A Simple Optimal Contention Resolution Scheme for Uniform Matroids

A common approach to solve a combinatorial optimization problem is to first solve a continous relaxation and then round the fractional solution. For the latter, the framework of contention resolution schemes (or CR schemes) introduced by Chekuri, Vondrak, and Zenklusen, has become 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 factor of $1 - e^{-k}k^k/k!$ for uniform matroids of rank $k$ (which matches the balancedness of $1-1/e$ for $k=1$), and show that this is optimal. While this bound can be obtained by combining previously known results, these require defining an exponential-sized linear program and 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.