arXiv subjects

Victor Reis

Publications and source records attributed to Victor Reis.

At least 19 recordsLinked to original sources

arXiv2026

Weighted Equitability and Matroid-Constrained Discrepancy

We prove weighted matroid equitability. Let $M=(E,\mathcal{I})$ be a matroid whose ground set can be partitioned into $k$ bases, and assign a nonnegative weight to every element. Then $E$ has a partition into $k$ bases such that the weights of any two bases differ by at most the largest element weight. We present two proofs based on the localized exchange theorem of Akrami, Liu, Raj, and V\'egh. The first is existential, while the second is constructive and leads to a strongly polynomial-time algorithm. As applications, we obtain an additive guarantee for matroid-constrained makespan minimization for identical machines and a strongly polynomial-time algorithm for finding an EF1 allocation under a matroid constraint and identical additive valuations. We further generalize the Beck--Fiala framework in discrepancy theory to settings with matroid constraints. Given a nonnegative matrix of column sparsity $\Delta$, we show that a fractional basis can be rounded to a basis of no larger cost while preserving every row sum within additive error $2\Delta$ times the largest matrix entry. Motivated by the $2$-sparse prefix Beck--Fiala conjecture, we formulate a conjecture on prefix-constrained matroid bases and prove a discrepancy bound $O(\log n)$. Finally, we give a counterexample to the weighted carpooling conjecture, thereby also disproving a conjecture by Morell and Skutella on single-source unsplittable flows with two-sided discrepancy bounds.

cs.DS↗

arXiv2026

Linear-size $\ell_1$ sparsifiers

We prove that for any matrix $A \in \mathbb{R}^{m \times n}$ and any $\varepsilon > 0$ there is a diagonal matrix $D \in \mathbb{R}_{\geq 0}^{m \times m}$ with at most $O(\frac{n}{\varepsilon^2})$ nonzero entries so that \[ (1-\varepsilon) \|Ax\|_1 \leq \|DAx\|_1 \leq (1+\varepsilon)\|Ax\|_1 \quad \forall x \in \mathbb{R}^n. \] In particular, for any zonotope $Z \subseteq \mathbb{R}^{n}$ there exists a zonotope $Z' \subseteq \mathbb{R}^{n}$ generated by at most $O(\frac{n}{\varepsilon^2})$ segments so that $(1-\varepsilon) Z \subseteq Z' \subseteq (1+\varepsilon) Z$. Previously, the best known bound was $O(\frac{n}{\varepsilon^2} \log n)$ due to Talagrand (1990).

math.MG↗

arXiv2026

Differentially Private Synthetic Data via APIs 4: Tabular Data

This paper investigates the problem of generating synthetic tabular data with differential privacy (DP) guarantees, enabling data sharing in sensitive domains. Despite extensive study, state-of-the-art methods often focus on minimizing low-order marginal query errors and overlook the challenges posed by high-order correlations. To address this gap, we extend the Private Evolution (PE) framework, originally developed for DP-compliant image and text synthesis, to tabular data. We introduce Tab-PE -- an algorithm for synthetic tabular data generation under DP constraints. Tab-PE iteratively improves a candidate dataset via an evolutionary process that leverages tabular-specialized operators to produce variations, privately scores them, and selects the highest-quality samples to retain and propagate. In contrast to the original PE, which relies on large foundation models, Tab-PE employs heuristic operators with significantly lower computational costs, making PE more practical and scalable for tabular data. Through extensive experiments on real-world and simulation datasets, we demonstrate that Tab-PE substantially outperforms prior baselines on datasets exhibiting high-order correlations. Compared to the best baseline -- AIM, Tab-PE improves classification accuracy by up to 10% while running 28 times faster.

cs.LG↗

arXiv2026

Optimal Vector Balancing for Zonotopes

A zonotope is a linear image of the cube $[-1,1]^m$ for some $m \in \mathbb{N}$. We show that there is a universal constant $C$ such that, for every zonotope $Z\subset \mathbb{R}^d$ and vectors $v_1,\dots,v_n\in Z$, there are signs $x_1,\dots,x_n\in\{-1,1\}$ with \[ \sum_{i=1}^n x_i v_i \in C\sqrt d\, Z. \] This resolves a 2002 question of Schechtman and generalizes Spencer's six standard deviations theorem, which corresponds to the case $Z=[-1,1]^d$.

math.MG↗

arXiv2025

Weighted Chairman Assignment and Flow-Time Scheduling

Given positive integers $m, n$, a fractional assignment $x \in [0,1]^{m \times n}$ and weights $d \in \mathbb{R}^n_{>0}$, we show that there exists an assignment $y \in \{0,1\}^{m \times n}$ so that for every $i\in[m]$ and $t\in [n]$, \[ \Big|\sum_{j \in [t]} d_j (x_{ij} - y_{ij}) \Big| < \max_{j \in [n]} d_j. \] This generalizes a result of Tijdeman (1973) on the unweighted version, known as the chairman assignment problem. This also confirms a special case of the single-source unsplittable flow conjecture with arc-wise lower and upper bounds due to Morell and Skutella (IPCO 2020). As an application, we consider a scheduling problem where jobs have release times and machines have closing times, and a job can only be scheduled on a machine if it is released before the machine closes. We give a $3$-approximation algorithm for maximum flow-time minimization.

cs.DS↗

arXiv2025

Struct-Bench: A Benchmark for Differentially Private Structured Text Generation

Differentially private (DP) synthetic data generation is a promising technique for utilizing private datasets that otherwise cannot be exposed for model training or other analytics. While much research literature has focused on generating private unstructured text and image data, in enterprise settings, structured data (e.g., tabular) is more common, often including natural language fields or components. Existing synthetic data evaluation techniques (e.g., FID) struggle to capture the structural properties and correlations of such datasets. In this work, we propose Struct-Bench, a framework and benchmark for evaluating synthetic datasets derived from structured datasets that contain natural language data. The Struct-Bench framework requires users to provide a representation of their dataset structure as a Context-Free Grammar (CFG). Our benchmark comprises 5 real-world and 2 synthetically generated datasets, each annotated with CFGs. We show that these datasets demonstrably present a great challenge even for state-of-the-art DP synthetic data generation methods. Struct-Bench also includes reference implementations of different metrics and a leaderboard, thereby providing researchers a standardized evaluation platform to benchmark and investigate privacy-preserving synthetic data generation methods. Further, we also present a case study showing how to use Struct-Bench to improve the synthetic data quality of Private Evolution (PE) on structured data. The benchmark and the leaderboard have been publicly made available at https://struct-bench.github.io.

cs.CL↗

arXiv2025

A Randomized Rounding Approach for DAG Edge Deletion

In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter $k$, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length $k$. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a $k$-approximation and showed that it is UGC-Hard to approximate better than $\lfloor 0.5k \rfloor$ for any constant $k \ge 4$ using a work of Svensson from 2012. The approximation ratio was improved to $\frac{2}{3}(k+1)$ by Klein and Wexler in 2016. In this work, we introduce a randomized rounding framework based on distributions over vertex labels in $[0,1]$. The most natural distribution is to sample labels independently from the uniform distribution over $[0,1]$. We show this leads to a $(2-\sqrt{2})(k+1) \approx 0.585(k+1)$-approximation. By using a modified (but still independent) label distribution, we obtain a $0.549(k+1)$-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below $0.542(k+1)$. Finally, we show a $0.5(k+1)$-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.

cs.DS↗

arXiv2023

Optimal Online Discrepancy Minimization

We prove that there exists an online algorithm that for any sequence of vectors $v_1,\ldots,v_T \in \mathbb{R}^n$ with $\|v_i\|_2 \leq 1$, arriving one at a time, decides random signs $x_1,\ldots,x_T \in \{ -1,1\}$ so that for every $t \le T$, the prefix sum $\sum_{i=1}^t x_iv_i$ is $10$-subgaussian. This improves over the work of Alweiss, Liu and Sawhney who kept prefix sums $O(\sqrt{\log (nT)})$-subgaussian, and gives a $O(\sqrt{\log T})$ bound on the discrepancy $\max_{t \in T} \|\sum_{i=1}^t x_i v_i\|_\infty$. Our proof combines a generalization of Banaszczyk's prefix balancing result to trees with a cloning argument to find distributions rather than single colorings. We also show a matching $\Omega(\sqrt{\log T})$ strategy for an oblivious adversary.

cs.DS↗

arXiv2023

Linear-Sized Sparsifiers via Near-Linear Time Discrepancy Theory

Discrepancy theory provides powerful tools for producing higher-quality objects which "beat the union bound" in fundamental settings throughout combinatorics and computer science. However, this quality has often come at the price of more expensive algorithms. We introduce a new framework for bridging this gap, by allowing for the efficient implementation of discrepancy-theoretic primitives. Our framework repeatedly solves regularized optimization problems to low accuracy to approximate the partial coloring method of [Rot17], and simplifies and generalizes recent work of [JSS23] on fast algorithms for Spencer's theorem. In particular, our framework only requires that the discrepancy body of interest has exponentially large Gaussian measure and is expressible as a sublevel set of a symmetric, convex function. We combine this framework with new tools for proving Gaussian measure lower bounds to give improved algorithms for a variety of sparsification and coloring problems. As a first application, we use our framework to obtain an $\widetilde{O}(m \cdot \epsilon^{-3.5})$ time algorithm for constructing an $\epsilon$-approximate spectral sparsifier of an $m$-edge graph, matching the sparsity of [BSS14] up to constant factors and improving upon the $\widetilde{O}(m \cdot \epsilon^{-6.5})$ runtime of [LeeS17]. We further give a state-of-the-art algorithm for constructing graph ultrasparsifiers and an almost-linear time algorithm for constructing linear-sized degree-preserving sparsifiers via discrepancy theory; in the latter case, such sparsifiers were not known to exist previously. We generalize these results to their analogs in sparsifying isotropic sums of positive semidefinite matrices. Finally, to demonstrate the versatility of our technique, we obtain a nearly-input-sparsity time constructive algorithm for Spencer's theorem (where we recover a recent result of [JSS23]).

cs.DS↗

arXiv2023

The Subspace Flatness Conjecture and Faster Integer Programming

In a seminal paper, Kannan and Lov\'asz (1988) considered a quantity $\mu_{KL}(\Lambda,K)$ which denotes the best volume-based lower bound on the covering radius $\mu(\Lambda,K)$ of a convex body $K$ with respect to a lattice $\Lambda$. Kannan and Lov\'asz proved that $\mu(\Lambda,K) \leq n \cdot \mu_{KL}(\Lambda,K)$ and the Subspace Flatness Conjecture by Dadush (2012) claims a $O(\log(2n))$ factor suffices, which would match the lower bound from the work of Kannan and Lov\'asz. We settle this conjecture up to a constant in the exponent by proving that $\mu(\Lambda,K) \leq O(\log^{3}(2n)) \cdot \mu_{KL} (\Lambda,K)$. Our proof is based on the Reverse Minkowski Theorem due to Regev and Stephens-Davidowitz (2017). Following the work of Dadush (2012, 2019), we obtain a $(\log(2n))^{O(n)}$-time randomized algorithm to solve integer programs in $n$ variables. Another implication of our main result is a near-optimal flatness constant of $O(n \log^{2}(2n))$, improving on the previous bound of $O(n^{4/3} \log^{O(1)} (2n))$.

math.OC↗

arXiv2022

The Vector Balancing Constant for Zonotopes

The vector balancing constant $\mathrm{vb}(K,Q)$ of two symmetric convex bodies $K,Q$ is the minimum $r \geq 0$ so that any number of vectors from $K$ can be balanced into an $r$-scaling of $Q$. A question raised by Schechtman is whether for any zonotope $K \subseteq \mathbb{R}^d$ one has $\mathrm{vb}(K,K) \lesssim \sqrt{d}$. Intuitively, this asks whether a natural geometric generalization of Spencer's Theorem (for which $K = B^d_\infty$) holds. We prove that for any zonotope $K \subseteq \mathbb{R}^d$ one has $\mathrm{vb}(K,K) \lesssim \sqrt{d} \log \log \log d$. Our main technical contribution is a tight lower bound on the Gaussian measure of any section of a normalized zonotope, generalizing Vaaler's Theorem for cubes. We also prove that for two different normalized zonotopes $K$ and $Q$ one has $\mathrm{vb}(K,Q) \lesssim \sqrt{d \log d}$. All the bounds are constructive and the corresponding colorings can be computed in polynomial time.

math.MG↗

arXiv2022

Approximate Carath\'eodory bounds via Discrepancy Theory

The approximate Carath\'eodory problem in general form is as follows: Given two symmetric convex bodies $P,Q \subseteq \mathbb{R}^m$, a parameter $k \in \mathbb{N}$ and $\mathbf{z} \in \textrm{conv}(X)$ with $X \subseteq P$, find $\mathbf{v}_1,\ldots,\mathbf{v}_k \in X$ so that $\|\mathbf{z} - \frac{1}{k}\sum_{i=1}^k \mathbf{v}_i\|_Q$ is minimized. Maurey showed that if both $P$ and $Q$ coincide with the $\| \cdot \|_p$-ball, then an error of $O(\sqrt{p/k})$ is possible. We prove a reduction to the vector balancing constant from discrepancy theory which for most cases can provide tight bounds for general $P$ and $Q$. For the case where $P$ and $Q$ are both $\| \cdot \|_p$-balls we prove an upper bound of $\sqrt{ \frac{\min\{ p, \log (\frac{2m}{k}) \}}{k}}$. Interestingly, this bound cannot be obtained taking independent random samples; instead we use the Lovett-Meka random walk. We also prove an extension to the more general case where $P$ and $Q$ are $\|\cdot \|_p$ and $\| \cdot \|_q$-balls with $2 \leq p \leq q \leq \infty$.

math.MG↗

arXiv2021

A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent

Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices $A_1, \dots, A_n \in \mathbb{R}^{m \times m}$, a Gaussian measure lower bound of $2^{-O(n)}$ for a scaling of the discrepancy body $\{x \in \mathbb{R}^n: \| \sum_{i=1}^n x_i A_i\| \leq 1\}$. We show this is equivalent to covering its polar with $2^{O(n)}$ translates of the cube $\frac{1}{n} B^n_\infty$, and construct such a cover via mirror descent. As applications of our framework, we show: $\bullet$ Matrix Spencer for Low-Rank Matrices. If the matrices satisfy $\|A_i\|_{\mathrm{op}} \leq 1$ and $\mathrm{rank}(A_i) \leq r$, we can efficiently find a coloring $x \in \{\pm 1\}^n$ with discrepancy $\|\sum_{i=1}^n x_i A_i \|_{\mathrm{op}} \lesssim \sqrt{n \log (\min(rm/n, r))}$. This improves upon the naive $O(\sqrt{n \log r})$ bound for random coloring and proves the matrix Spencer conjecture when $r m \leq n$. $\bullet$ Matrix Spencer for Block Diagonal Matrices. For block diagonal matrices with $\|A_i\|_{\mathrm{op}} \leq 1$ and block size $h$, we can efficiently find a coloring $x \in \{\pm 1\}^n$ with $\|\sum_{i=1}^n x_i A_i \|_{\mathrm{op}} \lesssim \sqrt{n \log (hm/n)}$. Using our proof, we reduce the matrix Spencer conjecture to the existence of a $O(\log(m/n))$ quantum relative entropy net on the spectraplex. $\bullet$ Matrix Discrepancy for Schatten Norms. We generalize our discrepancy bound for matrix Spencer to Schatten norms $2 \le p \leq q$. Given $\|A_i\|_{S_p} \leq 1$ and $\mathrm{rank}(A_i) \leq r$, we can efficiently find a partial coloring $x \in [-1,1]^n$ with $|\{i : |x_i| = 1\}| \ge n/2$ and $\|\sum_{i=1}^n x_i A_i\|_{S_q} \lesssim \sqrt{n \min(p, \log(rk))} \cdot k^{1/p-1/q}$, where $k := \min(1,m/n)$.

cs.DS↗

arXiv2021

A Tighter Relation Between Hereditary Discrepancy and Determinant Lower Bound

In seminal work, Lov\'asz, Spencer, and Vesztergombi [European J. Combin., 1986] proved a lower bound for the hereditary discrepancy of a matrix $A \in \mathbb{R}^{m \times n}$ in terms of the maximum $|\det(B)|^{1/k}$ over all $k \times k$ submatrices $B$ of $A$. We show algorithmically that this determinant lower bound can be off by at most a factor of $O(\sqrt{\log (m) \cdot \log (n)})$, improving over the previous bound of $O(\log(mn) \cdot \sqrt{\log (n)})$ given by Matou\v{s}ek [Proc. of the AMS, 2013]. Our result immediately implies $\mathrm{herdisc}(\mathcal{F}_1 \cup \mathcal{F}_2) \leq O(\sqrt{\log (m) \cdot \log (n)}) \cdot \max(\mathrm{herdisc}(\mathcal{F}_1), \mathrm{herdisc}(\mathcal{F}_2))$, for any two set systems $\mathcal{F}_1, \mathcal{F}_2$ over $[n]$ satisfying $|\mathcal{F}_1 \cup \mathcal{F}_2| = m$. Our bounds are tight up to constants when $m = O(\mathrm{poly}(n))$ due to a construction of P\'alv\"olgyi [Discrete Comput. Geom., 2010] or the counterexample to Beck's three permutation conjecture by Newman, Neiman and Nikolov [FOCS, 2012].

cs.DS↗

arXiv2021

Tight bounds on the Fourier growth of bounded functions on the hypercube

We give tight bounds on the degree $\ell$ homogenous parts $f_\ell$ of a bounded function $f$ on the cube. We show that if $f: \{\pm 1\}^n \rightarrow [-1,1]$ has degree $d$, then $\| f_\ell \|_\infty$ is bounded by $d^\ell/\ell!$, and $\| \hat{f}_\ell \|_1$ is bounded by $d^\ell e^{\binom{\ell+1}{2}} n^{\frac{\ell-1}{2}}$. We describe applications to pseudorandomness and learning theory. We use similar methods to generalize the classical Pisier's inequality from convex analysis. Our analysis involves properties of real-rooted polynomials that may be useful elsewhere.

cs.CC↗

arXiv2020

An Elementary Exposition of Pisier's Inequality

Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make the inequality and its proof more accessible, because we think they will find additional applications. We demonstrate this with a new type of restriction on the Fourier spectrum of bounded functions on the discrete cube.

math.FA↗

arXiv2020

Balancing Polynomials in the Chebyshev Norm

Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $\|p_i\|_\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in \{-1,1\}$ so that \[\Big\|\sum_{i=1}^n x_i p_i\Big\|_\infty < 30\sqrt{n}, \] where $\|p\|_\infty := \sup_{|x| \le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.

math.CA↗

arXiv2020

Vector Balancing in Lebesgue Spaces

A tantalizing conjecture in discrete mathematics is the one of Koml\'os, suggesting that for any vectors $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_2^m$ there exist signs $x_1, \dots, x_n \in \{ -1,1\}$ so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(1)$. It is a natural extension to ask what $\ell_q$-norm bound to expect for $\mathbf{a}_1,\ldots,\mathbf{a}_n \in B_p^m$. We prove that, for $2 \le p \le q \le \infty$, such vectors admit fractional colorings $x_1, \dots, x_n \in [-1,1]$ with a linear number of $\pm 1$ coordinates so that $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_q \leq O(\sqrt{\min(p,\log(2m/n))}) \cdot n^{1/2-1/p+ 1/q}$, and that one can obtain a full coloring at the expense of another factor of $\frac{1}{1/2 - 1/p + 1/q}$. In particular, for $p \in (2,3]$ we can indeed find signs $\mathbf{x} \in \{ -1,1\}^n$ with $\|\sum_{i=1}^n x_i\mathbf{a}_i\|_\infty \le O(n^{1/2-1/p} \cdot \frac{1}{p-2})$. Our result generalizes Spencer's theorem, for which $p = q = \infty$, and is tight for $m = n$. Additionally, we prove that for any fixed constant $\delta>0$, in a centrally symmetric body $K \subseteq \mathbb{R}^n$ with measure at least $e^{-\delta n}$ one can find such a fractional coloring in polynomial time. Previously this was known only for a small enough constant -- indeed in this regime classical nonconstructive arguments do not apply and partial colorings of the form $\mathbf{x} \in \{ -1,0,1\}^n$ do not necessarily exist.

cs.DS↗