On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems Citation for published version (APA): Hurkens, C. A. J., & Schrijver, A. (1989). On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems.

Abstract. The research domain of process discovery aims at constructing a process model (e.g. a Petri net) which is an abstract representation of an execution log. Such a Petri net should (1) be able to reproduce the log under consideration and (2) be independent of the number of cases in the log. In this paper, we present a process discovery algorithm where we use concepts taken from the language-based theory of regions, a wellknown Petri net research area. We identify a number of shortcomings of this theory from the process discovery perspective, and we provide solutions based on integer linear programming.

Time-indexed formulations for machine scheduling problems have received a great deal of attention; not only do the linear programming relaxations provide strong lower bounds, but they are good guides for approximation algorithms as well. Unfortunately, time-indexed formulations have one major disadvantage—their size. Even for relatively small instances the number of constraints and the number of variables can be large. In this paper, we discuss how Dantzig-Wolfe decomposition techniques can be applied to alleviate, at least partly, the difficulties associated with the size of time-indexed formulations. In addition, we show that the application of these techniques still allows the use of cut generation techniques.

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We develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables. The algorithm is based on lattice basis reduction. It first finds a short vector satisfying the system of diophantine equations, and a set of vectors belonging to the nullspace of the constraint matrix. Due to basis reduction, all these vectors are relatively short. The next step is to branch on linear combinations of the null-space vectors, which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists. The research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing. Our algorithm is tested with good results on real-life data, and on instances from the literature.