arXiv subjects

Thomas Rothvoss

Publications and source records attributed to Thomas Rothvoss.

At least 19 recordsLinked to original sources

arXiv2026

A note on the $\Sigma_2^P$-completeness of the Frobenius number

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $\Sigma_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

cs.CC↗

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

Nearly-Tight Bounds for Zonotope Containment and Beyond

We investigate the convex-body containment problem $\max\{s >0 : s Z \subseteq Q\}$, where the outer body $Q \subseteq \mathbb R^d$ is described by a membership oracle and the inner body $Z \subseteq \mathbb R^d$ is a zonotope. Our main result is a sampling-based $O(\sqrt{d})$-approximation algorithm for this problem that almost matches the lower bound of $\Omega(\sqrt{d/\log d})$ by Khot and Naor in the oracle model. Assuming zonotopes can be sparsified by a linear number of generators, which is referred to as Talagrand conjecture, our approach attains the optimal approximation factor of $\Theta(\sqrt{d/\log d})$. Our second main result is a proof of Talagrand's conjecture for $\Delta$-modular zonotopes whenever $\Delta$ is constant. Those zonotopes are of the form $Z = \{ Wx \colon \| x\|_\infty \leq 1\}$ where the non-zero $d \times d$ sub-determinants of $W$ are between $1$ and $\Delta$. This result establishes a connection between zonoid sparsification and spectral sparsification of Batson, Spielman and Srivastava. We complement these results with a universal $\Omega(\sqrt{d/\log d})$ lower bound holding for all zonotopes. Finally, we consider containment problems $\max\{s >0 : s K \subseteq Q\}$, for general convex bodies $K \subseteq \mathbb R^d$. A result of Nasz\'odi on approximating $K \subseteq \mathbb R^d$ by a polytope implies a $\Theta(d/\log d)$ approximation algorithm in polynomial time. We show the tightness of this approximation factor in the oracle model via a reduction to the circumradius computation. Our lower bound holds for centrally symmetric convex sets, implying that Barvinok's optimal $O(\sqrt{d})$-approximation of a centrally symmetric convex body by a polytope with a polynomial number of vertices cannot be computed in polynomial time.

cs.DS↗

arXiv2025

Excluding a Line Minor via Design Matrices and Column Number Bounds for the Circuit Imbalance Measure

For a real matrix $A \in \mathbb{R}^{d \times n}$ with non-collinear columns, we show that $n \leq O(d^4 \kappa_A)$ where $\kappa_A$ is the \emph{circuit imbalance measure} of $A$. The circuit imbalance measure $\kappa$ is a real analogue of $\Delta$-modularity for integer matrices, satisfying $\kappa_A \leq \Delta_A$ for integer $A$. The circuit imbalance measure has numerous applications in the context of linear programming (see Ekbatani, Natura and V{\'e}gh (2022) for a survey). Our result generalizes the $O(d^4 \Delta_A)$ bound of Averkov and Schymura (2023) for integer matrices and provides the first polynomial bound holding for all parameter ranges on real matrices. To derive our result, similar to the strategy of Geelen, Nelson and Walsh (2021) for $\Delta$-modular matrices, we show that real representable matroids induced by $\kappa$-bounded matrices are minor closed and exclude a rank $2$ uniform matroid on $O(\kappa)$ elements as a minor (also known as a line of length $O(\kappa)$). As our main technical contribution, we show that any simple rank $d$ complex representable matroid which excludes a line of length $l$ has at most $O(d^4 l)$ elements. This complements the tight bound of $(l-3)\binom{d}{2} + d$ for $l \geq 4$, of Geelen, Nelson and Walsh which holds when the rank $d$ is sufficiently large compared to $l$ (at least doubly exponential in $l$).

cs.DM↗

arXiv2025

DiscQuant: A Quantization Method for Neural Networks Inspired by Discrepancy Theory

Quantizing the weights of a neural network has two steps: (1) Finding a good low bit-complexity representation for weights (which we call the quantization grid) and (2) Rounding the original weights to values in the quantization grid. In this paper, we study the problem of rounding optimally given any quantization grid. The simplest and most commonly used way to round is Round-to-Nearest (RTN). By rounding in a data-dependent way instead, one can improve the quality of the quantized model significantly. We study the rounding problem from the lens of \emph{discrepancy theory}, which studies how well we can round a continuous solution to a discrete solution without affecting solution quality too much. We prove that given $m=\mathrm{poly}(1/\epsilon)$ samples from the data distribution, we can round all but $O(m)$ model weights such that the expected approximation error of the quantized model on the true data distribution is $\le \epsilon$ as long as the space of gradients of the original model is approximately low rank (which we empirically validate). Our proof, which is algorithmic, inspired a simple and practical rounding algorithm called \emph{DiscQuant}. In our experiments, we demonstrate that DiscQuant significantly improves over the prior state-of-the-art rounding method called GPTQ and the baseline RTN over a range of benchmarks on Phi3mini-3.8B and Llama3.1-8B. For example, rounding Phi3mini-3.8B to a fixed quantization grid with 3.25 bits per parameter using DiscQuant gets 64\% accuracy on the GSM8k dataset, whereas GPTQ achieves 54\% and RTN achieves 31\% (the original model achieves 84\%). We make our code available at https://github.com/jerry-chee/DiscQuant.

cs.LG↗

arXiv2025

A parameterized linear formulation of the integer hull

Let $A \in \mathbb{Z}^{m \times n}$ be an integer matrix with components bounded by $\Delta$ in absolute value. Cook et al.~(1986) have shown that there exists a universal matrix $B \in \mathbb{Z}^{m' \times n}$ with the following property: For each $b \in \mathbb{Z}^m$, there exists $t \in \mathbb{Z}^{m'}$ such that the integer hull of the polyhedron $P = \{ x \in \mathbb{R}^n \colon Ax \leq b\}$ is described by $P_I = \{ x \in \mathbb{R}^n \colon Bx \leq t\}$. Our \emph{main result} is that $t$ is an \emph{affine} function of $b$ as long as $b$ is from a fixed equivalence class of the lattice $D \cdot \mathbb{Z}^m$. Here $D \in \mathbb{N}$ is a number that depends on $n$ and $\Delta$ only. Furthermore, $D$ as well as the matrix $B$ can be computed in time depending on $\Delta$ and $n$ only. An application of this result is the solution of an open problem posed by Cslovjecsek et al.~(SODA 2024) concerning the complexity of \emph{2-stage-stochastic integer programming} problems. The main tool of our proof is the classical theory of \emph{Chv\'atal-Gomory cutting planes} and the \emph{elementary closure} of rational polyhedra.

cs.CC↗

arXiv2024

A Geometric Perspective on the Injective Norm of Sums of Random Tensors

Matrix concentration inequalities, intimately connected to the Non-Commutative Khintchine inequality, have been an important tool in both applied and pure mathematics. We study tensor versions of these inequalities, and establish non-asymptotic inequalities for the $\ell_p$ injective norm of random tensors with correlated entries. In certain regimes of $p$ and the tensor order, our tensor concentration inequalities are nearly optimal in their dimension dependencies. We illustrate our result with applications to problems including structured models of random tensors and matrices, tensor PCA, and connections to lower bounds in coding theory. Our techniques are based on covering number estimates as opposed to operator theoretic tools, which also provide a geometric proof of a weaker version of the Non-Commutative Khintchine inequality, motivated by a question of Talagrand.

math.PR↗

arXiv2023

Forall-exist statements in pseudopolynomial time

Given a convex set $Q \subseteq R^m$ and an integer matrix $W \in Z^{m \times n}$, we consider statements of the form $ \forall b \in Q \cap Z^m$ $\exists x \in Z^n$ s.t. $Wx \leq b$. Such statements can be verified in polynomial time with the algorithm of Kannan and its improvements if $n$ is fixed and $Q$ is a polyhedron. The running time of the best-known algorithms is doubly exponential in~$n$. In this paper, we provide a pseudopolynomial-time algorithm if $m$ is fixed. Its running time is $(m \Delta)^{O(m^2)}$, where $\Delta = \|W\|_\infty$. Furthermore it applies to general convex sets $Q$.

math.OC↗

arXiv2023

Stronger Coreset Bounds for Kernel Density Estimators via Chaining

We apply the discrepancy method and a chaining approach to give improved bounds on the coreset complexity of a wide class of kernel functions. Our results give randomized polynomial time algorithms to produce coresets of size $O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$ for the Gaussian and Laplacian kernels in the case that the data set is uniformly bounded, an improvement that was not possible with previous techniques. We also obtain coresets of size $O\big(\frac{1}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$ for the Laplacian kernel for $d$ constant. Finally, we give the best known bounds of $O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log(2\max\{1,\alpha\})}\big)$ on the coreset complexity of the exponential, Hellinger, and JS Kernels, where $1/\alpha$ is the bandwidth parameter of the kernel.

cs.LG↗

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

Polytopes with Bounded Integral Slack Matrices Have Sub-Exponential Extension Complexity

We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, \Delta\}$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log r$, where the rank of $f$ is defined to be the rank of the $A \times B$ matrix whose entries are the function values. As a corollary, we show that any $n$-dimensional polytope that admits a slack matrix with entries from $\{0,1,\dots,\Delta\}$ has extension complexity at most $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log n)$.

cs.DM↗

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

From approximate to exact integer programming

Approximate integer programming is the following: For a convex body $K \subseteq \mathbb{R}^n$, either determine whether $K \cap \mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity $c$. Approximate integer programming can be solved in time $2^{O(n)}$ while the fastest known methods for exact integer programming run in time $2^{O(n)} \cdot n^n$. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point $x^* \in (K \cap \mathbb{Z}^n)$ can be found in time $2^{O(n)}$, provided that the remainders of each component $x_i^* \mod{\ell}$ for some arbitrarily fixed $\ell \geq 5(n+1)$ of $x^*$ are given. The algorithm is based on a cutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a $2^{O(n)}n^n$ algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (2012) that is considerably more involved. Our algorithm also relies on a new asymmetric approximate Carath\'eodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equation-standard form $Ax = b, 0 \leq x \leq u, \, x \in \mathbb{Z}^n$ . Such a problem can be reduced to the solution of $\prod_i O(\log u_i +1)$ approximate integer programming problems. This implies, for example that knapsack or subset-sum problems with polynomial variable range $0 \leq x_i \leq p(n)$ can be solved in time $(\log n)^{O(n)}$. For these problems, the best running time so far was $n^n \cdot 2^{O(n)}$.

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

Approximate $\mathrm{CVP}$ in time $2^{0.802 \, n}$ -- now in any norm!

We show that a constant factor approximation of the shortest and closest lattice vector problem in any norm can be computed in time $2^{0.802\, n}$. This contrasts the corresponding $2^n$ time, (gap)-SETH based lower bounds for these problems that even apply for small constant approximation. For both problems, $\mathrm{SVP}$ and $\mathrm{CVP}$, we reduce to the case of the Euclidean norm. A key technical ingredient in that reduction is a twist of Milman's construction of an $M$-ellipsoid which approximates any symmetric convex body $K$ with an ellipsoid $\mathcal{E}$ so that $2^{\varepsilon n}$ translates of a constant scaling of $\mathcal{E}$ can cover $K$ and vice versa.

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↗

arXiv2021

On the Hardness of Scheduling With Non-Uniform Communication Delays

In the scheduling with non-uniform communication delay problem, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. One of the top ten open problems in scheduling theory, in the influential list by Schuurman and Woeginger and its latest update by Bansal, asks if the problem admits a constant factor approximation algorithm. In this paper, we answer the question in negative by proving that there is a logarithmic hardness for the problem under the standard complexity theory assumption that NP-complete problems do not admit quasi-polynomial time algorithms. Our hardness result is obtained using a surprisingly simple reduction from a problem that we call Unique Machine Precedence constraints Scheduling (UMPS). We believe that this problem is of central importance in understanding the hardness of many scheduling problems and conjecture that it is very hard to approximate. Among other things, our conjecture implies a logarithmic hardness of related machine scheduling with precedences, a long-standing open problem in scheduling theory and approximation algorithms.

cs.DS↗