Integer Programming and Conway's Game of Life
SIAM Review
Automating branch-and-bound for dynamic programs
PEPM '08 Proceedings of the 2008 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation
Propagation via lazy clause generation
Constraints
Journal of Artificial Intelligence Research
IJCAI'95 Proceedings of the 14th international joint conference on Artificial intelligence - Volume 1
MiniZinc: towards a standard CP modelling language
CP'07 Proceedings of the 13th international conference on Principles and practice of constraint programming
Using relaxations in maximum density still life
CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
Lazy clause generation reengineered
CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
Improving combinatorial optimization: extended abstract
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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The Maximum Density Still Life Problem (CSPLib prob032) is to find the maximum number of live cells that can fit in an nxn region of an infinite board, so that the board is stable under the rules of Conway@?s Game of Life. It is considered a very difficult problem and has a raw search space of O(2^n^^^2). Previous state of the art methods could only solve up to n=20. We give a powerful reformulation of the problem into one of minimizing ''wastage'' instead of maximizing the number of live cells. This reformulation allows us to compute very strong upper bounds on the number of live cells, which dramatically reduces the search space. It also gives us significant insights into the nature of the problem. By combining these insights with several powerful techniques: remodeling, lazy clause generation, bounded dynamic programming, relaxations, and custom search, we are able to solve the Maximum Density Still Life Problem for all n. This is possible because the Maximum Density Still Life Problem is in fact well behaved mathematically for sufficiently large n (around n200) and if such very large instances can be solved, then there exist ways to construct provably optimal solutions for all n from a finite set of base solutions. Thus we show that the Maximum Density Still Life Problem has a closed form solution and does not require exponential time to solve.