A polyhedral study of triplet formulation for single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183-190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral.     

Approximability results for the resource-constrained project scheduling problem with a single type of resources

In this paper, we consider the well-known resource-constrained project scheduling problem. We give some arguments that already a special case of this problem with a single type of resources is not approximable in polynomial time with an approximation ratio bounded by a constant. We prove that there exist instances for which the optimal makespan values for the non-preemptive and the preemptive problems have a ratio of O(logn), where n is the number of jobs. This means that there exist instances for which the lower bound of Mingozzi et al. has a bad relative error of O(logn), and the calculation of this bound is an NP-hard problem. In addition, we give a proof that there exists a type of instances for which known approximation algorithms with polynomial time complexity have an approximation ratio of at least equal to $O(\sqrt{n})$ , and known lower bounds have a relative error of at least equal to O(logn). This type of instances corresponds to the single machine parallel-batch scheduling problem 1|p?batch,b=∞|C _max.     

Bound reduction using pairs of linear inequalities

We describe a procedure to reduce variable bounds in mixed integer nonlinear programming (MINLP) as well as mixed integer linear programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction procedure extends the feasibility based bound reduction technique on linear functions, used in MINLP and MILP. However, it can also be seen as a special case of optimality based bound reduction, a method to infer variable bounds from an LP relaxation of the problem. For an LP relaxation with m constraints and n variables, there are O(m ²) pairs of constraints, and a naïve implementation of our bound reduction scheme has complexity O(n ³) for each pair. Therefore, its overall complexity O(m ² n ³) can be prohibitive for relatively large problems. We have developed a more efficient procedure that has complexity O(m ² n ²), and embedded it in two Open-Source solvers: one for MINLP and one for MILP. We provide computational results which substantiate the usefulness of this bound reduction technique for several instances.     

Lagrangean relaxation for a lower bound to a set partitioning problem with side constraints: properties and algorithms

The problem (P) addressed here is a special set partitioning problem with two additional non-trivial constraints. A Lagrangean Relaxation (LR_u) is proposed to provide a lower bound to the optimal solution to this problem. This Lagrangean relaxation is accomplished by a subgradient optimization procedure which solves at each iteration a special 0–1 knapsack problem (KP-k). We give two procedures to solve (KP-k), namely an implicity enumeration algorithm and a column generation method. The approach is promising for it provides feasible integer solutions to the side constraints that will hopefully be optimal to most of the instances of the problem (P). Properties of the feasible solutions to (KP-k) are highlighted and it is shown that the linear programming relaxation to this problem has a worst case time bound of order O(n³).     

Normal projection: deterministic and probabilistic algorithms

We consider the following problem： for a collection of points in an n-dimensional space, find a linear projection mapping the points to the ground field such that different points are mapped to different values. Such projections are called normal and are useful for making algebraic varieties into normal positions. The points may be given explicitly or implicitly and the coefficients of the projection come from a subset S of the ground field. If the subset S is small, this problem may be hard. This paper deals with relatively large S, a deterministic algorithm is given when the points are given explicitly, and a lower bound for success probability is given for a probabilistic algorithm from in the literature.     

Equivalence of some LP-based lower bounds for the Golomb ruler problem

The Golomb Ruler problem consists in finding n integers such that all possible differences are distinct and such that the largest difference is minimum. We review three lower bounds based on linear programming that have been proposed in the literature for this problem, and propose a new one. We then show that these 4 lower bounds are equal. Finally we discuss some computational experience aiming at identifying the easiest lower bound to compute in practice.     

A computational study and survey of methods for the single-row facility layout problem

The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerländer and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.     

Lower bounds for combinatorial problems on graphs

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is established in a unified way for a rather large class of problems.     

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.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Improved compact linearizations for the unconstrained quadratic 0-1 minimization problem

We present and compare three new compact linearizations for the quadratic 0-1 minimization problem, two of which achieve the same lower bound as does the “standard linearization”. Two of the linearizations require the same number of constraints with respect to Glover’s one, while the last one requires n additional constraints where n is the number of variables in the quadratic 0-1 problem. All three linearizations require the same number of additional variables as does Glover’s linearization. This is an improvement on the linearization of Adams, Forrester and Glover (2004) which requires n additional variables and 2n additional constraints to reach the same lower bound as does the standard linearization. Computational results show however that linearizations achieving a weaker lower bound at the root node have better global performances than stronger linearizations when solved by Cplex.     

An exact algorithm for large multiple knapsack problems

The Multiple Knapsack Problem (MKP) is the problem of assigning a subset of n items to m distinct knapsacks, such that the total profit sum of the selected items is maximized, without exceeding the capacity of each of the knapsacks. The problem has several applications in naval as well as financial management. A new exact algorithm for the MKP is presented, which is specially designed for solving large problem instances. The recursive branch-and-bound algorithm applies surrogate relaxation for deriving upper bounds, while lower bounds are obtained by splitting the surrogate solution into the m knapsacks by solving a series of Subset-sum Problems. A new separable dynamic programming algorithm is presented for the solution of Subset-sum Problems, and we also use this algorithm for tightening the capacity constraints in order to obtain better upper bounds. The developed algorithm is compared to the mtm algorithm by Martello and Toth, showing the benefits of the new approach. A surprising result is that large instances with n=100 000 items may be solved in less than a second, and the algorithm has a stable performance even for instances with coefficients in a moderately large range.     

On multiple caps in finite projective spaces

In this paper, we consider new results on (k, n)-caps with n > 2. We provide a lower bound on the size of such caps. Furthermore, we generalize two product constructions for (k, 2)-caps to caps with larger n. We give explicit constructions for good caps with small n. In particular, we determine the largest size of a (k, 3)-cap in PG(3, 5), which turns out to be 44. The results on caps in PG(3, 5) provide a solution to four of the eight open instances of the main coding theory problem for q = 5 and k = 4.     

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ? 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.     

Improved approximation for non-preemptive single machine flow-time scheduling with an availability constraint

We study the problem of scheduling n non-preemptable jobs on a single machine which is not available for processing during a given time period. The objective is to minimize the sum of the job completion times. The best known approximation algorithm for this NP-hard problem has a relative worst-case error bound of 17.6%. We present a parametric O(nlog n)-algorithm H with which better worst-case error bounds can be obtained. The best error bound calculated for the algorithm in the paper is 7.4%. In a computational experiment, we test the algorithm with the performance guarantee set to 10.2%. It turns out that randomly generated instances with up to 1000 jobs can be solved with a mean (maximum) error of 0.31% (3.18%) and a mean (maximum) computation time of 0.8 (9.7) seconds.     

The number of 2-sat functions

Our aim in this paper is to address the following question: of the 2^{2 n} Boolean functions onn variables, how many are expressible as 2-SAT formulae? In other words, we wish to count the number of different instances of 2-SAT, counting two instances as equivalent if they have the same set of satisfying assignments. Viewed geometrically, we are asking for the number of subsets of then-dimensional discrete cube that are unions of (n-2)-dimensional subcubes.There is a trivial upper bound of 2^4(n/2), the number of 2-SAT formulae. There is also an obvious lower bound of 2^(n/2), corresponding to the monotone 2-SAT formulae. Our main result is that, rather surprisingly, this lower bound gives the correct speed: the number of 2-SAT functions is 2^(1+0(1)) n ^{2 2}.     

A new lower bound for the linear knapsack problem with general integer variables

It is well known that the linear knapsack problem with general integer variables (LKP) is NP-hard. In this paper we first introduce a special case of this problem and develop an O(n) algorithm to solve it. We then show how this algorithm can be used efficiently to obtain a lower bound for a general instance of LKP and prove that it is at least as good as the linear programming lower bound. We also present the results of a computational study that show that for certain classes of problems the proposed bound on average is tighter than other bounds proposed in the literature.