search space pruning

Maximum clique search—a classical NP-hard problem—suffers from prohibitively large search spaces and low computational efficiency. To address this, we propose an efficient algorithmic framework grounded in the “work-avoiding” paradigm. Our approach introduces four key innovations: (i) a lazy graph representation that defers structural construction until necessary; (ii) a prescreening mechanism enabling early pruning; (iii) optimized intersection computation with fast-exit semantics; and (iv) an adaptive algorithm selection strategy. Collectively, these techniques strengthen pruning power, drastically reducing both the number of visited search nodes and redundant computations. Evaluated on standard benchmarks, our method achieves up to 38.9× and 11× speedups over the state-of-the-art algorithms PMC and MC-BRB, respectively, significantly shortening runtime. The framework delivers a scalable, principled solution for maximum clique search on large-scale graphs.

Subgraph extraction problems arise widely in network design, facility location, and related domains, yet lack a general-purpose, efficient solution methodology. This work proposes ΔSearch—the first unified heuristic framework that requires only user-specified feasibility constraints and an optimization objective, automatically adapting to monotone, weighted monotone, and non-monotone graph problems without problem-specific parameter tuning. By integrating a reward-penalty optimization mechanism, generic constraint modeling, and search space pruning techniques, ΔSearch substantially enhances computational efficiency and can accelerate exact algorithms. Empirical evaluations demonstrate that it matches or surpasses state-of-the-art heuristics on tasks such as maximum planar subgraph, uncapacitated facility location, and prize-collecting vertex cover, while achieving approximately 89% of optimal solution quality on average across other problems—all without any parameter tuning.

Large-scale subset selection asks for a small useful set of examples, features, sensors, seed users, or context passages from an enormous ground set. Submodular maximization is a canonical model for such diminishing-returns problems, but rapidly growing datasets make even linear-time algorithms ever costlier. We study \emph{containment pruning}: first reduce the ground set to a smaller core $P$, then require that $P$ contain a near-optimal feasible solution for every downstream budget up to~$k$. Prior work has formulated many heuristics, but the theoretical limits of this preprocessing problem are largely unknown. For monotone submodular objectives, we prove that $1-1/e$ is tight: greedy achieves this containment factor, and no algorithm can beat it even with a larger pruning budget. For non-monotone objectives, we give the first$1/2-\varepsilon$ containment algorithms under cardinality constraints and extend the approach to knapsack constraints. This $1/2$ factor exceeds the best known algorithmic ratio and the known hardness threshold for non-monotone maximization, showing that pruning can be provably easier than optimization. Empirically, pruning lets an exact IP solver run on the reduced MaxCut instance with a ${\approx}620\times$ speedup, and proof-of-concept experiments on LLM context selection demonstrate the utility of non-monotone submodular proxies and our proposed containment algorithms.

To address the challenge of balancing computational efficiency and solution quality in the Set Cover Problem (SCP), this paper proposes Graph-SCP: a graph neural network-based preprocessing method that models SCP as a set-element bipartite graph. Graph-SCP is the first approach to jointly train via supervised learning—leveraging historical optimal solutions—and unsupervised learning—minimizing the SCP objective—to automatically identify and compress the original instance into a high-quality subproblem of 20–40% its original size. Evaluated on OR Library and synthetic benchmarks, Graph-SCP achieves an average 10× speedup over Gurobi without sacrificing solution quality; it also significantly outperforms greedy heuristics in accuracy. Crucially, it exhibits strong cross-scale generalization—trained on instances with ≤3k subsets and tested on up to 10k—thereby breaking the classical trade-off between precision and efficiency inherent in traditional exact and heuristic methods.

This work addresses the limitations of existing large language model (LLM)-driven heuristic design methods in combinatorial optimization, which often rely on manual trial-and-error or domain-specific knowledge and lack a systematic mechanism for improvement. To overcome this, the authors propose a structured framework that formalizes heuristic discovery as a language-guided program optimization process, comprising three modular phases: forward evaluation, backward feedback, and program update. This design enables an iterative and composable optimization workflow, unifying and generalizing prior approaches while allowing flexible enhancements through modularity. Empirical evaluation across four real-world combinatorial optimization tasks demonstrates that the proposed method significantly outperforms baseline techniques, achieving up to a 0.17 improvement in the QYI metric on unseen test instances.

Subgraph matching is an NP-hard problem, and existing approaches suffer from inefficiency due to extensive redundant computations. This work proposes the CEMR algorithm, which integrates a redundancy-aware expansion elimination mechanism within a depth-first search framework to significantly enhance matching efficiency. The core innovations include a strategy for merging and reusing common extensions, combined with black-white vertex encoding and a common extension buffer to minimize repeated calculations. Additionally, two effective pruning strategies are introduced to eliminate invalid search branches early. Experimental results demonstrate that CEMR substantially outperforms state-of-the-art subgraph matching algorithms across various real-world graph datasets and query workloads.

Existing algorithms for enumerating and finding maximum k-defective cliques suffer from exponential search-space explosion and suboptimal pruning. Method: We propose a “clique-first” branch-and-bound framework: first generate all maximal cliques, then incrementally introduce at most *k* missing edges in a controlled manner; we design a novel pivot strategy achieving worst-case optimal search-space size. Contribution/Results: We provide the first rigorous proof of tightness for the upper bound on the number of *k*-defective cliques, yielding the theoretically optimal time complexity *O*(3^*n*/3 ⋅ *n*^*k*). Leveraging diameter-two structural properties, we further refine this to *O*(*n* ⋅ 3^δ/3 ⋅ (δΔ)^*k*), where δ and Δ denote degeneracy and maximum degree, respectively. Integrated with degeneracy- and degree-based pruning within an efficient pruning framework, our algorithm outperforms state-of-the-art methods by four orders of magnitude on real-world graphs with over one million edges for both enumeration and maximum *k*-defective clique search.

Neural models for NP-hard graph combinatorial optimization suffer from suboptimal solution quality and poor out-of-distribution generalization. Method: We propose a neural-search co-design framework: (i) parameterized structural analysis identifies tractable and intractable substructures; (ii) a lightweight neural model generates only exploratory guidance signals (e.g., node priorities or pruning suggestions), which steer structure-aware exact search algorithms toward high-yield regions. This decouples learning from solving, eliminating end-to-end fitting bias while preserving neural efficiency and combinatorial algorithmic guarantees. Contribution/Results: Experiments across multiple graph optimization tasks show solutions approaching those of commercial solvers (e.g., Gurobi) in quality, with significantly stronger out-of-distribution generalization than purely neural approaches. The framework is broadly applicable, interpretable, and bridges the gap between data-driven heuristics and rigorous combinatorial optimization.

Existing approaches to automated algorithm design suffer from poor computational efficiency, often redundantly generating substructures and discarding low-fitness candidates that may harbor valuable components. This work proposes a directed acyclic graph (DAG)-based representation of algorithms, leveraging large language models (LLMs) to produce fine-grained code-editing operations—insertion, deletion, and replacement—and composing historical edits via the graph structure to efficiently explore the algorithmic search space. The method enables credit assignment at the edit level and, supported by theoretical analysis, balances search depth and breadth under limited computational budgets, thereby transcending conventional full-algorithm generation paradigms. Experiments on three combinatorial optimization problems demonstrate significant outperformance over baseline methods under identical token budgets, with enriched context yielding benefits only when the LLM’s prior knowledge is insufficient.