arXiv subjects

Trevor Vaughn

Publications and source records attributed to Trevor Vaughn.

4 recordsLinked to original sources

arXiv2026

A Simple Las Vegas Algorithm for Sparse Nonnegative Convolution

Let $A, B \in \mathbb{Z}_{\ge 0}^n$ be nonnegative vectors and let $t = |\operatorname{supp}(A \star B)|$. We give a Las Vegas algorithm that computes $A \star B$ in $O(t \log t)$ expected time. More generally, for every $0 < δ\le \frac{1}{2}$, the algorithm terminates within $O(t \log t \log \frac{1}δ)$ time with probability at least $1 - δ$. The algorithm uses dense convolution, linear hashing, and the length reduction of \cite{BFN22}. Its main ingredient is a carry-free representation of the indices as vectors of constant dimension $d$ whose coordinates have size $O(t / \log t)$. We can then take our hash function to be the inner product with a random element of $\mathbb{F}_p^d$ for a prime $p$ of size $Ω(t / \log t)$: this preserves addition and gives collision probability exactly $1/p$, while identities regarding the moments of the vectors identify and recover the isolated terms as in \cite{BFN22}. Our expected running time matches that of Jin and Xu~\cite{JX24} while using substantially different tools and yielding a simpler algorithm. Note that their algorithm also terminates within $O(t \log t)$ time with probability at least $1 - \frac{1}{t}$, while our tail bound is weaker.

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arXiv2026

Faster Minimum k-Cut I: Simple and Sparse Weighted Graphs

The minimum $k$-cut problem asks for the fewest edges whose removal leaves an input graph with at least $k$ connected components. Previously, the best algorithm for simple graphs ran in $O_k(n^{(1-\varepsilon)k+O(1)})$ time~\cite{HL22}, showing that the $n^k$ barrier can be broken up to a polynomial overhead. We give the first $\widetilde O_k(n^{ck})$-time algorithm for Minimum $k$-Cut on simple graphs for an absolute constant $c<1$. More precisely, the running times are $\widetilde O(n^2)$ for $k=3$, $\widetilde O(n^{55/19})$ for $k=4$, and $\widetilde O(n^{4.112007})$ for $k=5$; for every $k\ge6$, the running time is \[ k^{O(k^2)}n^{1+(6k-6)\frac{k-1.749614}{7k-10}}(\log n)^{O(k^2)}, \] whose exponent is $\frac67k-0.132\ldots+O(1/k)$. The algorithm combines three ingredients. First, for weighted Minimum $k$-Cut we give a randomized \[ k^{O(k^2)}n^{k-2}(m+n)\log^3(n) \] -time algorithm: it perturbs the edge weights so that any minimum $k$-cut has a side with boundary strictly smaller than average. These then cut few edges of some tree in a logarithmic-size sample from a tree packing with high probability. After we enumerate them, we recursively compute $(k-1)$-cuts to complete them to the $k$-cuts of which they were a part. A variant of the perturbation and processing the entire packing support give a deterministic $k^{O(k^2)}n^{k+O(1)}$-time variant. Second, for cut size $s$, we give an improved FPT algorithm using a near-linear-time construction of an $(O(s\log^2 n\log\log n),s)$ edge-unbreakable tree decomposition with $O(s\log^2 n\log\log n)$ adhesion; this also gives a near-linear-time approximation algorithm for Minimum $k$-Cut. Third, we refine the border/island framework of~\cite{HL22}, using rectangular matrix multiplication to recover singleton islands and balancing it against the improved FPT algorithm.

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arXiv2026

Faster Minimum k-Cut II: Near-Optimal and Deterministic for Weighted Graphs

The Minimum $k$-Cut problem asks for a minimum-weight set of edges whose removal leaves an undirected weighted graph with at least $k$ connected components. We consider only $k \ge 3$. Under the Max-Weight Clique conjecture, weighted Minimum $k$-Cut requires $n^{k-1-o(1)}$ time for every fixed $k$. The fastest previous algorithm for weighted graphs ran in $n^{k-2}(m+n)(\log n)^{O(k^2)}$ randomized time~\cite{LV26}; for $k=3$, this gave an $\widetilde O(nm)$-time algorithm. We give randomized and deterministic algorithms matching the conditional lower bound in the exponent. On an $n$-vertex, $m$-edge weighted graph, our randomized algorithm runs with high probability in \begin{equation*} k^{O(k^2)}n^{k-1}\log^2n \end{equation*} time. Our deterministic algorithm runs in \begin{equation*} k^{O(k^2)}n^{k-1}\log^{O(1)}n \end{equation*} time. In particular, weighted Minimum $3$-Cut can be solved in $O(n^2 \log^2 n)$ randomized time and in $\widetilde O(n^2)$ deterministic time. The algorithms have two main components. First, we give a faster algorithm for weighted Minimum $3$-Cut. After handling optima with a very small side and optima with two light sides, the remaining optimum has a unique structured side. Tree packing reduces its completion to a batched collection of $2$-respecting cut problems. Second, we reduce Minimum $k$-Cut to Minimum $3$-Cut by enumerating a bounded family of light-cut candidates and recursively completing either side of each candidate. If the enumeration produces too many cuts, then we can instead produce an optimum $k$-cut directly. We derandomize the $3$-cut algorithm using a deterministic near-minimum-cut skeleton, and derandomize the reduction using a specialized $4$-cut algorithm using the skeleton, the constructive light-cut bounds, and the deterministic spectral sparsifier of \cite{BSS12}.

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arXiv2026

Deterministic Spectral Sparsification in Almost-Linear Time for Dense Graphs

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 $0<\varepsilon\le1/2$. Assuming $m,\varepsilon^{-1}\le n^{O(1)}$ 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 $α_{r+1}=3-1/(α_r-1)$, starting from $α_0=ω$. After $O(\log n)$ levels, the exponent is $2+O(1/\log n)$, yielding $m^{1+o(1)}+\widetilde O_{\varepsilon}(n^2)$ time.

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