Contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak and Zenklusen, are a class of randomized rounding algorithms for converting a fractional solution to a relaxation for a down-closed constraint family into an integer solution. A CR scheme takes a fractional point $x$ in a relaxation polytope, rounds each coordinate $x_i$ independently to get a possibly non-feasible set, and then drops some elements in order to satisfy the constraints. Intuitively, a contention resolution scheme is $c$-balanced if every element $i$ is selected with probability at least $c \cdot x_i$. It is known that general matroids admit a $(1-1/e)$-balanced CR scheme, and that this is (asymptotically) optimal. This is in particular true for the special case of uniform matroids of rank one. In this work, we provide a simple and explicit monotone CR scheme for uniform matroids of rank $k$ on $n$ elements with a balancedness of $1 - \binom{n}{k}\:\left(1-\frac{k}{n}\right)^{n+1-k}\:\left(\frac{k}{n}\right)^k$, and show that this is optimal. As $n$ grows, this expression converges from above to $1 - e^{-k}k^k/k!$. While this asymptotic bound can be obtained by combining previously known results, these require defining an exponential-sized linear program, as well as using random sampling and the ellipsoid algorithm. Our procedure, on the other hand, has the advantage of being simple and explicit. This scheme extends naturally into an optimal CR scheme for partition matroids.
翻译:由 Chekuri、 Vondrak 和 Zenklusen 推出的 内容解析方案 ( 或 CR 方案 ) 由 Chekuri、 Vondrak 和 Zenklusen 推出的 内容解析方案 ( 或 CR 方案 ) 是一个随机化的圆算法, 将一个分解解决方案转换成一个自下而上约束式的解析方案 。 一个 CR 方案在一个放松的多功能中采用一个分数点 $x美元, 以独立协调$x美元, 以获得一个可能的不可行的数据集, 然后为了满足这些限制而降低某些元素。 直觉地说, 如果选择每个元素, 美元的可能性至少是 $c\ c\ cddddd, 将一个分解的分解法, 而一般的配机体接受$( 1 / / e) 美元 美元 美元 美元 =\\\\\\\\\\\\\ k r\ k make make rodeal scheal scheal) a schedeal.