On Rounding on the Hypersimplex
We study correlated rounding on the hypersimplex, the base polytope of the uniform matroid. For each point \(x\) in the hypersimplex, the goal is to sample a \(k\)-subset \(A(x)\) with marginals \(x\), while coupling the samples for all choices of \(x\) so that nearby inputs produce nearby sets. We give conditional constant-stretch results for the natural maximum-entropy sequential scheme, based on a conjectured spectral property of the covariance matrix of the maximum-entropy distribution over \(k\)-subsets; this conjecture has been extensively tested numerically. Under this property, the scheme samples the maximum-entropy \(k\)-subset distribution with prescribed marginals using a common random ordering and common uniform thresholds. For every \(x,y\in[0,1]^n\) with \(\sum_i x_i=\sum_i y_i=k\), it satisfies \[ \mathbb{E}\!\left[|A(x)\triangle A(y)|\right] \le 6\|x-y\|_1 . \] Thus, conditional on the spectral hypothesis, the previous \(O(\log k)\) bound for hypersimplex correlated rounding would improve to a constant and the open question raised by Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc would have a positive answer. By adding dummy coordinates, the same conditional result gives stretch at most \(12\) for the at-most-\(k\) polytope.