Andreas Göbel

Panagiotis Aivasiliotis, Andreas Göbel, Matthias Lanzinger et al.

The Weisfeiler--Leman (WL) algorithm is one of the most influential heuristics for the graph isomorphism problem. The expressive power of WL has been extensively studied in the contexts of descriptive complexity, logics, graph neural networks, and the theory of homomorphism indistinguishabily. Notably, two graphs are indistinguishable by the $k$-dimensional WL algorithm if and only if they are indistinguishable by homomorphism-counts from graphs of treewidth at most $k$. An intrinsic question is to find a natural version of the WL algorithm for relational structures of higher arity admitting an equivalent characterisation via homomorphism indistinguishability along bounded generalised hypertree width (GHW). Scheidt and Schweikardt solved this for $k=1$ by defining the RCR algorithm and showing indistinguishability from $α$-acyclic structures. In this work, we resolve this for all $k\ge1$: we develop $k$-RCR and show that two structures $\mathcal{A}$ and $\mathcal{B}$ are insdistinguishable by $k$-RCR if and only if they have the same homomorphism-counts from all structures $\mathcal{C}$ of generalised hypertreewidth $\le k$. Moreover, we introduce a ``fractional'' version of $k$-RCR and show that two structures are insdistinguishable by fractional $k$-RCR if and only if they have the same homomorphism-counts from all structures with (a variant of) fractional hypertreewidth at most $k$. Last, we develop $k$-HyperOWL, the first relational WL algorithm operating directly on a relational structure. We show that $k$-HyperOWL is as expressive as $k$-RCR and that, given a structure $\mathcal{A}$, $k$-HyperOWL can compute $t$ iterative refinements in time $O(t|\mathcal{A}|^{k+1})$. Moreover, the colouring produced by $k$-HyperOWL can be used as a constructive preprocessing routine for counting homomorphisms from structures of generalised hypertreewidth $\le k$.

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.