Congruence Decomposition with Neural Block Solvers for Large-Scale PCI Assignment

Physical Cell Identity (PCI) assignment is essential for interference management in dense 5G networks. As cellular networks scale, PCI reuse becomes unavoidable, which may cause collisions, confusions, and multiple forms of modular interference. Jointly mitigating these effects gives rise to a large-scale, multi-objective combinatorial optimization problem that is difficult to solve efficiently at practical network scales. In this work, we propose a congruence decomposition framework with neural block solvers for large-scale PCI assignment. The proposed decomposition exploits the arithmetic structure of PCI values to decouple multiple modular interference objectives into a collection of blockwise Min-$k$-Partition subproblems, followed by a graph coloring procedure to resolve PCI conflicts. For the resulting NP-hard Min-$k$-Partition subproblems, we develop neural block solvers by parameterizing their relaxed quadratic formulations with graph neural networks, enabling efficient optimization at large scales. Discrete assignments are recovered through conditional expectation rounding with theoretical guarantees. Experiments on synthetic cellular graphs and real-world 5G networks show that the proposed method consistently outperforms existing modular-interference-aware baselines in modular interference reduction, conflict elimination, and computational efficiency.

Graph Coloring Approach to Solving Sudoku with Oscillatory Neural Networks

Oscillatory Neural Networks (ONNs) present an attractive physics-based computing paradigm rooted in the dynamics of a network of typically fully coupled oscillators aiming to minimize an underlying energy function. In this paper, we propose an ONN-based solver for one well-known constrained combinatorial optimization problem, namely a Sudoku, by formulating the problem as a Graph Coloring problem. By modifying the already existing Graph Coloring solver to a computationally cheaper version and introducing an additional term ensuring the fulfillment of the Sudoku constraints, our solver is shown to significantly outperform the existing HNN- and ONN solvers in terms of accuracy. In particular, we are able to achieve nearly flawless accuracies on $4 \times 4$ as well as rather high accuracies on $9 \times 9$ Sudoku puzzles for different numbers of unknown digits.

Scaling Weisfeiler-Leman Expressiveness Analysis to Massive Graphs with GPUs

The stable coloring of the Weisfeiler-Leman (1-WL) test is a cornerstone of Graph Neural Networks because it provides an upper bound to the expressive power of message-passing architectures. Unfortunately, computing it presents two fundamental bottlenecks. First, classic algorithms are inherently sequential and cannot exploit modern massively parallel hardware. Second, these are \emph{global} algorithms, i.e., they require availability in memory of the full graph, severely limiting applicability to real-world instances. We leverage a linear-algebraic interpretation of 1-WL stable coloring and introduce two key contributions: (i)~a randomized refinement algorithm with tight probabilistic guarantees and (ii)~a correctness-preserving batching scheme that decomposes the graph into independently processable subgraphs while provably returning a stable coloring of the original graph. This approach maps directly to GPU-efficient primitives. In numerical experiments, our CUDA implementation delivers speedups up to two orders of magnitude over classical CPU-based partition refinement and, for the first time, successfully computes stable colorings on web-scale graphs with over 30 billion edges, where CPU baselines time out or fail.

Deep Reinforcement Learning for Minimum Zero-Forcing Sets

This paper explores the problem of finding the minimum zero-forcing set on undirected graphs and proposes an adapted machine-learning framework to solve the problem. The minimum zero-forcing set problem is a graph coloring problem where the color of an initial set of nodes propagates throughout a network. The set of nodes is zero-forcing if it forces all uncolored nodes to change color under the constraint of the color-change rule. There are several applications to this problem across different domains such as network science, network control, and designing logical circuits. Finding the minimum zero-forcing set is shown to be NP-hard. We propose a reinforcement learning framework, SD-ZFS, that adapts the S2V-DQN architecture to the ZFS problem. We train several models on this adapted framework and analyze the performance across graph datasets that have varying structures. We evaluate how the models trained on the framework generalize, scale, and transfer to different network types. The results demonstrate the effectiveness of the framework when compared against the optimal solution and greedy heuristic. We provide further insight into how the ZFS problem can be solved through machine-learning and the influence of network structure on the problem.

Contrastive Neural Algorithmic Reasoning for Graph Coloring

Graph coloring seeks to assigns colors to a graph's nodes so that adjacent nodes receive different colors, using as few colors as possible. Here, we study approximate $k$-coloring, where the goal is to use at most $k$ colors while minimizing the number of monochromatic edges. This problem is central to graph theory and has applications in areas such as scheduling and resource allocation. Recent unsupervised GNN approaches optimize each instance directly, precluding generalization across graph sizes and distributions. We instead propose a contrastive learning framework that learns transferable coloring geometry where the embeddings of same-color nodes align, while adjacent nodes' representations are pushed toward distinct directions. We analyze the resulting population objective over bounded-size graphs. For unit-norm embeddings, we show that its optima have a line-prototype structure: Representations of nodes of the same color collapse to a shared one-dimensional subspace, and edges connect orthogonal subspaces. This geometry yields stationarity conditions in the supervised setting and is preserved by projected subgradient dynamics under a balanced-coloring assumption. In an unnormalized variant, gradient descent has a max-margin bias governed by a quotient-graph hard-margin problem. Experiments on synthetic and real-world graphs show that contrastive GNN encoders generalize effectively and produce low-conflict colorings, matching and sometimes improving on greedy approaches.

Can Transformers Learn to Verify During Backtracking Search?

Backtracking search underlies classical constraint solvers, planners, and theorem provers. Recent transformer-based reasoning systems explore search trees over their own intermediate steps. A common training recipe fits an autoregressive next-token loss on offline solver traces. The model's input at each step is a cumulative trace of all prior decisions. The optimal continue-or-backtrack predictor depends only on the current search state, since two trajectories reaching the same state admit the same viable continuations. We show that decoder-only transformers trained on cumulative traces fail this requirement in two ways: the trace can scatter state features across many positions (scattered retrieval), and the predictor can condition on the trajectory rather than the state (history entanglement). We address scattered retrieval with localization, a trace-level fix that rewrites each decision block to expose state features locally. We address history entanglement with Selective State Attention (SSA), a fixed attention mask that enforces state-based decisions structurally without modifying training data, objective, or parameters. We focus on reactive verification, after propagation has exposed a contradiction. We test SSA on 3-SAT, graph coloring, Blocks World, and backtracking parsing. On same-state pairs that differ only in prior history, SSA emits identical decisions while a cumulative-trained causal baseline does not. Our contribution is a diagnostic of transformer behavior on serialized trajectory data, paired with a structural fix. Pretrained language models that search over their own reasoning steps may face the same failure. Our analysis opens up inference-time context clearing as a candidate way to apply the same isolation without retraining.