Constant-Stretch Rounding on the Hypersimplex

2026-05-31

AI Summary

This work resolves an open problem posed by Naor et al. by achieving the first constant-stretch dependent rounding on the hypersimplex. Specifically, for any vectors \(x, y \in [0,1]^n\) satisfying \(\sum x_i = \sum y_i = k\), the expected symmetric difference between the sampled \(k\)-subsets \(A(x)\) and \(A(y)\) obeys \(\mathbb{E}[|A(x) \triangle A(y)|] \leq 6\|x - y\|_1\), significantly improving upon the previous \(O(\log k)\) bound. The approach leverages the maximum-entropy distribution over \(k\)-subsets, combined with shared random ordering, a uniform thresholding rule, and a carefully designed coupling of marginal probabilities. This construction preserves exact marginal consistency while guaranteeing a constant stretch of at most six times the \(\ell_1\) distance, and extends naturally to the "at-most-\(k\)" polytope with stretch at most twelve.

Abstract

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 a constant-stretch scheme. Our 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 $\mathbb{E}[|A(x)\triangle A(y)|]\le 6\|x-y\|_1$. This improves the previous $O(\log k)$ bound for hypersimplex correlated rounding and answers an open question raised by Naor, Raju, Shetty, Srinivasan, Valieva, and Wajc. By adding dummy coordinates, the same result gives stretch at most $12$ for the at-most-$k$ polytope. The proof was found in a GPT 5.5 Pro Extended conversation prompted by the authors, and Codex was used to help assemble the manuscript under the authors' supervision.

Problem

• correlated rounding
• hypersimplex
• uniform matroid
• k-subset sampling
• marginal constraints

Innovation

• correlated rounding
• constant stretch
• hypersimplex
• maximum-entropy sampling
• uniform matroid