arXiv subjects

Anatole Dahan

Publications and source records attributed to Anatole Dahan.

4 recordsLinked to original sources

arXiv2026

Subgroup Accessibility in Group Order Logic

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

cs.LO↗

arXiv2026

Robust Graph Isomorphism, Quadratic Assignment and VC Dimension

We present an additive $\varepsilon n^{2}$-approximation algorithm for the Graph Edit Distance problem (GED) on graphs of VC dimension $d$ running in time $n^{O(d/\varepsilon^{2})}$. In particular, this recovers a previous result by Arora, Frieze, and Kaplan [Math. Program. 2002] who gave an $\varepsilon n^{2}$-approximation running in time $n^{O(\log n/\varepsilon^{2})}$. Similar to the work of Arora et al., we extend our results to arbitrary Quadratic Assignment problems (QAPs) by introducing a notion of VC dimension for QAP instances, and giving an $\varepsilon n^{2}$-approximation for QAPs with bounded weights running in time $n^{O(\varepsilon^{-2}(d + \log\varepsilon^{-1}))}$. As a particularly interesting special case, we further study the problem $\varepsilon$-$\mathsf{GI}$, which entails determining if two graphs $G,H$ over $n$ vertices are isomorphic, when promised that if they are not, their graph edit distance is at least $\varepsilon n^{2}$. We show that the standard Weisfeiler--Leman algorithm of dimension $O(\varepsilon^{-1}d\log(\varepsilon^{-1}))$ solves this problem on graphs of VC dimension $d$. We also show that dimension $O(\varepsilon^{-1}\log n)$ suffices on arbitrary $n$-vertex graphs, while $k$-WL fails on instances at distance $Ω(n^{2}/k)$.

cs.DS↗

arXiv2025

Group Order Logic

We introduce an extension of fixed-point logic ($\mathsf{FP}$) with a group-order operator ($\mathsf{ord}$), that computes the size of a group generated by a definable set of permutations. This operation is a generalization of the rank operator ($\mathsf{rk}$). We show that $\mathsf{FP} + \mathsf{ord}$ constitutes a new candidate logic for the class of polynomial-time computable queries ($\mathsf{P}$). As was the case for $\mathsf{FP} + \mathsf{rk}$, the model-checking of $\mathsf{FP} + \mathsf{ord}$ formulae is polynomial-time computable. Moreover, the query separating $\mathsf{FP} + \mathsf{rk}$ from $\mathsf{P}$ exhibited by Lichter in his recent breakthrough is definable in $\mathsf{FP} + \mathsf{ord}$. Precisely, we show that $\mathsf{FP} + \mathsf{ord}$ canonizes structures with Abelian colors, a class of structures which contains Lichter's counter-example. This proof involves expressing a fragment of the group-theoretic approach to graph canonization in the logic $\mathsf{FP}+ \mathsf{ord}$.

cs.LO↗

arXiv2020

Relativization of Gurevich's Conjectures

Gurevich (1988) conjectured that there is no logic for $\textsf{P}$ or for $\textsf{NP}\cap \textsf{coNP}$. For the latter complexity class, he also showed that the existence of a logic would imply that $\textsf{NP} \cap \textsf{coNP}$ has a complete problem under polynomial time reductions. We show that there is an oracle with respect to which $\textsf P$ does have a logic and $\textsf P \ne\textsf{NP}$. We also show that a logic for $\textsf{NP} \cap \textsf{coNP}$ follows from the existence of a complete problem and a further assumption about canonical labelling. For intersection classes $Σ^p_n \cap Π^p_n$ higher in the polynomial hierarchy, the existence of a logic is equivalent to the existence of complete problems.

cs.LO↗