Deterministic Spectral Sparsification in Almost-Linear Time for Dense Graphs

Abstract: A spectral sparsifier of a weighted graph is a reweighted subgraph whose Laplacian quadratic form approximates that of the original graph. Let $G$ be a positively weighted $n$-vertex, $m$-edge multigraph, let $0&lt;\varepsilon\le1/2$. Assuming $m,\varepsilon^{-1}\le n<sup>{O(1)}$ and the ratio of maximum to minimum weight is polynomially bounded, we deterministically construct a $(1\pm\varepsilon)$-spectral sparsifier with [ O!\left(n\varepsilon^{-2}\log^{24+o(1)}n\right) ] edges in [ m^{1+o(1)}+O!\left(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n\right) ] time. The construction has two main ingredients. First, we sparsify an approximately regular expander by partitioning its edges into few matchings and viewing their normalized Laplacians as an isotropic family of positive semidefinite matrices. Rather than sample from this family and apply matrix Chernoff, we select matchings deterministically using a pessimistic estimator. We evaluate the resulting conditional-expectation scores in two ways to produce two algorithms: using dense matrix multiplication and sparsely using polynomial approximations to the inverse square root and matrix exponential. Deterministic expander decomposition, along with replacing vertices by fixed expander graphs to achieve approximate regularity, extends these algorithms to general graphs. Second, a recursive blocking scheme applies the dense algorithm to smaller subgraphs and the sparse algorithm to their union, balancing their costs. Reusing the resulting algorithm as the dense algorithm gives $α<em>{r+1}=3-1/(α_r-1)$, starting from $α_0=ω$. After $O(\log n)$ levels, the exponent is $2+O(1/\log n)$, yielding $$m^{{1+o(1)}+\widetilde} O</em>{\varepsilon}(n<sup>2)$$ time.

• The paper develops the first deterministic algorithm that constructs a (1±ε)-spectral sparsifier with O(nε⁻² log²⁴⁺ᵒ⁽¹⁾n) edges in m¹⁺ᵒ⁽¹⁾ + O(n²ε⁻⁹ᐟ² log¹¹³ᐟ²⁺ᵒ⁽¹⁾n) time.
• The method derandomizes matrix Chernoff bounds through conditional expectation, inexact score evaluation, deterministic expander decomposition, and recursive compression to approach the natural n² time barrier for dense inputs.
• The result nearly matches randomized sparsification in both size and runtime while highlighting open challenges, including reducing substantial polylogarithmic overheads, improving ε-dependence, and lowering space requirements.

Overview

This paper gives the first deterministic, nearly-quadratic-time algorithm for constructing linear-sized spectral sparsifiers of dense graphs. Given a positively weighted $n$-vertex, $m$-edge multigraph with $m,\varepsilon^{-1}\le n^{O(1)}$ and polynomially bounded weight ratio, the authors deterministically construct a $(1\pm\varepsilon)$-spectral sparsifier with $O(n\varepsilon^{-2}\log^{24+o(1)}n)$ edges in time $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$ (2608.13910).

The deterministic setting has lagged substantially behind the randomized one. Randomized constructions achieve $O(n\varepsilon^{-2})$ edges in $\widetilde O(m)$ time (2608.13910), and recent discrepancy-minimization methods give the same sparsity in $\widetilde O(m\varepsilon^{-7/2})$ randomized time (2608.13910). Prior deterministic methods were either the slow BSS barrier argument or large-polynomial-time discrepancy walks (2608.13910). This paper closes most of that gap: the output is within polylogarithmic factors of optimal sparsity, and the runtime is within polylogarithmic factors of $n^2$ for dense graphs, where $m$0 is the natural barrier since the input itself may contain $m$1 edges.

Deterministic matrix Chernoff via conditional expectation

The technical core is a derandomization of the matrix Chernoff bound. Given positive semidefinite matrices $m$2 with $m$3 and $m$4, the procedure selects, with repetition, $m$5 indices such that $m$6, deterministically.

The proof applies the method of conditional expectation to the two-sided trace-exponential pessimistic estimator of Wigderson and Xiao. At each step, the candidate-dependent part of the one-step estimator bound is a linear score $m$7, whose average over all candidates equals the current estimator. Hence some candidate does not increase the estimator, and after $m$8 steps the estimator, which dominates the failure indicator, remains below its initial value of $m$9. Two features make this practical. First, the authors prove an inexact-score lemma: additive score errors of order $m,\varepsilon^{-1}\le n^{O(1)}$0 per candidate preserve the guarantee, so approximate matrix-function evaluation suffices. Second, because the matchings partition the edge set, all candidate scores are computed in a single edge scan once the score matrix is formed.

Sparsifying a regular expander

For a graph with conductance $m,\varepsilon^{-1}\le n^{O(1)}$1, non-loop weights at most $m,\varepsilon^{-1}\le n^{O(1)}$2, and minimum weighted degree $m,\varepsilon^{-1}\le n^{O(1)}$3, the edges are partitioned into $m,\varepsilon^{-1}\le n^{O(1)}$4 matchings via a deterministic $m,\varepsilon^{-1}\le n^{O(1)}$5-matching decomposition based on Euler-tour splitting. Conjugating each matching Laplacian by $m,\varepsilon^{-1}\le n^{O(1)}$6 and by $m,\varepsilon^{-1}\le n^{O(1)}$7, where $m,\varepsilon^{-1}\le n^{O(1)}$8 is orthogonal to $m,\varepsilon^{-1}\le n^{O(1)}$9, yields matrices $(1\pm\varepsilon)$0 with $(1\pm\varepsilon)$1 and $(1\pm\varepsilon)$2. Deterministic matrix Chernoff then selects $(1\pm\varepsilon)$3 matchings whose reweighted union is a $(1\pm\varepsilon)$4-sparsifier.

Two score-evaluation implementations are given. The dense implementation explicitly diagonalizes the normalized Laplacian using a deterministic approximate eigendecomposition (2608.13910), forms the inverse square root via an integral representation, and evaluates the two matrix exponentials by shifted diagonalization. Its cost is $(1\pm\varepsilon)$5 per iteration, independent of edge count. The sparse implementation avoids diagonalization: the inverse square root is approximated by a degree-$(1\pm\varepsilon)$6 Chebyshev-type polynomial (proved in an appendix via the integral representation of $(1\pm\varepsilon)$7 and truncated quadrature), and the exponentials by polynomials of degree $(1\pm\varepsilon)$8. Applying these polynomials costs $(1\pm\varepsilon)$9 per matrix–vector product, and the score matrix is materialized column by column in $O(n\varepsilon^{-2}\log^{24+o(1)}n)$0 time per expander.

From expanders to general graphs

Two reductions extend the expander routine to arbitrary graphs. Degree regularization splits each high-degree vertex into a fiber of constant-degree copies wired together by an auxiliary constant-conductance expander, pads degrees with self-loops, and preserves conductance up to constants; contracting the fibers maps sparsifiers of the regularized graph back to sparsifiers of the original, because the fiber structure satisfies $O(n\varepsilon^{-2}\log^{24+o(1)}n)$1. Deterministic expander decomposition (2608.13910) then partitions the graph into clusters of conductance $O(n\varepsilon^{-2}\log^{24+o(1)}n)$2 with at most half the edges crossing clusters. Combined with low-degree peeling (which discards fewer than $O(n\varepsilon^{-2}\log^{24+o(1)}n)$3 of the residual edges while guaranteeing the minimum-degree hypothesis), each round sparsifies at least $O(n\varepsilon^{-2}\log^{24+o(1)}n)$4 of the residual edges, so $O(n\varepsilon^{-2}\log^{24+o(1)}n)$5 rounds suffice, with geometrically decreasing residual sizes. Edge-disjoint composition turns the per-cluster guarantees into one global $O(n\varepsilon^{-2}\log^{24+o(1)}n)$6-approximation. Weighted graphs are handled by dyadic weight-class decomposition; edge-dependent costs sum over classes, and the weight spread of outputs grows by only $O(n\varepsilon^{-2}\log^{24+o(1)}n)$7 dyadic classes.

The resulting single-scale bounds are $O(n\varepsilon^{-2}\log^{24+o(1)}n)$8 edges, with dense time $O(n\varepsilon^{-2}\log^{24+o(1)}n)$9 and sparse time $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$0 (2608.13910). The dense implementation's weakness is the $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$1 term (with $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$2); the sparse implementation's weakness is the factor $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$3, which can be $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$4.

Recursive self-improvement to the quadratic barrier

The two implementations are combined recursively. At level $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$5, the vertex set is partitioned into blocks of size $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$6; each block-pair graph (at most $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$7 vertices) is sparsified by the level-$$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$8 algorithm, and the union of these sparsifiers—having only $$m^{1+o(1)} + O(n^2\varepsilon^{-9/2}\log^{113/2+o(1)}n)$$9 edges—is compressed by the sparse implementation. Balancing the costs $O(n\varepsilon^{-2})$0 and $O(n\varepsilon^{-2})$1 yields the recurrence $O(n\varepsilon^{-2})$2 from $O(n\varepsilon^{-2})$3. Writing $O(n\varepsilon^{-2})$4, the reciprocals satisfy $O(n\varepsilon^{-2})$5, giving the closed form

$O(n\varepsilon^{-2})$6

so after $O(n\varepsilon^{-2})$7 levels, $O(n\varepsilon^{-2})$8. The per-level accuracy $O(n\varepsilon^{-2})$9 ensures the accumulated multiplicative error stays within $\widetilde O(m)$0, and each level adds only $\widetilde O(m)$1 dyadic weight classes. This yields the main theorem: $\widetilde O(m)$2 edges in $\widetilde O(m)$3 deterministic time. The exponent gap shrinks as $\widetilde O(m)$4, and the paper notes the construction is compatible with the current best bound $\widetilde O(m)$5 (2608.13910).

Limitations and open questions

The paper's guarantees carry several qualifications stated by the authors. The running time and sparsity bounds hold under the assumptions that $\widetilde O(m)$6, $\widetilde O(m)$7, and the weight ratio $\widetilde O(m)$8 are polynomially bounded in $\widetilde O(m)$9; the dyadic weight-span of the output grows by $\widetilde O(m\varepsilon^{-7/2})$0 classes, so graphs with super-polynomial weight ratios fall outside the analysis. The polylogarithmic overheads are substantial—$\widetilde O(m\varepsilon^{-7/2})$1 in sparsity and $\widetilde O(m\varepsilon^{-7/2})$2 in time—driven largely by the conductance parameter $\widetilde O(m\varepsilon^{-7/2})$3 of the deterministic expander decomposition and by the $\widetilde O(m\varepsilon^{-7/2})$4 recursion levels; tightening these exponents is not addressed. The output sparsity remains a factor $\widetilde O(m\varepsilon^{-7/2})$5 from the optimal $\widetilde O(m\varepsilon^{-7/2})$6, and the paper does not determine whether deterministic near-linear-in-$\widetilde O(m\varepsilon^{-7/2})$7 time is achievable, which would require circumventing the $\widetilde O(m\varepsilon^{-7/2})$8 cost of the dense score evaluations on dense inputs. The sparse implementation also requires $\widetilde O(m\varepsilon^{-7/2})$9 space, which may be restrictive. Whether the conditional-expectation framework extends to other matrix-Chernoff-based primitives (e.g., spectral sketches or sparsifiers of submatrices) is left unexplored.

Conclusion

The paper establishes that linear-sized spectral sparsifiers can be constructed deterministically in $n^2$0 time, matching the randomized state of the art up to polylogarithmic factors in both sparsity and runtime. The combination of pessimistic-estimator derandomization of matrix Chernoff, two complementary score-evaluation regimes, and a recursive self-improvement that drives the vertex exponent to the quadratic barrier constitutes a clean and broadly applicable algorithmic template. The remaining polylogarithmic gaps—in output size, in runtime, and in the dependence on $n^2$1—define the natural next targets for this line of work.