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Aim of this book is a computer proof for the MIRUP (modified integer roundup property) conjecture for Bin Packing in low dimension. Bin Packing is a well known combinatorial optimization problem (BPP), which appears as subprobem in many applications like scheduling. Two algorithms for an algorithmic approach to the MIRUP conjecture are presented. One works with generating possible knapsacks by picking subsets of possible patterns. It computes all relevant instances and checks MIRUP directly up to dimension 7. The running time for dimension 6 is less than a minute. A parallelised version takes less than 10 days by usage of 10 computation nodes for dimension 7. The other algorithm generates the knapsack instances using separating hyperplanes. It runs less than 5 hours on a single computation node to check dimension 6.