graph software developers

Abstract: Graph generative models commonly use graph-specific Transformer architectures to jointly update node and edge features. However, these architectures do not directly benefit from recent hardware-aware attention implementations, limiting their scalability. The adaptation is, however, non-trivial: edge and graph-level features must be injected into pairwise attention scores, graph masks must be respected, and dynamic edge updates remain a major source of quadratic memory usage. We propose an edge-aware attention layer for graph generation that incorporates structural information through head-wise score modulation, while updating edge representations with a lightweight residual mechanism that retains attention-score dependent edge updates. This layer can be effectively implemented using PyTorch's FlexAttention. We achieve similar generation quality to our baseline while reducing peak GPU memory by around 65% and substantially improving training and sampling throughput. These results suggest that the proposed architecture is a practical step toward scalable graph foundation models.

Abstract: For a fixed graph $T$ and a fixed nonempty family $\mathcal{F}$ of graphs, let $\operatorname{ex}(G, T, \mathcal{F})$ denote the maximum number of copies of $T$ in an $\mathcal{F}$-free subgraph of a graph $G$. Alon and Shikhelman~[Algorithmica, 2022] conjectured that if $e(T)> 0$ and no member of $\mathcal{F}$ is a subgraph of a blow-up of $T$, then for every $\varepsilon>0$, approximating $\operatorname{ex}(G, T, \mathcal{F})$ within an additive error $n^{v(T)-\varepsilon}$ is NP-hard for an input graph $G$ on $n$ vertices. The previously established general cases for this conjecture include $\operatorname{ex}(G, K_2, \mathcal{F})$ by Alon, Shapira and Sudakov~[Ann. of Math., 2009], and $\operatorname{ex}(G, K_m, K_k)$ by Alon and Shikhelman. In this paper, we confirm this conjecture for every bipartite graph $T$ and every clique $T$. Our reductions use iterative leaf additions, symmetric rooted extensions and random transversals to encode the maximum number of edges in a $(k-1)$-colorable subgraph of $G$, where $k=\min_{F\in \mathcal{F}}χ(F)$.

Abstract: The $k$-core of a graph is its (unique) largest subgraph with minimum degree at least $k$. For any $k \geq 3$, deciding whether a given vertex belongs to the $k$-core is a P-complete problem, meaning that it is inherently sequential and highly unlikely to admit efficient parallel algorithms, even on graphs of maximum degree $k+1$. This paper investigates alternative parameterizations of the $k$-core problem to identify conditions under which it can be placed into sub-polynomial complexity classes. We prove that the problem is in para-NC$^{2+ε}$ when parameterized by treewidth, and in para-NC$^3$ when parameterized by $k$ on chordal graphs. Furthermore, we introduce a novel NC$^{3}$ algorithm for interval graphs when $k = \mathcal{O}(\lg v(G))$, which relies on an improved parameterization by pathwidth. Finally, we establish corresponding lower bounds, demonstrating that, even with these parameterizations, computing the $k$-core remains L-hard, meaning it requires at least logarithmic space. These findings explore the boundary of parallel tractability for the $k$-core problem by highlighting the graph parameters that make it inherently sequential.