A common approach to tackle a combinatorial optimization problem is to first solve a continuous relaxation and then round the obtained fractional solution. For the latter, the framework of contention resolution schemes (or CR schemes), introduced by Chekuri, Vondrak, and Zenklusen, is a general and successful tool. 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 independence constraints. Intuitively, a CR 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 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. Moreover, this scheme generalizes into an optimal CR scheme for partition matroids.
翻译:解决组合优化问题的常见方法是首先解决连续放松,然后绕过获得的分解解决方案。对于后者而言,由 Chekuri、 Vondrak 和 Zenklusen 引入的争议解决方案(或CR 方案)框架(或CR 方案)是一个普遍和成功的工具。 CR 方案在一个放松的聚点中采用一个分点 $xx美元,每个回合独立地协调$x_i美元,以获得一个可能不可行的数据集,然后降低一些元素以满足独立限制。 直观地说, 如果每个元素都选择了美元或CR(或CR 方案 ) 的争议解决方案(或CRC 方案 ) 框架, 是一个普通的工具。 对于一等级的统一机的特例来说, 尤其如此。 在这项工作中, 我们提供了一个简单和明确的单一的 CRP 方案, 其平衡值在 1 - binom@n\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\