Dimension-Free Correlated Sampling for the Hypersimplex

TL;DR

The paper tackles correlated sampling on the hypersimplex $\Delta_{n,k}$, where the goal is to round input vectors to sets of size at most $k$ while preserving marginals and achieving small expected disagreement across inputs. It introduces a recursive, dimension-reducing composition that projects points to a lower-dimensional polytope, applies a base sampler, and lifts the result back, yielding an $O(\log k)$-stretch independent of $n$. The construction achieves sublinear, input-sparsity time, near-linear parallel depth, and dynamic update capabilities, while also preserving submodular objectives via negative association. The authors demonstrate applications to online paging, metric multi-labeling, and swift submodular welfare reallocation, illustrating the broad impact of dimension-free correlated sampling for the hypersimplex and hinting at potential constant-stretch regimes under favorable conditions.

Abstract

Sampling from multiple distributions so as to maximize overlap has been studied by statisticians since the 1950s. Since the 2000s, such correlated sampling from the probability simplex has been a powerful building block in disparate areas of theoretical computer science. We study a generalization of this problem to sampling sets from given vectors in the hypersimplex, i.e., outputting sets of size (at most) some $k$ in $[n]$, while maximizing the sampled sets' overlap. Specifically, the expected difference between two output sets should be at most $α$ times their input vectors' $\ell_1$ distance. A value of $α=O(\log n)$ is known to be achievable, due to Chen et al.~(ICALP'17). We improve this factor to $O(\log k)$, independent of the ambient dimension~$n$. Our algorithm satisfies other desirable properties, including (up to a $\log^* n$ factor) input-sparsity sampling time, logarithmic parallel depth and dynamic update time, as well as preservation of submodular objectives. Anticipating broader use of correlated sampling algorithms for the hypersimplex, we present applications of our algorithm to online paging, offline approximation of metric multi-labeling and swift multi-scenario submodular welfare approximating reallocation.

Figures (5)

• Figure 1: Red nodes highlight the path $P$ from the root $r$ to the leaf representing coordinate $1$ in $T$. The blue node is not on $P$ and the input from this node to $z_3$ is the same for both inputs $\mathbf x$ and $\mathbf y$.
• Figure 2: The standard binary encoding of nodes where the root is labeled with $1$. The left child of node $i$ is labeled $2i$ and the right child is labeled $2i + 1$.
• Figure 3: Binary tree where node $i$ is associated with tuple $Encode(i)$.
• Figure 4: \ref{['alg:lca']} on binary strings $x = 1110101$ and $y = 1011101$ to find their LCA, $1101$.
• Figure 5: \ref{['alg:node-rounding']} is called at internal nodes (colored red) that have non-zero valued leaves (colored blue) in both subtrees when these internal nodes are popped. The order in which the internal nodes are popped in the above example is $1, 4, 3, 2, 6, 7, 5$.

Theorems & Definitions (77)

• Definition 1.0
• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 3.1
• proof
• proof
• Definition 3.3
• proof