batched partition refinement

Traditional 1-WL stable coloring algorithms struggle to scale to web-scale graphs due to their inherently sequential nature and high memory overhead. This work addresses this limitation by leveraging the linear-algebraic formulation of 1-WL coloring to propose the first randomized refinement algorithm with rigorous probabilistic correctness guarantees. Furthermore, we introduce a provably correct graph batching strategy that preserves coloring accuracy while enabling efficient mapping of computations onto GPU primitives. By integrating randomized algorithms, graph partitioning, and CUDA-based parallelism, our approach successfully achieves stable coloring on web-scale graphs with over 30 billion edges—attaining up to two orders of magnitude speedup over CPU baselines, whereas conventional methods either time out or fail entirely at this scale.

Canonical labeling of highly symmetric random circulant graphs—including their directed variants—remains challenging, as conventional combinatorial approaches such as color refinement fail on vertex-transitive graphs due to symmetry-induced indistinguishability. Method: This paper introduces a novel hybrid framework integrating color refinement with vertex individualization. Its core innovation lies in deriving a unique canonical label solely from the counts of walks of all lengths from each vertex to the individualized vertex. The method unifies Tinhofer’s canonicalization procedure, the 2-dimensional Weisfeiler–Leman algorithm, and walk-counting analysis to overcome refinement stagnation caused by automorphic symmetry. Results: Experiments demonstrate efficient canonical labeling for almost all random circulant (and circulant directed) graphs, along with construction of their canonical Cayley representations. This significantly advances the theoretical frontiers of graph isomorphism testing and encoding of symmetric graphs.

Graph isomorphism testing lacks a known polynomial-time algorithm, yet combinatorial heuristics—such as color refinement—perform efficiently on most practical instances. This paper initiates the application of smoothed analysis to this problem, systematically studying isomorphism testing and canonical labeling under random perturbations and on Erdős–Rényi random graphs (G(n,p)). Leveraging probabilistic analysis, combinatorial graph theory, and the one-dimensional Weisfeiler–Leman (WL) algorithm, we rigorously prove that, under the (G(n,1/2)) model, naive color refinement produces a unique canonical labeling with probability (1 - o(1)), thereby solving graph isomorphism correctly with high probability in polynomial time. Our key contribution is the first smoothed analysis framework justifying the empirical efficiency of classical heuristic algorithms; moreover, we elevate the success probability of the WL algorithm from constant to asymptotically almost sure—establishing its robustness under realistic input distributions.

This paper establishes a tight lower bound on the minimum number of iterations required by the $k$-dimensional Weisfeiler–Leman ($k$-WL) algorithm to decide graph isomorphism. Addressing the long-standing open question—posed in 2001—of whether $k$-WL iteration complexity can exceed linear time, the authors construct a family of CFI graphs and combine WL coloring dynamics analysis with logical characterizations of graphs. They prove that, for general $n$-vertex graphs, $Omega(n^{k/2})$ iterations are necessary and sufficient for $k$-WL to achieve full coarsest equitable partition refinement. This resolves the open problem definitively. Moreover, the result yields the first quantitative connections between WL iteration depth and fundamental concepts in theoretical computer science: definability in first-order logic with counting, expressive power of graph neural networks, and computational complexity. The work thus provides a foundational theoretical basis for graph representation learning.

This paper studies the dynamic edge coloring problem: maintaining a deterministic $(1+varepsilon)Delta$-edge coloring in an $n$-vertex graph subject to edge insertions and deletions, where $Delta$ denotes the current maximum degree—thereby breaking the classical $2Delta-1$ bound of greedy static algorithms. We propose the first deterministic dynamic edge coloring algorithm, whose core innovation is a shallow degree-splitter hierarchy that simultaneously achieves low update overhead and strong chromatic guarantees. Via amortized analysis and an exponentially small amortized update time construction, our algorithm achieves an amortized update time of $2^{ ilde{O}(sqrt{log n})}$. When $varepsilon^{-1}$ is subpolynomial, this bound is strictly subpolynomial—marking a substantial improvement over all prior deterministic algorithms.

This paper addresses the problem of efficiently maintaining a (Δ+1)-coloring in dynamic graphs under edge insertions and deletions, where Δ denotes an upper bound on the current maximum degree. We propose the first unified dynamic coloring framework supporting sequential, parallel, and distributed models. Our method leverages randomized local recoloring combined with batched synchronization. It achieves, for the first time, expected constant-time complexity across all three models: worst-case O(1) update time in the sequential model; expected O(1) work per update and poly(log n) depth in the parallel model; and O(1) uncolored nodes with convergence in O(log n) rounds in the distributed model. The algorithm natively supports batch updates and ensures correctness and efficiency via message pruning and rigorous convergence analysis.

This work addresses the flow decomposition problem on general directed graphs (including cyclic ones), breaking the decade-long restriction to directed acyclic graphs (DAGs). We propose the first unifying framework based on dominator trees, pioneering their use for detecting safe edge sequences in cyclic graphs: we establish that the longest safe sequence corresponds to an expansion of common leaf nodes across two dominator trees and identify it in linear time. Integrated with mixed-integer linear programming (MILP), our framework leverages dominator-tree preprocessing to fix safe path variables a priori, drastically reducing model size and eliminating costly linearizations of nonlinear terms. Evaluated on four bacterial genome datasets, our approach achieves up to 1000× speedup; over 90% of instances are solved within 30 seconds. This significantly enhances both solvability and efficiency of minimum flow decomposition and minimum absolute error models.

This work addresses the computational challenge of uniformly sampling proper $k$-colorings of graphs when the maximum degree $\Delta$ is large. The authors propose a novel approach based on partial rejection sampling (PRS), which introduces tunable soft coloring constraints that are progressively tightened to achieve exact uniform sampling. By integrating a recursive divide-and-conquer strategy, the original problem is decomposed into $O(\log n)$ independent subproblems of reduced size, each solvable in parallel by any exact sampler. This method is the first to combine soft coloring with PRS, enabling parallelization and achieving a runtime of $O(L^{\log^* n} \cdot n\Delta)$ when the number of relaxation levels $L$ is independent of $n$, improving upon the best-known algorithms that require $k > 3\Delta$. Empirical evidence suggests $L$ is likely constant, indicating potential for linear-time performance.

This work addresses the challenge of efficiently approximating k-graphlet distributions in massive graphs, a task hindered by conventional methods that require loading the entire graph into memory and thus suffer from poor scalability. The authors propose a streaming sampling algorithm that operates with only O(n^{1+c}) memory and performs a constant number—specifically O(1/c)—of graph traversals, thereby breaking the prior Ω(log n) lower bound on traversal complexity. This approach achieves near-optimal asymptotic performance in both memory usage and traversal cost. Leveraging streaming processing, subgraph sampling, and probabilistic approximation techniques, the algorithm substantially outperforms existing methods on both real-world and synthetic graphs, delivering speedups of several orders of magnitude, particularly on moderately dense graphs.

This work addresses a central challenge in the semi-streaming model: achieving efficient deterministic vertex coloring with few colors and near-linear memory. The paper presents the first deterministic semi-streaming algorithm that, using only Õ(n) memory, computes an O(Δ)-coloring in O(√log Δ) rounds, where Δ denotes the maximum degree of the graph. By integrating multi-pass edge-stream processing with a novel coloring strategy, this approach simultaneously achieves a number of colors linear in Δ and a number of rounds sublogarithmic in Δ—marking the first such result to break the longstanding trade-off between round complexity and color count that has constrained prior deterministic algorithms.