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LBCAD Report Summary

Project ID: 616787
Funded under: FP7-IDEAS-ERC
Country: Czech Republic

Mid-Term Report Summary - LBCAD (Lower bounds for combinatorial algorithms and dynamic problems)

The project so far led to several new discoveries and many other results which substantially deepen our understanding of computational complexity. The most suprising is the discovery of the concept of catalytic computation which runs counter to the usual lower-bound intuition and improves our understanding of the role of space in computation. We obtained a set of new algorithms for the edit distance problem, a problem that recently attracts a lot of attention in proving conditional lower bounds. We formulated a new approach for measuring space of optimal strategies in (mathematical) game theory. We made a progress on lower bounds in communication complexity, designed a combinatorial model for Boolean matrix multiplication and proved lower bounds in that model. We proved several results shedding new light on structure of graphs and their algorithms.

We briefly describe the new concept of catalytic computation. Catalytic computation is an algorithmic technique in which we use memory occupied by data unrelated to the problem, e.g., someone else's data, to perform useful computation without destroying the data. This allows an algorithm to use much less of its own working memory than one would usually expect. For example, we can solve reachability in directed graphs in polynomial time using only O(log n) bits of our own working memory while using polynomial size extra memory regardless of what the extra memory contains (be it, e.g., incompressible or encrypted data). Upon finishing, the algorithm restores the original content of the extra memory. The best known polynomial time algorithms for the same problem that do not use the extra memory require almost linear working space. At this point we do not know the full potential of the concept and the algorithmic techniques but it is already clear that this substantially changes our fundamental understanding of space needed for computation.

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Czech Republic
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