arXiv subjects

Moritz Stargalla

Publications and source records attributed to Moritz Stargalla.

9 recordsLinked to original sources

arXiv2026

Parameterized Complexity of $L_p$-Lipschitz Constants for Input Convex Neural Networks and $L_p$-Norm Maximization over Zonotopes

Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.

cs.CC↗

arXiv2026

Shallower ReLU Network Representations via Exact Linear Algebra

We study the depth required by ReLU networks to exactly represent piecewise linear functions, focusing specifically on the maximum function. This problem has recently received significant attention in both the ML and TCS literature. We prove that $\max_n(x)=\max\{x_1,\ldots,x_n\}$ is exactly representable with two hidden layers for every $n\leq 12$. Previously, this was only known up to $n\leq5$ [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26]. We obtain our constructions through an exact computer-assisted search within a space of candidate solutions: After a symmetry reduction, we obtain a finite system of linear equations over $\mathbb{Q}$ such that any solution yields a valid representation of the maximum function. The resulting constructions have a structured first hidden layer, which enables recursive substitution into deeper networks. This yields an exact ReLU representation of $\max_n$ with at most $\lceil \log_6(n/2) \rceil+1$ hidden layers. Consequently, every continuous piecewise-linear function on $\mathbb{R}^d$ admits an exact representation with at most $\lceil\log_6((d+1)/2)\rceil+1$ hidden layers; in particular, two hidden layers suffice for $d\leq 11$. Again, these results improve upon [Bakaev, Brunck, Hertrich, Stade, Yehudayoff, STOC'26], who proved analogous logarithmic bounds with base three.

cs.LG↗

arXiv2026

OptFSST: Optimized FSST String Compression

Strings account for a substantial fraction of data in modern analytical systems, making lightweight compression with fast random access an important building block for efficient query processing. Fast Static Symbol Table (FSST) addresses this need by replacing frequent byte sequences with compact codes while preserving independent decompression of individual strings. However, FSST's compression effectiveness is limited by its greedy symbol selection and greedy encoding strategy, leaving encoding gains on the table. We present OptFSST, an optimized FSST variant that improves its compression factors while preserving its static-symbol-table design and random-access decompression. OptFSST optimally encodes the text using dynamic programming given a symbol table. Additionally, we show that a generalized version of the symbol-table selection problem is NP-hard when the alphabet is part of the input, motivating heuristic table construction for field-level compressors. Hence, we add in OptFSST (i) an additional frequency counter that accelerates the discovery of longer symbols and (ii) a pruning strategy that removes redundant and conflicting symbol candidates during table construction. We also extend the same techniques to FSST12, yielding OptFSST12. Our evaluation on 92 real-world string datasets shows that OptFSST improves the compression factors of FSST and FSST12 by up to 47.7% and 91.5%, with an average improvement of 7.3% and 17.0%, respectively, while retaining the fine-grained random-access properties. Notably, OptFSST12 improves FSST12's decompression speed by $1.2\times$ on average.

cs.DB↗

arXiv2026

Tropical Circuits with Scalar Multiplication Gates

We study tropical circuits with scalar multiplication gates, that is, algebraic circuits whose gates implement $\max$, $+$, or multiplication with a positive constant. For such circuits, we prove exponential size lower bounds for computing maximum weight directed spanning trees and maximum weight bipartite perfect matchings. As a corollary, we obtain an exponential size separation between monotone and non-monotone maxout neural networks, which generalize the popularly used ReLU neural networks. One conclusion from this is that neural network models with enforced convexity constraints, such as input-convex neural networks (ICNNs), sometimes need to be exponentially larger than their unrestricted counterparts in order to express the same functions.

cs.CC↗

arXiv2026

An Adaptive Variable Neighborhood Search for a Family of Set Covering Routing Problems with an Application in Disaster Relief Operations

This paper studies a variant of the Set Covering Routing Problem (SCRP) motivated by post-disaster humanitarian logistics. We consider a hybrid distribution concept in which the majority of transportation is performed by helicopters, while ground transport is limited to the last mile, addressing severe accessibility constraints in disaster-affected regions. The resulting problem integrates landing site location, routing, and covering decisions, incorporating features of the Multi-Vehicle Covering Tour Problem (m-CTP) and the Vehicle Routing with Demand Allocation Problem (VRDAP) in a facility-capacitated, multi-depot setting. Due to the computational complexity of the problem, we develop an Adaptive Variable Neighborhood Search (AVNS) that combines established routing operators with novel mechanisms for covering decisions. The performance of the proposed approach is evaluated on benchmark instances for the related m-CTP and VRDAP problems, demonstrating competitive solution quality compared to problem-specific state-of-the-art approaches. Furthermore, we apply our AVNS to a real-world case study based on the 2024 flash floods in Afghanistan. The results highlight the practical relevance of the proposed framework and provide managerial insights into effective distribution strategies for disaster response operations.

math.OC↗

arXiv2025

Parameterized Hardness of Zonotope Containment and Neural Network Verification

Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posed by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that, for all $\ell\ge 2$, deciding positivity (and thus surjectivity) of a function $f:\mathbb{R}^d\to\mathbb{R}$ computed by an $\ell$-layer ReLU network is W[$\ell-1$]-hard when parameterized by the input dimension $d$. The case $\ell=2$ implies that zonotope non-containment (a problem that is of independent interest in computational geometry, control theory, and robotics) is W[1]-hard with respect to the ambient dimension $d$. Moreover, we show that approximating the maximum within any multiplicative factor and computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in $\ell$-layer networks is NP-hard and W[$\ell-1$]-hard with respect to $d$. For $\ell\ge 3$, approximating the $L_p$-Lipschitz constant is NP- and W[$\ell-2$]-hard. We further show that the above problems are NP- and W[$t$]-hard (for all $t\ge 1$) with respect to $\ell$ for constant $d$. Notably, our hardness results imply that the naive enumeration-based methods for these fundamental problems running in $n^{(\ell-1) d}\cdot\operatorname{poly}(N)$ time are all essentially optimal under the Exponential Time Hypothesis.

cs.CC↗

arXiv2025

Optimized Qubit Routing for Commuting Gates via Integer Programming

Quantum computers promise to outperform their classical counterparts at certain tasks. However, existing quantum devices are error-prone and restricted in size. Thus, effective compilation methods are crucial to exploit limited quantum resources. In this work, we address the problem of qubit routing for commuting gates, which arises, for example, during the compilation of the well-known Quantum Approximate Optimization Algorithm. We propose a two-step decomposition approach based on integer programming, which is guaranteed to return an optimal solution. To justify the use of integer programming, we prove NP-hardness of the underlying optimization problem. Furthermore, we derive asymptotic upper and lower bounds on the quality of a solution. We develop several integer programming models and derive linear descriptions of related polytopes, which generalize to applications beyond this work. Finally, we conduct a computational study showing that our approach outperforms existing heuristics in terms of quality and exact methods in terms of runtime.

math.OC↗

arXiv2025

The Computational Complexity of Counting Linear Regions in ReLU Neural Networks

An established measure of the expressive power of a given ReLU neural network is the number of linear regions into which it partitions the input space. There exist many different, non-equivalent definitions of what a linear region actually is. We systematically assess which papers use which definitions and discuss how they relate to each other. We then analyze the computational complexity of counting the number of such regions for the various definitions. Generally, this turns out to be an intractable problem. We prove NP- and #P-hardness results already for networks with one hidden layer and strong hardness of approximation results for two or more hidden layers. Finally, on the algorithmic side, we demonstrate that counting linear regions can at least be achieved in polynomial space for some common definitions.

cs.CC↗

arXiv2024

An Adaptive Variable Neighborhood Search for a Family of Set Covering Routing Problems with an Application in Disaster Relief Operations

This paper studies a variant of the Set Covering Routing Problem (SCRP) motivated by post-disaster humanitarian logistics. We consider a hybrid distribution concept in which the majority of transportation is performed by helicopters, while ground transport is limited to the last mile, addressing severe accessibility constraints in disaster-affected regions. The resulting problem integrates landing site location, routing, and covering decisions, incorporating features of the Multi-Vehicle Covering Tour Problem (m-CTP) and the Vehicle Routing with Demand Allocation Problem (VRDAP) in a facility-capacitated, multi-depo setting. Due to the computational complexity of the problem, we develop an Adaptive Variable Neighborhood Search (AVNS) that combines established routing operators with novel mechanisms for covering decisions. The performance of the proposed approach is evaluated on benchmark instances for the related m-CTP and VRDAP problems, demonstrating competitive solution quality compared to problem-specific state-of-the-art approaches. Furthermore, we apply our AVNS to a real-world case study based on the 2024 flash floods in Afghanistan. The results highlight the practical relevance of the proposed framework and provide managerial insights into effective distribution strategies for disaster response operations.

math.OC↗