arXiv subjects

Leo Wennmann

Publications and source records attributed to Leo Wennmann.

6 recordsLinked to original sources

arXiv2026

Distributed Santa Claus via Global Rounding

In this paper, we consider the Santa Claus problem in the CONGEST model. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. Notably, each gift can have a different value. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we obtain the first results the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\hat \Theta(\sqrt n+D)$ rounds, where our $\widetilde\Omega(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.

cs.DS↗

arXiv2026

Non-Additive Discrepancy: Coverage Functions in a Beck-Fiala Setting

Recent concurrent work by Dupr\'{e} la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

math.CO↗

arXiv2025

A Simple Algorithm for Trimmed Multipoint Evaluation

Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and total degree $D$, the goal is to evaluate it on a natural class of input points. This problem arises as a key subroutine in recent algorithmic results [Dinur; SODA '21], [Dell, Haak, Kallmayer, Wennmann; SODA '25]. It is known that trimmed multipoint evaluation can be solved in near-linear time [van der Hoeven, Schost; AAECC '13] by a clever yet somewhat involved algorithm. We give a simple recursive algorithm that avoids heavy computer-algebraic machinery, and can be readily understood by researchers without specialized background.

cs.DS↗

arXiv2025

Cost Preserving Dependent Rounding for Allocation Problems

We present a dependent randomized rounding scheme, which rounds fractional solutions to integral solutions satisfying certain hard constraints on the output while preserving Chernoff-like concentration properties. In contrast to previous dependent rounding schemes, our algorithm guarantees that the cost of the rounded integral solution does not exceed that of the fractional solution. Our algorithm works for a class of assignment problems with restrictions similar to those of prior works. In a non-trivial combination of our general result with a classical approach from Shmoys and Tardos [Math. Programm.'93] and more recent linear programming techniques developed for the restricted assignment variant by Bansal, Sviridenko [STOC'06] and Davies, Rothvoss, Zhang [SODA'20], we derive a O(log n)-approximation algorithm for the Budgeted Santa Claus Problem. In this new variant, the goal is to allocate resources with different values to players, maximizing the minimum value a player receives, and satisfying a budget constraint on player-resource allocation costs.

cs.DS↗

arXiv2024

Solving Polynomial Equations Over Finite Fields

We present a randomized algorithm for solving low-degree polynomial equation systems over finite fields faster than exhaustive search. In order to do so, we follow a line of work by Lokshtanov, Paturi, Tamaki, Williams, and Yu (SODA 2017), Bj\"orklund, Kaski, and Williams (ICALP 2019), and Dinur (SODA 2021). In particular, we generalize Dinur's algorithm for $\mathbb{F}_2$ to all finite fields, in particular the "symbolic interpolation" of Bj\"orklund, Kaski, and Williams, and we use an efficient trimmed multipoint evaluation and interpolation procedure for multivariate polynomials over finite fields by Van der Hoeven and Schost (AAECC 2013). The running time of our algorithm matches that of Dinur's algorithm for $\mathbb{F}_2$ and is significantly faster than the one of Lokshtanov et al. for $q>2$. We complement our results with tight conditional lower bounds that, surprisingly, we were not able to find in the literature. In particular, under the strong exponential time hypothesis, we prove that it is impossible to solve $n$-variate low-degree polynomial equation systems over $\mathbb{F}_q$ in time $O((q-\varepsilon)^{n})$. As a bonus, we show that under the counting version of the strong exponential time hypothesis, it is impossible to compute the number of roots of a single $n$-variate low-degree polynomial over $\mathbb{F}_q$ in time ${O((q-\varepsilon)^{n})}$; this generalizes a result of Williams (SOSA 2018) from $\mathbb{F}_2$ to all finite fields.

cs.CC↗

arXiv2024

Minimizing Tardy Processing Time on a Single Machine in Near-Linear Time

In this work we revisit the elementary scheduling problem $1||\sum p_j U_j$. The goal is to select, among $n$ jobs with processing times and due dates, a subset of jobs with maximum total processing time that can be scheduled in sequence without violating their due dates. This problem is NP-hard, but a classical algorithm by Lawler and Moore from the 60s solves this problem in pseudo-polynomial time $O(nP)$, where $P$ is the total processing time of all jobs. With the aim to develop best-possible pseudo-polynomial-time algorithms, a recent wave of results has improved Lawler and Moore's algorithm for $1||\sum p_j U_j$: First to time $\tilde O(P^{7/4})$ [Bringmann, Fischer, Hermelin, Shabtay, Wellnitz; ICALP'20], then to time $\tilde O(P^{5/3})$ [Klein, Polak, Rohwedder; SODA'23], and finally to time $\tilde O(P^{7/5})$ [Schieber, Sitaraman; WADS'23]. It remained an exciting open question whether these works can be improved further. In this work we develop an algorithm in near-linear time $\tilde O(P)$ for the $1||\sum p_j U_j$ problem. This running time not only significantly improves upon the previous results, but also matches conditional lower bounds based on the Strong Exponential Time Hypothesis or the Set Cover Hypothesis and is therefore likely optimal (up to subpolynomial factors). Our new algorithm also extends to the case of $m$ machines in time $\tilde O(P^m)$. In contrast to the previous improvements, we take a different, more direct approach inspired by the recent reductions from Modular Subset Sum to dynamic string problems. We thereby arrive at a satisfyingly simple algorithm.

cs.DS↗