稀疏几何集系统与Beck-Fiala猜想
Sparse Geometric Set Systems and the Beck-Fiala Conjecture
摘要：
我们调查的组合差异的几何集系统有界浅细胞的复杂性在\emph{Beck-Fiala}设置，其中每个点属于最多$t$范围。对于具有浅胞复杂度$ψ(m,k)=g(m)k^{c}$的集合系统，其中$(i)$ $g(m) = o(m^{\varepsilon})$对于任何$\varepsilon\in (0,1],$ $(ii)$ $ψ$在$m$中是非减的，并且$(iii)$ $c>0$独立于$m$和$k$，我们得到了\[ O\left(\sqrt{\left(\log n+\left(t^{c}g(n)\right)^{\frac{1}{1+c}}\right)\log n}\right).\]对于$t=ω(\log^2 n)$，在几种情况下，例如对于$\mathbb{R}^2$中的点和半平面/圆盘/伪圆盘的集合系统，$\mathbb{R}^3$中的点和直方等，这些边界是$o(\sqrt{t})$，它验证（并改进）Beck和Fiala~ \emph{(Disc. Appl. Math., 1981)}的约束边界。我们的界限是通过显示存在的\emph{matchings with low crossing number}，使用的乘法权重更新方法的Welzl \emph{(SoCG, 1988)}，连同最近的界的穆斯塔法\emph{(Disc. Comp. Geom., 2015)}上\emph{shallow packings}的集合系统在其浅细胞的复杂性。对于浅胞复杂度为$ψ(m,k)=m^{c_1}g(m)k^{c}$的集合系统，我们得到了交叉数至多为\[ O\left(\left(n^{c_1}g(n)t^{c}\right)^{\frac{1}{1+c_1+c}}\right).\]的匹配。
英文摘要：
We investigate the combinatorial discrepancy of geometric set systems having bounded shallow cell complexity in the \emph{Beck-Fiala} setting, where each point belongs to at most $t$ ranges. For set systems with shallow cell complexity $ψ(m,k)=g(m)k^{c}$, where $(i)$ $g(m) = o(m^{\varepsilon})$ for any $\varepsilon\in (0,1],$ $(ii)$ $ψ$ is non-decreasing in $m$, and $(iii)$ $c>0$ is independent of $m$ and $k$, we get a discrepancy bound of \[ O\left(\sqrt{\left(\log n+\left(t^{c}g(n)\right)^{\frac{1}{1+c}}\right)\log n}\right).\] For $t=ω(\log^2 n)$, in several cases, such as for set systems of points and half-planes / disks / pseudo-disks in $\mathbb{R}^2$, points and orthants in $\mathbb{R}^3$ etc., these bounds are $o(\sqrt{t})$, which verifies (and improves upon) the conjectured bound of Beck and Fiala~\emph{(Disc. Appl. Math., 1981)}. Our bounds are obtained by showing the existence of \emph{matchings with low crossing number}, using the multiplicative weights update method of Welzl \emph{(SoCG, 1988)}, together with the recent bound of Mustafa \emph{(Disc. Comp. Geom., 2015)} on \emph{shallow packings} of set systems in terms of their shallow cell complexity. For set systems of shallow cell complexity $ψ(m,k)=m^{c_1}g(m)k^{c}$, we obtain matchings with crossing number at most \[ O\left(\left(n^{c_1}g(n)t^{c}\right)^{\frac{1}{1+c_1+c}}\right).\] These are of independent interest.
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