‐constrained minimum spanning tree problems by constraint programming

The diameter‐constrained minimum spanning tree problem consists in finding a minimum spanning tree of a given graph, subject to the constraint that the maximum number of edges between any two vertices in the tree is bounded from above by a given constant. This problem typically models network design applications where all vertices communicate with each other at a minimum cost, subject to a given quality requirement. We propose alternative formulations using constraint programming that circumvent weak lower bounds yielded by most mixed‐integer programming formulations. Computational results show that the proposed formulation, combined with an appropriate search procedure, solves larger instances and is faster than other approaches in the literature.     

Computing minimal doubly resolving sets of graphs

In this paper we consider the minimal doubly resolving sets problem (MDRSP) of graphs. We prove that the problem is NP-hard and give its integer linear programming formulation. The problem is solved by a genetic algorithm (GA) that uses binary encoding and standard genetic operators adapted to the problem. Experimental results include three sets of ORLIB test instances: crew scheduling, pseudo-boolean and graph coloring. GA is also tested on theoretically challenging large-scale instances of hypercubes and Hamming graphs. Optimality of GA solutions on smaller size instances has been verified by total enumeration. For several larger instances optimality follows from the existing theoretical results. The GA results for MDRSP of hypercubes are used by a dynamic programming approach to obtain upper bounds for the metric dimension of large hypercubes up to 2⁹⁰$2^{90}$ nodes, that cannot be directly handled by the computer.     

A comparative study of formulations and solution methods for the discrete ordered p-median problem

This paper presents several new formulations for the Discrete Ordered Median Problem (DOMP) based on its similarity with some scheduling problems. Some of the new formulations present a considerably smaller number of constraints to define the problem with respect to some previously known formulations. Furthermore, the lower bounds provided by their linear relaxations improve the ones obtained with previous formulations in the literature even when strengthening is not applied. We also present a polyhedral study of the assignment polytope of our tightest formulation showing its proximity to the convex hull of the integer solutions of the problem. Several resolution approaches, among which we mention a branch and cut algorithm, are compared. Extensive computational results on two families of instances, namely randomly generated and from Beasley's OR-library, show the power of our methods for solving DOMP.     

Two-machine flow shop scheduling with linear decreasing job deterioration

In this paper, we consider a two-machine flow shop scheduling problem with deteriorating jobs. By a deteriorating job, we mean that the processing time is a decreasing function of its execution start time. A proportional linear decreasing deterioration function is assumed. The objective is to find a sequence that minimizes total completion time. Optimal solutions are obtained for some special cases. For the general case, several dominance properties and some lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational results for randomly generated problem instances are presented, which show that the heuristic algorithm effectively and efficiently in obtaining near-optimal solutions.     

Derivation of algorithms for cutwidth and related graph layout parameters

In this paper, we investigate algorithms for some related graph parameters. Each of these asks for a linear ordering of the vertices of the graph (or can be formulated as such), and constructive linear time algorithms for the fixed parameter versions of the problems have been published for several of these. Examples are cutwidth, pathwidth, and directed or weighted variants of these. However, these algorithms have complicated technical details. This paper attempts to present ideas in these algorithms in a different more easily accessible manner, by showing that the algorithms can be obtained by a stepwise modification of a trivial hypothetical non-deterministic algorithm. The methodology is applied to rederive known results for the cutwidth and the pathwidth problem, and obtain new results for several variants of these problems, like directed and weighted variants of cutwidth and modified cutwidth.     

Finding optimal hardware/software partitions

Most previous approaches to hardware/software partitioning considered heuristic solutions. In contrast, this paper presents an exact algorithm for the problem based on branch-and-bound. Several techniques are investigated to speed up the algorithm, including bounds based on linear programming, a custom inference engine to make the most out of the inferred information, advanced necessary conditions for partial solutions, and different heuristics to obtain high-quality initial solutions. It is demonstrated with empirical measurements that the resulting algorithm can solve highly complex partitioning problems in reasonable time. Moreover, it is about ten times faster than a previous exact algorithm based on integer linear programming. The presented methods can also be useful in other related optimization problems.     

A column generation approach to the heterogeneous fleet vehicle routing problem

We consider a vehicle routing problem with a heterogeneous fleet of vehicles having various capacities, fixed costs and variable costs. An approach based on column generation (CG) is applied for its solution, hitherto successful only in the vehicle routing problem with time windows. A tight integer programming model is presented, the linear programming relaxation of which is solved by the CG technique. A couple of dynamic programming schemes developed for the classical vehicle routing problem are emulated with some modifications to efficiently generate feasible columns. With the tight lower bounds thereby obtained, the branch-and-bound procedure is activated to obtain an integer solution. Computational experience with the benchmark test instances confirms that our approach outperforms all the existing algorithms both in terms of the quality of solutions generated and the solution time.     

A GRASP with path‐relinking and restarts heuristic for the prize‐collecting generalized minimum spanning tree problem

Given a graph with its vertex set partitioned into a set of groups, nonnegative costs associated to its edges, and nonnegative prizes associated to its vertices, the prize‐collecting generalized minimum spanning tree problem consists in finding a subtree of this graph that spans exactly one vertex of each group and minimizes the sum of the costs of the edges of the tree less the prizes of the selected vertices. It is a generalization of the NP‐hard generalized minimum spanning tree optimization problem. We propose a GRASP (greedy randomized adaptive search procedure) heuristic for its approximate solution, incorporating path‐relinking for search intensification and a restart strategy for search diversification. The hybridization of the GRASP with path‐relinking and restarts heuristic with a data mining strategy that is applied along with the GRASP iterations, after the elite set is modified and becomes stable, contributes to making the heuristic more robust. The computational experiments show that the heuristic developed in this work found very good solutions for test problems with up to 439 vertices. All input data for the test instances and detailed numerical results are made available from Mendeley Data.     

A column generation heuristic for a dynamic generalized assignment problem

This paper studies the dynamic generalized assignment problem (DGAP) which extends the well-known generalized assignment problem by considering a discretized time horizon and by associating a starting time and a finishing time with each task. Additional constraints related to warehouse and yard management applications are also considered. Three linear integer programming formulations of the problem are introduced. The strongest one models the problem as an origin–destination integer multi-commodity flow problem with side constraints. This model can be solved quickly for instances of small to moderate size. However, because of its computer memory requirements, it becomes impractical for larger instances. Hence, a column generation algorithm is used to compute lower bounds by solving the linear program (LP) relaxation of the problem. This column generation algorithm is also embedded in a heuristic aimed at finding feasible integer solutions. Computational experiments on large-scale instances show the effectiveness of the proposed approach.     

A branch-and-bound algorithm to minimize total weighted completion time on identical parallel machines with job release dates

In this paper, we consider an identical parallel machine scheduling problem with release dates. The objective is to minimize the total weighted completion time. This problem is known to be strongly NP-hard. We propose some dominance properties and two lower bounds. We also present an efficient heuristic. A branch-and-bound algorithm, in which the heuristic, the lower bounds and the dominance properties are incorporated, is proposed and tested on a large set of randomly generated instances.     

Exact and heuristic procedures for single machine scheduling with quadratic earliness and tardiness penalties

The present paper studies the single machine, no-idle-time scheduling problem with a weighted quadratic earliness and tardiness objective. We investigate the relationship between this problem and the assignment problem, and we derive two lower bounds and several heuristic procedures based on this relationship. Furthermore, the applicability of the time-indexed integer programming formulation is investigated. The results of a computational experiment on a set of randomly generated instances show (1) the high-quality results of the proposed heuristics, (2) the low optimality gap of one of the proposed lower bounds and (3) the applicability of the integer programming formulation to small and medium size cases of the problem.     

Two‐node‐connected star problem

In this study, a two‐node‐connected star problem (2NCSP) is introduced. We are given a simple graph and internal and external costs for each link of the graph. The goal is to find the minimum‐cost spanning subgraph, where the core is two‐node‐connected and the remaining external nodes are connected to the core. First, we show that the 2NCSP belongs to the class of NP‐hard computational problems. Therefore, a greedy randomized adaptive search procedure (GRASP) heuristic is developed, enriched with a variable neighborhood descent (VND). The neighborhood structures include exact integer linear programming models to find the best paths and two‐node‐connected replacements, as well as a shaking operation in order to prevent being trapped in a local minima. The ring star problem (RSP) represents a relevant model in network optimization, where the core is a ring instead of an arbitrary two‐node‐connected graph. We contrast our GRASP/VND methodology with a previous reference work on the RSP in order to highlight the effectiveness of our heuristic. The heuristic is competitive, and the best results produced for several instances so far are under study. In this study, a discussion of the results and trends for future work are provided.     

A note on “event-based MILP models for resource-constrained project scheduling problems”

Recently, new mixed integer linear programming formulations for the resource-constrained project scheduling problem were proposed by Koné et al. [3]. Unfortunately, the presentation of the first new model (called start/end-based formulation SEE) was not correct. More precisely, a set of necessary constraints representing the relative positioning of start and end events of activities was unintentionally omitted in the paper although it was present in the integer program used for the computational experiments. After presenting a counterexample showing the incorrectness, we provide a disaggregated and an aggregated variant of the set of necessary constraints, the disaggregated formulation yielding in theory a better linear programming relaxation. We present computational results showing that although the linear programming relaxations of both formulations yield equivalently poor lower bounds, the disaggregated formulation shows in average a better performance for integer solving of a well-known set of 30-activity instances.