A computational comparison of algorithms for the inventory routing problem

The inventory routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Each day a fleet of trucks is dispatched over a set of routes to resupply a subset of the customers. In this paper, we describe and compare algorithms for this problem defined over a short planning period, e.g. one week. These algorithms define the set of customers to be serviced each day and produce routes for a fleet of vehicles to service those customers. Two algorithms are compared in detail, one which first allocates deliveries to days and then solves a vehicle routing problem and a second which treats the multi-day problem as a modified vehicle routing problem. The comparison is based on a set of real data obtained from a propane distribution firm in Pennsylvania. The solutions obtained by both procedures compare quite favorably with those in use by the firm.Part of this work was performed while this author was visiting the University of Waterloo.     

Geometric algorithms for the minimum cost assignment problem

We consider the minimum-cost λ-assignment problem, which is equivalent to the minimum-weight one-to-many matching problem on a complete bipartite graph Γ = (A, B), where A and B have n and k nodes (n ? k), respectively. Formulating the problem geometrically, we given an O(kn + k^2.5n^0.5 log^1.5 n) time randomized algorithm, which is better than the existing O(kn² + n² log n) time algorithm if n > k log k.     

Approximated algorithms for the minimum dilation triangulation problem

The complexity status of the minimum dilation triangulation (MDT) problem for a general point set is unknown. Therefore, we focus on the development of approximated algorithms to find high quality triangulations of minimum dilation. For an initial approach, we design a greedy strategy able to obtain approximate solutions to the optimal ones in a simple way. We also propose an operator to generate the neighborhood which is used in different algorithms: Local Search, Iterated Local Search, and Simulated Annealing. Besides, we present an algorithm called Random Local Search where good and bad solutions are accepted using the previous mentioned operator. For the experimental study we have created a set of problem instances since no reference to benchmarks for these problems were found in the literature. We use the sequential parameter optimization toolbox for tuning the parameters of the SA algorithm. We compare our results with those obtained by the OV-MDT algorithm that uses the obstacle value to sort the edges in the constructive process. This is the only available algorithm found in the literature. Through the experimental evaluation and statistical analysis, we assess the performance of the proposed algorithms using this operator.     

Approximation algorithms for the minimum rainbow subgraph problem

We consider the minimum rainbow subgraph problem (MRS): given a graph G, whose edges are coloured with p colours. Find a subgraph F⊆G of G of minimum order and with p edges such that each colour occurs exactly once. For graphs with maximum degree Δ(G) there is a greedy polynomial-time approximation algorithm for the MRS problem with an approximation ratio of Δ(G). In this paper we present a polynomial-time approximation algorithm with an approximation ratio of for Δ≥2.     

A semidefinite programming approach to the hypergraph minimum bisection problem

The hypergraph minimum bisection (HMB) problem is the problem of partitioning the vertices of a hypergraph into two sets of equal size so that the total weight of hyperedges crossing the sets is minimized. HMB is an NP-hard problem that arises in numerous applications – for example, in digital circuit design. Although many heuristics have been proposed for HMB, there has been no known mathematical program that is equivalent to HMB. As a means of shedding light on HMB, we study the equivalent, complement problem of HMB (called CHMB), which attempts to maximize the total weight of non-crossing hyperedges. We formulate CHMB as a quadratically constrained quadratic program, considering its semidefinite programming relaxation and providing computational results on digital circuit partitioning benchmark problems. We also provide results towards an approximation guarantee for CHMB.     

Edge-swapping algorithms for the minimum fundamental cycle basis problem

We consider the problem of finding a fundamental cycle basis with minimum total cost in an undirected graph. This problem is NP-hard and has several interesting applications. Since fundamental cycle bases correspond to spanning trees, we propose a local search algorithm, a tabu search and variable neighborhood search in which edge swaps are iteratively applied to a current spanning tree. We also present a mixed integer programming formulation of the problem whose linear relaxation yields tighter lower bounds than other known formulations. Computational results obtained with our algorithms are compared with those from the best available constructive heuristic on several types of graphs. This article extends the conference paper (Amaldi et al. 2004).     

The minimum distance superset problem: formulations and algorithms

The partial digest problem consists in retrieving the positions of a set of points on the real line from their unlabeled pairwise distances. This problem is critical for DNA sequencing, as well as for phase retrieval in X-ray crystallography. When some of the distances are missing, this problem generalizes into a “minimum distance superset problem”, which aims to find a set of points of minimum cardinality such that the multiset of their pairwise distances is a superset of the input. We introduce a quadratic integer programming formulation for the minimum distance superset problem with a pseudo-polynomial number of variables, as well as a polynomial-size integer programming formulation. We investigate three types of solution approaches based on an available integer programming solver: (1) solving a linearization of the pseudo-polynomial-sized formulation, (2) solving the complete polynomial-sized formulation, or (3) performing a binary search over the number of points and solving a simpler feasibility or optimization problem at each step. As illustrated by our computational experiments, the polynomial formulation with binary search leads to the most promising results, allowing to optimally solve most instances with up to 25 distance values and 8 solution points.     

A branch-and-cut algorithm for the equicut problem

We describe an algorithm for solving the equicut problem on complete graphs. The core of the algorithm is a cutting-plane procedure that exploits a subset of the linear inequalities defining the convex hull of the incidence vectors of the edge sets that define an equicut. The cuts are generated by several separation procedures that will be described in the paper. Whenever the cutting-plane procedure does not terminate with an optimal solution, the algorithm uses a branch-and-cut strategy. We also describe the implementation of the algorithm and the interface with the LP solver. Finally, we report on computational results on dense instances with sizes up to 100 nodes.     

Rank-two update algorithms for the minimum volume enclosing ellipsoid problem

We consider the problem of computing a (1+ε)-approximation to the minimum volume enclosing ellipsoid (MVEE) of a given set of m points in R ⁿ . Based on the idea of sequential minimal optimization (SMO) method, we develop a rank-two update algorithm. This algorithm computes an approximate solution to the dual optimization formulation of the MVEE problem, which updates only two weights of the dual variable at each iteration. We establish that this algorithm computes a (1+ε)-approximation to MVEE in O(mn ³/ε) operations and returns a core set of size O(n ²/ε) for ε∈(0,1). In addition, we give an extension of this rank-two update algorithm. Computational experiments show the proposed algorithms are very efficient for solving large-scale problem with a high accuracy.     

Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem

Consider the problem of computing a (1+?)-approximation to the minimum volume axis-aligned ellipsoid (MVAE) enclosing a set of m points in Rⁿ. We first provide an extension and improvement to algorithm proposed in Kumar and Y?ld?r?m (2008) [5] (the KY algorithm) for the MVAE problem. The main challenge of the MVAE problem is that there is no closed form solution in the line search step (beta). Therefore, the KY algorithm proposed a certain choice of beta that leads to their complexity and core set results in solving the MVAE problem. We further analyze the line search step to derive a new beta, relying on an analysis of up to the fourth order derivative. This choice of beta leads to the improved complexity and core set results. The second modification is given by incorporating “away steps” into the first one at each iteration, which obtains the same complexity and core set results as the first one. In addition, since the second modification uses the idea of “dropping points”, it has the potential to compute smaller core sets in practice. Some numerical results are given to show the efficiency of the modified algorithms.     

The node capacitated graph partitioning problem: A computational study

In this paper we consider the problem ofk-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-and-cut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in finite element methods and partitioning problems associated with electronic circuit layout and compiler design. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

A branch-and-cut method for the obnoxious p-median problem

The obnoxious p-median (OpM) problem is the repulsive counterpart of the ore known attractive p-median problem. Given a set I of cities and a set J of possible locations for obnoxious plants, a p-cardinality subset Q of J is sought, such that the sum of the distances between each city of I and the nearest obnoxious site in Q is maximised. We formulate (OpM) as a {0,1} linear programming problem and propose three families of valid inequalities whose separation problem is polynomial. We describe a branch-and-cut approach based on these inequalities and apply it to a set of instances found in the location literature. The computational results presented show the effectiveness of these inequalities for (OpM). The work of the first author has been partially supported by the Coordinated Project C.A.M.P.O. and that of the third author by a short mobility grant, both of the Italian National Research Council.     

Valid inequalities and branch-and-cut for the clique pricing problem

Motivated by an application in highway pricing, we consider the problem that consists in setting profit-maximizing tolls on a clique subset of a multicommodity transportation network. We formulate the problem as a linear mixed integer program and propose strong valid inequalities, some of which define facets of the two-commodity polyhedron. The numerical efficiency of these inequalities is assessed by embedding them within a branch-and-cut framework.     

A branch-and-cut algorithm for the plant-cycle location problem

The Plant-Cycle Location Problem (PCLP) is defined on a graph G=(I∪J, E), where I is the set of customers and J is the set of plants. Each customer must be served by one plant, and the plant must be opened to serve customers. The number of customers that a plant can serve is limited. There is a cost of opening a plant, and of serving a customer from an open plant. All customers served by a plant are in a cycle containing the plant, and there is a routing cost associated to each edge of the cycle. The PCLP consists in determining which plants to open, the assignment of customers to plants, and the cycles containing each open plant and its customers, minimizing the total cost. It is an NP-hard optimization problem arising in routing and telecommunications. In this article, the PCLP is formulated as an integer linear program, a branch-and-cut algorithm is developed, and computational results on real-world data and randomly generated instances are presented. The proposed approach is able to find optimal solutions of random instances with up to 100 customers and 100 potential plants, and of instances on real-world data with up to 120 customers and 16 potential plants.     

A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (ℓ,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (ℓ,S) inequalities to a general class of valid inequalities, called the inequalities, and we establish necessary and sufficient conditions which guarantee that the inequalities are facet-defining. A separation heuristic for inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. This research has been supported in part by the National Science Foundation under Award number DMII-0121495.     

A comparison of two algorithms for the assignment problem

State-of-the-art computational results have shown that the shortest augmenting path (SAP) methods are more efficient than other primal-dual and primal-simplex based methods for solving the linear assignment problem on uniprocessor computers. There is, however, some controversy concerning their merits when compared with Bertsekas' auction algorithm on multiprocessor computers. In this study we investigate the performance of these competing methods on the Alliant FX/8. For each method, theoretical motivation, sources of parallelism and computational results are presented.     

Correlative Sparsity in Primal-Dual Interior-Point Methods for LP,SDP, and SOCP

Exploiting sparsity has been a key issue in solving large-scale optimization problems. The most time-consuming part of primal-dual interior-point methods for linear programs, second-order cone programs, and semidefinite programs is solving the Schur complement equation at each iteration, usually by the Cholesky factorization. The computational efficiency is greatly affected by the sparsity of the coefficient matrix of the equation which is determined by the sparsity of an optimization problem (linear program, semidefinite program or second-order cone program). We show if an optimization problem is correlatively sparse, then the coefficient matrix of the Schur complement equation inherits the sparsity, and a sparse Cholesky factorization applied to the matrix results in no fill-in. S. Kim’s research was supported by Kosef R01-2005-000-10271-0 and KRF-2006-312-C00062.