InductWave: Inductive Multi-Hop Logical Query Answering on Knowledge Graphs

Knowledge graph question answering (KGQA) requires navigating from topic entities to an answer several relations away. Recent methods prompt a frontier LLM to explore the graph through a retrieval tool, but their reliance on frontier-scale inference makes them costly to deploy. We present Search-on-Graph-R1 (\sogrone{}), which internalizes this navigation into a compact 8B model through supervised fine-tuning (SFT) followed by reinforcement learning (RL). Our central idea is to scaffold a frontier teacher with each question's gold SPARQL query, so the teacher traverses a known answer-bearing path with a live \texttt{Search} tool rather than having to discover the path itself. Since every call executes against a live Freebase server, the resulting trajectories are grounded in the knowledge graph by construction. On WebQSP, CWQ, and GrailQA, \sogrone{} at 8B surpasses every frozen frontier-LLM system in our comparison and posts the strongest results on CWQ of any system we compare against. It does so using no auxiliary module at inference and no LLM judge during training. Isolating each training stage shows that SFT and RL contribute complementary gains, our approach transfers across model families, and RL learns to reach answers in fewer \texttt{Search} calls than its SFT initialization.

Knowledge Graphs (KGs) are widely used to mitigate the limitations of Large Language Models (LLMs), such as outdated knowledge and hallucinations. Existing LLM-KG integration frameworks typically rely on predefined operators to retrieve factual knowledge from KGs and inject it into prompts for answer generation. This paradigm faces two critical bottlenecks: 1) Inflexibility: The predefined operators are limited in scope and thus lack sufficient compositional expressiveness to fully capture the complex semantics required by KG questions. 2) Unscalability: Direct injection of factual knowledge into prompts limits scalability in handling large-scale factual knowledge. To address these two bottlenecks, we propose Code-on-Graph (CoG), a programmatic reasoning framework for LLM-KG integration. Specifically, given the factual knowledge retrieved at each reasoning step, CoG first identifies the corresponding KG schemas and represents these schemas as Python classes, which serve as abstract interfaces to the retrieved facts. It then generates executable code grounded in these classes, with the retrieved facts instantiated as objects of the corresponding classes during execution. This design enables flexible code-based reasoning while avoiding the direct injection of large-scale factual knowledge into prompts. Experiments on WebQSP, CWQ, and GrailQA demonstrate that CoG outperforms prior state-of-the-art models by up to 10.5%.

Large language models (LLMs) excel at reasoning but struggle with knowledge-intensive questions due to limited context and parametric knowledge. However, existing methods that rely on finetuned LLMs or GNN retrievers are limited by dataset-specific tuning and scalability on large or unseen graphs. We propose the LLM-KGFR collaborative framework, where an LLM works with a structured retriever, the Knowledge Graph Foundation Retriever (KGFR). KGFR encodes relations using LLM-generated descriptions and initializes entities based on their roles in the question, enabling zero-shot generalization to unseen KGs. To handle large graphs efficiently, it employs Asymmetric Progressive Propagation (APP)- a stepwise expansion that selectively limits high-degree nodes while retaining informative paths. Through node-, edge-, and path-level interfaces, the LLM iteratively requests candidate answers, supporting facts, and reasoning paths, forming a controllable reasoning loop. Experiments demonstrate that LLM-KGFR achieves strong performance while maintaining scalability and generalization, providing a practical solution for KG-augmented reasoning.