Solving Assembly Line Balancing Problems by Combining IP and CP

Assembly line balancing problems consist in partitioning the work necessary to assemble a number of products among different stations of an assembly line. We present a hybrid approach for solving such problems, which combines constraint programming and integer programming.Comment: 10 pages, Sixth Annual Workshop of the ERCIM Working Group on Constraints, Prague, June 200

Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.Comment: 21 page

An integrated approach for requirement selection and scheduling in software release planning

It is essential for product software companies to decide which requirements should be included in the next release and to make an appropriate time plan of the development project. Compared to the extensive research done on requirement selection, very little research has been performed on time scheduling. In this paper, we introduce two integer linear programming models that integrate time scheduling into software release planning. Given the resource and precedence constraints, our first model provides a schedule for developing the requirements such that the project duration is minimized. Our second model combines requirement selection and scheduling, so that it not only maximizes revenues but also simultaneously calculates an on-time-delivery project schedule. Since requirement dependencies are essential for scheduling the development process, we present a more detailed analysis of these dependencies. Furthermore, we present two mechanisms that facilitate dynamic adaptation for over-estimation or under-estimation of revenues or processing time, one of which includes the Scrum methodology. Finally, several simulations based on real-life data are performed. The results of these simulations indicate that requirement dependency can significantly influence the requirement selection and the corresponding project plan. Moreover, the model for combined requirement selection and scheduling outperforms the sequential selection and scheduling approach in terms of efficiency and on-time delivery. \u

Optimizing Constrained Subtrees of Trees

Given a tree G = (V, E) and a weight function defined on subsets of its nodes, we consider two associated problems. The first, called the "rooted subtree problem", is to find a maximum weight subtree, with a specified root, from a given set of subtrees. The second problem, called "the subtree packing problem", is to find a maximum weight packing of node disjoint subtrees chosen from a given set of subtrees, where the value of each subtree may depend on its root. We show that the complexity status of both problems is related, and that the subtree packing problem is polynomial if and only if each rooted subtree problem is polynomial. In addition we show that the convex hulls of the feasible solutions to both problems are related: the convex hull of solutions to the packing problem is given by "pasting together" the convex hulls of the rooted subtree problems. We examine in detail the case where the set of feasible subtrees rooted at node i consists of all subtrees with at most k nodes. For this case we derive valid inequalities, and specify the convex hull when k < 4