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：争用解决机制（CR机制）由Chekuri、Vondrak和Zenklusen提出，是一类随机舍入算法，用于将下行封闭约束族松弛的分数解转化为整数解。CR机制取松弛多面体中的一个分数点$x$，独立舍入每个坐标$x_i$以生成可能不可行的集合，然后丢弃部分元素以满足约束。直观上，如果每个元素$i$被选中的概率至少为$c \cdot x_i$，则称争用解决机制是$c$-平衡的。已知一般拟阵具有$(1-1/e)$-平衡的CR机制，且该结果是（渐近）最优的，这对秩为1的均匀拟阵特例同样成立。本文针对$n$个元素上秩为$k$的均匀拟阵，提出一种简单且显式的单调CR机制，其平衡度为$1 - \binom{n}{k}\:\left(1-\frac{k}{n}\right)^{n+1-k}\:\left(\frac{k}{n}\right)^k$，并证明该结果是最优的。随着$n$增大，该表达式从上方向$1 - e^{-k}k^k/k!$收敛。尽管该渐近界可通过组合已知结果得到，但需定义指数大小的线性规划，并使用随机采样和椭球算法。相比之下，我们的方法具有简单显式的优势。该机制可自然推广为划分拟阵的最优CR机制。