Hypercube Cut Sparsification Bounds

• Hypercube cut sparsification is defined as constructing a subgraph that preserves every cut within a (1 ± ε) factor while reducing the edge count.
• Upper bounds are achieved via leverage score sampling in the Boolean hypercube, yielding O(n log n/ε²) edges, which is matched by tight lower bounds.
• Recent advancements include applying streaming and linear sketching methods, demonstrating that even highly symmetric graphs require robust sparsification techniques.

A hypercube cut sparsification bound characterizes the minimum number of edges required in a reweighted subgraph (sparsifier) of the $d$-dimensional Boolean hypercube $Q_d$ so that all cuts in the graph are preserved up to a multiplicative $(1 \pm \epsilon)$ factor. Cut sparsification for hypercubes is of both theoretical and algorithmic importance, as it provides canonical lower and upper bounds for edge sparsification in highly structured and high-dimensional regular graphs. This article covers the tight bounds, structural properties, and principal techniques relevant to the hypercube cut sparsification problem, including both static and streaming/sketching frameworks.

1. Formal Definition and Problem Statement

Given an undirected, weighted graph $G = (V, E, w)$ and $\epsilon \in (0,1)$, a $(1 \pm \epsilon)$-cut sparsifier is a subgraph $H = (V, E', w')$ (with $E' \subseteq E$ and possibly reweighted edges) such that for every $S \subset V$,

$$\sum_{(u,v)\in E(H)} w'_H(u,v) |1_S(u) - 1_S(v)| = (1 \pm \epsilon) \sum_{(u,v)\in E} w(u,v)|1_S(u) - 1_S(v)|.$$

For the $d$-dimensional hypercube $Q_d$ on $n=2^d$ vertices, every vertex has degree $d$ and the total number of edges is $m = nd/2$.

The objective is to determine, as a function of $n$ and $\epsilon$, the minimum possible size (number of edges) of any such cut sparsifier for $Q_d$, and to identify tight upper and lower bounds.

2. Upper Bounds: Existence and Construction

For general graphs, the Batson–Spielman–Srivastava (BSS) framework guarantees that every $n$-vertex graph admits a $(1 \pm \epsilon)$-cut sparsifier with $O(n/\epsilon^2)$ edges via leverage score sampling, i.e., by sampling each edge in proportion to its effective resistance and reweighting appropriately.

Specializing to $Q_d$, the sparsification bound is: $|E(H)| = O\left( \frac{n}{\epsilon^2} \right)$ for some absolute constant $C$ independent of $d$ or $n$, achievable by the BSS technique or any equivalent spectral sparsification approach where the Laplacian quadratic form is preserved for all indicator vectors of cuts (Filtser et al., 2015). Effective resistances in $Q_d$ are uniformly bounded due to its regularity and symmetry, and every edge has large cut-strength $\lambda_e = d$, enabling efficient leverage-score estimation and sampling rates.

A more refined sampling-based analysis for $Q_d$ leverages the following edge sampling probability for each edge $e$: $p_e = C \cdot \frac{\log n}{\epsilon^2 d}.$ Given $m = d2^{d-1}$, the expected size of the sparsifier is

$$|E'| = m \cdot p_e = (d2^{d-1}) \cdot O \left( \frac{\log n}{\epsilon^2 d} \right) = O\left( \frac{n \log n}{\epsilon^2} \right).$$

In pseudopolynomial or $\tilde{O}$-notation (hiding polylogarithmic factors), the bound is $\tilde{O}(n/\epsilon^2)$ (Khanna et al., 2024).

3. Lower Bounds and Tightness

Matching lower bounds for sparsification of $Q_d$ demonstrate that these upper bounds cannot be improved even for the cube's high symmetry:

• For all $1/\sqrt{n} \ll \epsilon \leq 1/2$, there exist $n$-vertex graphs (in particular, $Q_d$ as a special case) for which any subgraph $H$ that preserves all cuts within $(1 \pm \epsilon)$ must have

$|E(H)| = \Omega\left( \frac{n}{\epsilon^2} \right)$

(Filtser et al., 2015). The proof uses a disjoint union of bipartite block gadgets embedded in $Q_d$ and an information-theoretic communication argument: any sparsifier with fewer edges will fail to preserve some cut, violating the specified guarantee.

• For $Q_d$, the structure alone does not yield improved constants or dependencies beyond the general bound; embedding regular bipartite structures within the cube shows that the lower bound is tight up to constant factors and polylogarithmic corrections (Filtser et al., 2015, Khanna et al., 2024).

4. Linear Sketching, Streaming, and Algorithmic Bounds

Recent work extends sparsification to streaming and dynamic models, where the sparsifier must be recoverable from a linear sketch of the incidence matrix. For general $n$-vertex, $m$-edge, $r$-uniform hypergraphs, a sketch of size

$\tilde{O}(n r \log m / \epsilon^2)$

bits suffices to recover a $(1 \pm \epsilon)$-sparsifier of size $\tilde{O}(n/\epsilon^2)$. For $Q_d$ (with $r=2, n=2^d, m = d2^{d-1}$): $\tilde{O}(n \log n / \epsilon^2)$ bits are sufficient to store a sketch and reconstruct a sparsifier with $\tilde{O}(n/\epsilon^2)$ edges. Lower bounds show that $\Omega(n \log m / \log n)$ bits are necessary, yielding tightness up to $\log n$ factors (Khanna et al., 2024).

A summary table:

Setting	Upper Bound (edges/bits)	Lower Bound
Static (edges)	$O(n/\epsilon^2)$	$\Omega(n/\epsilon^2)$
$Q_d$ static (edges)	$O(n \log n/\epsilon^2)$ ($\tilde{O}(n/\epsilon^2)$)	$\Omega(n/\epsilon^2)$
Sketching/Streaming (bits)	$\tilde{O}(n \log n/\epsilon^2)$	$\Omega(n \log n/\log n)$

5. Proof Methods and Structural Insights

The principal proof technique for upper bounds involves leverage score (effective resistance) or $k$-cut-strength based sampling, capitalizing on the Laplacian structure of $Q_d$. Key steps include:

• Viewing the hypercube as an electrical network and sampling edges with rates inversely proportional to their edge strengths or effective resistances.
• Utilizing Chernoff and union bounds over $O(n^2)$ cuts (i.e., over all possible cuts $S \subset V$), guaranteeing with high probability that every cut is preserved multiplicatively.
• The resulting sampled subgraph achieves approximation via its Laplacian, ensuring all cuts are preserved to the desired accuracy.

Lower bounds use construction of hard instances via embedding block gadgets or invoking information/communication-theoretic arguments that exploit the necessity of retaining sufficient information about the cut structure. For streaming and sketching, lower bounds derive from communication complexity reductions.

6. Specialization to the Boolean Hypercube

A key feature of the hypercube case is that, despite the cube's high regularity, no improved dependency on $\epsilon$ or $d$ (beyond logarithmic factors) is available compared to general graphs of the same vertex size. Specifically, for $Q_d$:

• Each edge is present in $d$ edge-disjoint cuts of size $d$ ($k$-cut-strength $\lambda_e = d$), so uniform treatment via general sparsification machinery is valid (Khanna et al., 2024).
• The minimum cut in $Q_d$ is of size $d$, so any $(1 \pm \epsilon)$-sparsifier must contain $\Omega(n)$ edges.
• The explicit formula for the optimal sparsifier size, up to polylogarithmic factors, is

$$|E'| = \Theta \left( \frac{n \log n}{\epsilon^2} \right ).$$

The $\log n$ factor arises from union-bounding over all $O(2^n)$ cuts.

No dependence on the cubic structure (such as symmetries or dimension) leads to sharper constants; the general lower bound for graph sparsification applies directly and tightly to $Q_d$.

7. Connections and Consequences

The hypercube cut sparsification bound underpins several broader results in graph algorithms and complexity:

• The tightness and uniform applicability of the $O(n/\epsilon^2)$ law for regular expanders and highly symmetric families.
• Implications for valued CSPs, such as 2LIN and 2SAT, where predicates reduce to cut-like structures, with sparsifiability governed by the same bounds (Filtser et al., 2015).
• Streaming algorithms and linear sketching for dynamic graphs rely on the bit-complexity bounds for cut sparsifiers, pushing the frontier of efficient graph processing (Khanna et al., 2024).
• The apparent resistance of the bound to improvement (i.e., the lack of logarithmic factor savings for $Q_d$) illustrates structural universality: combinatorial expansion and uniformity do not, by themselves, yield easier sparsification.

This suggests that, within the family of highly regular and high-dimensional graphs, cut sparsification complexity remains dominated by the number of vertices and the required precision parameter $\epsilon$, rather than the specific internal symmetry or dimensionality of the graph.