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- W2558885861 abstract "A well-known theorem of Spencer shows that any set system with n sets over n elements admits a coloring of discrepancy $$O(sqrt{n})$$ . While the original proof was non-constructive, recent progress brought polynomial time algorithms by Bansal, Lovett and Meka, and Rothvoss. All those algorithms are randomized, even though Bansal’s algorithm admitted a complicated derandomization. We propose an elegant deterministic polynomial time algorithm that is inspired by Lovett-Meka as well as the Multiplicative Weight Update method. The algorithm iteratively updates a fractional coloring while controlling the exponential weights that are assigned to the set constraints. A conjecture by Meka suggests that Spencer’s bound can be generalized to symmetric matrices. We prove that $$n times n$$ matrices that are block diagonal with block size q admit a coloring of discrepancy $$O(sqrt{n} cdot sqrt{log (q)})$$ . Bansal, Dadush and Garg recently gave a randomized algorithm to find a vector x with entries in $$lbrace {-1,1rbrace }$$ with $$Vert AxVert _{infty } le O(sqrt{log n})$$ in polynomial time, where A is any matrix whose columns have length at most 1. We show that our method can be used to deterministically obtain such a vector." @default.
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- W2558885861 date "2017-01-01" @default.
- W2558885861 modified "2023-10-16" @default.
- W2558885861 title "Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method" @default.
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- W2558885861 doi "https://doi.org/10.1007/978-3-319-59250-3_31" @default.
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