First-order integer programming for MAP problems

Finding the most probable (MAP) model in SRL frameworks such as Markov logic and Problog can, in principle, be solved by encoding the problem as a `grounded-out' mixed integer program (MIP). However, useful first-order structure disappears in this process motivating the development of first-order MIP approaches. Here we present mfoilp, one such approach. Since the syntax and semantics of mfoilp is essentially the same as existing approaches we focus here mainly on implementation and algorithmic issues. We start with the (conceptually) simple problem of using a logic program to generate a MIP instance before considering more ambitious exploitation of first-order representations.Comment: corrected typo

Integer programming methods for special college admissions problems

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones

Cutting plane methods for general integer programming

Integer programming (IP) problems are difficult to solve due to the integer restrictions imposed on them. A technique for solving these problems is the cutting plane method. In this method, linear constraints are added to the associated linear programming (LP) problem until an integer optimal solution is found. These constraints cut off part of the LP solution space but do not eliminate any feasible integer solution. In this report algorithms for solving IP due to Gomory and to Dantzig are presented. Two other cutting plane approaches and two extensions to Gomory's algorithm are also discussed. Although these methods are mathematically elegant they are known to have slow convergence and an explosive storage requirement. As a result cutting planes are generally not computationally successful

The parallel approximability of a subclass of quadratic programming

In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version

Heuristic branch-and-price for building long term trainee schedules.

Branch-and-price is an increasingly important technique for solving large integer programming models. Staff scheduling has been a particularly fruitful area since these problems typically exhibit a decomposable structure. Beside computational efficiency branch-and-price produces two other important advantages in comparison with pure integer programming. Firstly, it often allows for a more accurate problem statement since many constraints which are hard to formulate in the integer program could be easily incorporated in the column generator. Secondly, a branch-and-price algorithm can easily be turned into an effective heuristic when optimality is no major concern. We illustrate these advantages for a medical trainee scheduling problem encountered at Oogziekenhuis Gasthuisberg Leuven and present some computational results together with implementation issues.Advantages; Area; Branch-and-price; Constraint; Efficiency; Heuristic; Integer programming; Model; Models; Problems; Research; Scheduling; Staff scheduling; Structure;

Algorithms for Highly Symmetric Linear and Integer Programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title slightly change