Andreas Göbel

Panagiotis Aivasiliotis, Andreas Göbel, Marc Roth

We study the complexity of the parameterised counting constraint satisfaction problem: given a set of constraints over a set of variables and a positive integer $k$, how many ways are there to assign $k$ variables to 1 (and the others to 0) such that all constraints are satisfied. Existing work has so far exclusively focused on restricted settings such as finding and counting homomorphisms between relational structures due to Grohe (JACM 2007) and Dalmau and Jonsson (TCS 2004), or the case of finite constraint languages due to Creignou and Vollmer (SAT 2012), and Bulatov and Marx (SICOMP 2014). In this work, we tackle a more general setting of Valued Parameterised Counting Constraint Satisfaction Problems (VCSPs) with infinite constraint languages. In this setting we are able to model significantly more general problems such as (weighted) parameterised factor problems on hypergraphs and counting weight-$k$ solutions of systems of linear equations, not captured by existing complexity classifications. We express parameterised VCSPs as parameterised Holant problems on uniform hypergraphs, and we establish complete and explicit complexity dichotomy theorems. For resolving the $\mathrm{P}$ vs. $\#\mathrm{P}$ question, we mainly rely on hypergraph gadgets, the existence of which we prove using properties of degree sequences necessary for realisability in uniform hypergraphs. For the $\mathrm{FPT}$ vs. $\#\mathrm{W}[1]$ question, we build upon the recently established combinatorial toolkit for parameterised holants on the special case of graphs by Aivasiliotis et al. (ICALP 2025) and also rely on an extension of the framework of the homomorphism basis due to Curticapean, Dell and Marx (STOC 17) to uniform hypergraphs. As a technical highlight, we also employ Curticapean's "CFI Filters'' (SODA 2024) to establish polynomial-time algorithms for isolating vectors in the homomorphism basis.

Francesco Quinzan, Vanja Doskoč, Andreas Göbel et al.

Several large-scale machine learning tasks, such as data summarization, can be approached by maximizing functions that satisfy submodularity. These optimization problems often involve complex side constraints, imposed by the underlying application. In this paper, we develop an algorithm with poly-logarithmic adaptivity for non-monotone submodular maximization under general side constraints. The adaptive complexity of a problem is the minimal number of sequential rounds required to achieve the objective. Our algorithm is suitable to maximize a non-monotone submodular function under a $p$-system side constraint, and it achieves a $(p + O(\sqrt{p}))$-approximation for this problem, after only poly-logarithmic adaptive rounds and polynomial queries to the valuation oracle function. Furthermore, our algorithm achieves a $(p + O(1))$-approximation when the given side constraint is a $p$-extendible system. This algorithm yields an exponential speed-up, with respect to the adaptivity, over any other known constant-factor approximation algorithm for this problem. It also competes with previous known results in terms of the query complexity. We perform various experiments on various real-world applications. We find that, in comparison with commonly used heuristics, our algorithm performs better on these instances.

Vanja Doskoč, Tobias Friedrich, Andreas Göbel et al.

We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature. In contrast to other heuristics, this requires no problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size. In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings. Additionally, we study this problem in a dynamic setting, by which knapsacks change during the optimization process. We modify our greedy algorithm to avoid a complete restart at each constraint update. This modification retains the approximation guarantees of the static case. We evaluate our results experimentally on a video summarization and sensor placement task. We show that our proposed algorithm competes with the state-of-the-art in static settings. Furthermore, we show that in dynamic settings with tight computational time budget, our modified greedy yields significant improvements over starting the greedy from scratch, in terms of the solution quality achieved.