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- W2951858218 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 $|Ax|_{infty} leq O(sqrt{log n})$ 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.
- W2951858218 created "2019-06-27" @default.
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- W2951858218 date "2016-11-26" @default.
- W2951858218 modified "2023-09-27" @default.
- W2951858218 title "Deterministic Discrepancy Minimization via the Multiplicative Weight Update Method" @default.
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