اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدUnambiguous DNFs and Alon-Saks-Seymour
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We exhibit an unambiguous k-DNF formula that requires CNF width $tilde{Omega}(k^2)$, which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon--Saks--Seymour problem in graph theory (posed in 1991), which asks: How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query and communication complexity.
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In 1992 Mansour proved that every size-$s$ DNF formula is Fourier-concentrated on $s^{O(loglog s)}$ coefficients. We improve this to $s^{O(loglog k)}$ where $k$ is the read number of the DNF. Since $k$ is always at most $s$, our bound matches Mansour
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