arXiv subjects

Simon Döring

Publications and source records attributed to Simon Döring.

6 recordsLinked to original sources

arXiv2026

The Classical Weisfeiler-Leman Algorithm Stabilizes in $O(n)$ Rounds

The classical Weisfeiler-Leman algorithm (also known as the $2$-dimensional Weisfeiler-Leman algorithm) is a simple combinatorial algorithm that was originally designed as a heuristic for the graph isomorphism problem. However, it has also numerous connections to other areas such as algebraic graph theory, logics, proof complexity, combinatorial optimization and machine learning. We prove that the classical Weisfeiler-Leman algorithm terminates after $5(n-1)$ iterations. This improves over the previous best upper bound of $O(n \log n)$ by Lichter, Ponomarenko and Schweitzer [LICS 2019], and asymptotically matches the known lower bound of $Ω(n)$ by Fürer [ICALP 2001]. Additionally, building on our results for the $2$-dimensional case, we obtain an improved upper bound of $O(n^{k-1}/(k-2)! + n^{k-2})$ on the number of iterations performed by the $k$-dimensional Weisfeiler-Leman algorithm, for every $k \geq 3$. Our arguments actually hold for a larger class of sequences of colorings of $k$-tuples; in this larger class our upper bounds are essentially tight for all $k \geq 3$.

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arXiv2025

The Complexity of Finding and Counting Subtournaments

We study the complexity of counting and finding small tournament patterns inside large tournaments. Given a fixed tournament $T$ of order $k$, we write ${\#}\text{IndSub}_{\text{To}}(\{T\})$ for the problem whose input is a tournament $G$ and the task is to compute the number of subtournaments of $G$ that are isomorphic to $T$. Previously, Yuster [Yus25] obtained that ${\#}\text{IndSub}_{\text{To}}(\{T\})$ is hard to compute for random tournaments $T$. We consider a new approach that uses linear combinations of subgraph-counts [CDM17] to obtain a finer analysis of the complexity of ${\#}\text{IndSub}_{\text{To}}(\{T\})$. We show that for all tournaments $T$ of order $k$ the problem ${\#}\text{IndSub}_{\text{To}}(\{T\})$ is always at least as hard as counting $\lfloor 3k/4 \rfloor$-cliques. This immediately yields tight bounds under ETH. Further, we consider the parameterized version of ${\#}\text{IndSub}_{\text{To}}(\mathcal{T})$ where we only consider patterns $T \in \mathcal{T}$ and that is parameterized by the pattern size $|V(T)|$. We show that ${\#}\text{IndSub}_{\text{To}}(\mathcal{T})$ is ${\#}W[1]$-hard as long as $\mathcal{T}$ contains infinitely many tournaments.

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arXiv2025

Counting Small Induced Subgraphs: Scorpions Are Easy but Not Trivial

We consider the parameterized problem $\#$IndSub$(Φ)$ for fixed graph properties $Φ$: Given a graph $G$ and an integer $k$, this problem asks to count the number of induced $k$-vertex subgraphs satisfying $Φ$. Dörfler et al. [Algorithmica 2022] and Roth et al. [SICOMP 2024] conjectured that $\#$IndSub$(Φ)$ is $\#$W[1]-hard for all non-meager properties $Φ$, i.e., properties that are nontrivial for infinitely many $k$. This conjecture has been confirmed for several restricted types of properties, including all hereditary properties [STOC 2022] and all edge-monotone properties [STOC 2024]. In this work, we refute this conjecture by showing that scorpion graphs, certain $k$-vertex graphs which were introduced more than 50 years ago in the context of the evasiveness conjecture, can be counted in time $O(n^4)$ for all $k$. A simple variant of this construction results in graph properties that achieve arbitrary intermediate complexity assuming ETH. We formulate an updated conjecture on the complexity of $\#$IndSub$(Φ)$ that correctly captures the complexity status of scorpions and related constructions.

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arXiv2025

Can You Link Up With Treewidth?

In a fundamental paper in parameterized complexity theory, Marx [ToC '10] constructed $k$-vertex graphs $H$ of maximum degree $3$ such that $n^{o(k /\log k)}$ time algorithms for detecting colorful $H$-subgraphs would refute the Exponential-Time Hypothesis (ETH). This result is widely used to obtain almost-tight conditional lower bounds for parameterized problems under ETH. We give a new and fully self-contained proof of this result that further simplifies a recent work by Karthik et al. [SOSA 2024]. In our proof, we introduce a novel graph parameter of independent interest, the linkage capacity $γ(H)$, and show that detecting colorful $H$-subgraphs in time $n^{o(γ(H))}$ refutes ETH. Then, we use a simple construction of communication networks credited to Beneš to obtain $k$-vertex graphs of maximum degree $3$ and linkage capacity $Ω(k / \log k)$, avoiding arguments involving expander graphs, which were required in previous papers. We also show that every graph $H$ of treewidth $t$ has linkage capacity $Ω(t / \log t)$, thus recovering a stronger result shown by Marx [ToC '10] with a simplified proof. Additionally, we obtain new tight lower bounds on the complexity of colorful subgraph detection for certain types of patterns by analyzing their linkage capacity: We prove that almost all $k$-vertex graphs of polynomial average degree $Ω(k^β)$ for $β> 0$ have linkage capacity $Θ(k)$, which implies tight lower bounds for finding such patterns $H$. As an application of these results, we also obtain tight lower bounds for counting small induced subgraphs having a fixed property $Φ$, improving bounds from, e.g., [Roth et al., FOCS 2020].

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arXiv2024

From Graph Properties to Graph Parameters: Tight Bounds for Counting on Small Subgraphs

A graph property is a function $Φ$ that maps every graph to {0, 1} and is invariant under isomorphism. In the $\#IndSub(Φ)$ problem, given a graph $G$ and an integer $k$, the task is to count the number of $k$-vertex induced subgraphs $G'$ with $Φ(G')=1$. $\#IndSub(Φ)$ can be naturally generalized to graph parameters, that is, to functions $Φ$ on graphs that do not necessarily map to {0, 1}: now the task is to compute the sum $\sum_{G'} Φ(G')$ taken over all $k$-vertex induced subgraphs $G'$. This problem setting can express a wider range of counting problems (for instance, counting $k$-cycles or $k$-matchings) and can model problems involving expected values (for instance, the expected number of components in a subgraph induced by $k$ random vertices). Our main results are lower bounds on $\#IndSub(Φ)$ in this setting, which simplify, generalize, and tighten the recent lower bounds of Döring, Marx, and Wellnitz [STOC'24] in various ways. (1) We show a lower bound for every nontrivial edge-monotone graph parameter $Φ$ with finite codomain (not only for parameters that take value in {0, 1}). (2) The lower bound is tight: we show that, assuming ETH, there is no $f(k)n^{o(k)}$ time algorithm. (3) The lower bound applies also to the modular counting versions of the problem. (4) The lower bound applies also to the multicolored version of the problem. We can extend the #W[1]-hardness result to the case when the codomain of $Φ$ is not finite, but has size at most $(1 - \varepsilon)\sqrt{k}$ on $k$-vertex graphs. However, if there is no bound on the size of the codomain, the situation changes significantly: for example, there is a nontrivial edge-monotone function $Φ$ where the size of the codomain is $k$ on $k$-vertex graphs and $\#IndSub(Φ)$ is FPT.

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arXiv2023

Counting Small Induced Subgraphs with Edge-monotone Properties

We study the parameterized complexity of #IndSub($Φ$), where given a graph $G$ and an integer $k$, the task is to count the number of induced subgraphs on $k$ vertices that satisfy the graph property $Φ$. Focke and Roth [STOC 2022] completely characterized the complexity for each $Φ$ that is a hereditary property (that is, closed under vertex deletions): #IndSub($Φ$) is #W[1]-hard except in the degenerate cases when every graph satisfies $Φ$ or only finitely many graphs satisfy $Φ$. We complement this result with a classification for each $Φ$ that is edge monotone (that is, closed under edge deletions): #IndSub($Φ$) is #W[1]-hard except in the degenerate case when there are only finitely many integers $k$ such that $Φ$ is nontrivial on $k$-vertex graphs. Our result generalizes earlier results for specific properties $Φ$ that are related to the connectivity or density of the graph. Further, we extend the #W[1]-hardness result by a lower bound which shows that #IndSub($Φ$) cannot be solved in time $f(k) \cdot |V(G)|^{o(\sqrt{\log k/\log\log k})}$ for any function $f$, unless the Exponential-Time Hypothesis (ETH) fails. For many natural properties, we obtain even a tight bound $f(k) \cdot |V(G)|^{o(k)}$; for example, this is the case for every property $Φ$ that is nontrivial on $k$-vertex graphs for each $k$ greater than some $k_0$.

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