Curriculum based course timetabling: Optimal solutions to the Udine benchmark instances

We present an integer programming approach to the university course timetabling problem, in which weekly lectures have to be scheduled and assigned to rooms. Students" curricula impose restrictions as to which courses may be scheduled in parallel. Besides some hard constraints (no two courses in the same room at the same time, etc.), there are several soft constraints in practice which give a convenient structure to timetables; these should be met as well as possible

Sorting with Complete Networks of Stacks

Knuth introduced the problem of sorting with a sequence of stacks. Tarjan extended this idea to sorting with acyclic networks of stacks (and queues), where items to be sorted move from a source through the network to a sink while they may be stored temporarily at nodes (the stacks). Both characterized which permutations are sortable; but complexity of sorting was not an issue

Tools for primal degenerate linear programs: IPS, DCA, and PE

ABSTRACT: This paper describes three recent tools for dealing with primal degeneracy in linear programming. The first one is the improved primal simplex (IPS) algorithm which turns degeneracy into a possible advantage. The constraints of the original problem are dynamically partitioned based on the numerical values of the current basic variables. The idea is to work only with those constraints that correspond to nondegenerate basic variables. This leads to a row-reduced problem which decreases the size of the current working basis. The main feature of IPS is that it provides a nondegenerate pivot at every iteration of the solution process until optimality is reached. To achieve such a result, a negative reduced cost convex combination of the variables at their bounds is selected, if any. This pricing step provides a necessary and sufficient optimality condition for linear programming. The second tool is the dynamic constraint aggregation (DCA), a constructive strategy specifically designed for set partitioning constraints. It heuristically aims to achieve the properties provided by the IPS methodology. We bridge the similarities and differences of IPS and DCA on set partitioning models. The final tool is the positive edge (PE) rule. It capitalizes on the compatibility definition to determine the status of a column vector and the associated variable during the reduced cost computation. Within IPS, the selection of a compatible variable to enter the basis ensures a nondegenerate pivot, hence PE permits a trade-off between strict improvement and high, reduced cost degenerate pivots. This added value is obtained without explicitly computing the updated column components in the simplex tableau. Ultimately, we establish tight bonds between these three tools by going back to the linear algebra framework from which emanates the so-called concept of subspace basis

Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

Dual variable based fathoming in dynamic programs for column generation

In this note, we aim at reducing the state space of dynamic programming algorithms used as column generators in solving the linear programming relaxation of set partitioning problems arising from practical applications. We propose a simple generic lower bounding criterion based on the respective dual optimal solution of the restricted master program. Key words: Dynamic programming, column generation, state space reduction

Combinatorially Simple Pickup and Delivery Paths

Pickup and delivery problems discussed in the literature often allow for only particularly simple solutions in terms of the sequence of visited locations. We study the very simplest pickup and delivery paths which are concatenations of short patterns visiting one or two requests. This restricted variant, still N P-hard, is close to the traveling salesman problem with the additional choice of what patterns to visit. We compare the number of restricted and unrestricted paths, and evaluate their respective path lengths. We conclude with two polynomially solvable cases

Dual Variable Based Fathoming in Dynamic Programs for Column Generation

In column generation schemes, particularly those proposed for set partitioning type problems, dynamic programming algorithms are applied to solve the respective pricing subproblem. In addition to traditional dominance criteria for state space reduction, we develop a simple generic lower bounding criterion based on the dual optimal solution of the restricted master problem. Key words: Dynamic programming; Column generation; Linear programming Column generation is a prominent---and often the solely applicable---method to cope with linear programming problems with a colossal number of variables. In recent years we have been witnessing the optimal solution of truly large problems, but still the need for faster algorithms remains, especially in the various practical application areas. The pricing subproblem usually constitutes the crucial part of a column generation scheme, not uncommonly for the reason of being a hard combinatorial optimization problem by itself. Although a dynamic program..

Combinatorially Simple Pickup and Delivery Paths

Pickup and delivery problems discussed in the literature are often constrained to particularly simple solutions in terms of the sequence of visited locations. We study the very simplest pickup and delivery paths which are concatenations of short patterns visiting one or two requests. This restricted variant, still N P -hard, is close to the traveling salesman problem with the additional choice of what patterns to visit. We compare the number of restricted and unrestricted paths, and evaluate their respective path lengths. We conclude with two polynomially solvable cases