perform problem reductions

In constraint programming, high-level modeling languages (e.g., Essence) introduce significant symmetry through abstract structures such as nested sets, severely degrading solver efficiency. To address this, we propose a lightweight, structure-aware symmetry breaking method operating at the underlying representation level. Rather than generating numerous explicit symmetry-breaking constraints, our approach matrix-encodes abstract structures and integrates an optimized variable ordering strategy tailored to indistinguishable objects—achieving incomplete yet highly effective symmetry elimination. Compared to Akgün et al. (2025), our method substantially reduces constraint set size and search redundancy. Empirical evaluation on diverse symmetry-rich benchmark problems demonstrates marked improvements in solving speed, while maintaining practicality and scalability across problem sizes and structural complexity.

This study addresses the unclear practical efficacy of automatically generated polynomial symmetry-breaking constraints in integer linear programming across different solvers. The authors systematically evaluate the performance of mainstream mathematical programming and SMT solvers when handling such constraints, comparing three strategies: native quadratic handling, internal reformulation, and explicit linearization. Their experiments reveal that the effectiveness of symmetry breaking is highly solver-dependent, advocating for a solver-aware evaluation paradigm. The findings indicate that compact families of quadratic symmetry-breaking constraints generally enhance solver performance, whereas excessive linearization, overly large breaking sets, or inappropriate reformulations often lead to model bloating or search degradation, thereby diminishing or even reversing potential benefits.

This study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

This work addresses the automatic construction of rotationally symmetric point sets in discrete geometry, targeting extremal symmetric configurations for the Erdős–Szekeres problem and minimizing the number of singular points in the “everywhere-unbalanced-points” problem. Methodologically, we introduce the first SAT encoding that directly incorporates rotational symmetry constraints and design a novel local-search feasibility solver to circumvent the ∃ℝ-completeness barrier; geometric realizability is rigorously verified via computational geometry techniques. Key contributions are: (1) the first known rotationally symmetric extremal configuration for the Erdős–Szekeres problem; (2) a reduction of the minimum number of singular points in the everywhere-unbalanced-points problem from 23 to 21—the current best-known bound; and (3) a new paradigm integrating SAT-based combinatorial search with geometric realizability verification.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

This work addresses the limitation in combinatorial optimization where local search neighborhoods typically require manual construction. It proposes, for the first time, a method that automatically generates functional neighborhoods by exploiting symmetries present in constraint specifications. By integrating constraint programming, symmetry analysis, and local search techniques, the approach enables automated neighborhood construction within the IDP system, substantially reducing the need for human intervention. Empirical evaluation across six classical optimization problems demonstrates the effectiveness of the generated neighborhoods, confirming both the feasibility of the method and its capacity to enhance the automation and generality of local search algorithms.

This work investigates the expressive power of higher-order algorithms—such as k-consistency, Sherali–Adams linear programming relaxations, and Affine Integer Programming (AIP)—for solving Constraint Satisfaction Problems (CSPs) and their promise variants. By introducing a polymorphic minion structure over templates, the authors establish a unified framework that fully characterizes the capabilities of these algorithms in terms of minion homomorphisms. They further propose a novel hierarchy of orthogonal relaxations based on vectors over ℤₚ, proving its equivalence to AIP-ℤₚ and showing it constitutes a new class of semidefinite programming–like methods. This approach successfully solves the smallest known CSP counterexample resistant to BLP+AIP—the D₄-group CSP—and decides solvability of systems of linear equations modulo p² at prime order p.

This work addresses the computational inefficiency of maximum common subgraph (MCS) solvers, which stems from the vast search space and inadequate handling of graph symmetries. We propose the first dual symmetry-breaking framework that simultaneously identifies modular symmetries in both the variable graph and the value graph based on local neighborhood structures. By leveraging equivalence class reasoning, our approach prunes isomorphic subtrees while preserving solution optimality. This method represents the first systematic integration of symmetry handling across both graphs, substantially enhancing pruning efficiency. Experimental results on standard MCS benchmarks demonstrate that our approach outperforms the current state-of-the-art RRSplit algorithm, solving more instances and significantly reducing both computation time and search space.

This work addresses the absence of a unified, verifiable catalog for small-scale fast matrix multiplication algorithms scattered across diverse domains with inconsistent formats and naming conventions. The authors construct a comprehensive algorithm repository covering all instances up to size 32×32×32, supporting multiple number fields and commutative variants. By introducing the notion of “non-overlappingness,” they clearly distinguish between discovering novel bilinear kernels and composing existing ones, thereby decoupling algorithm discovery from composition and resolving attribution disputes in the literature. Leveraging a state-of-the-art closure-search framework augmented with techniques such as axis flipping, Kronecker products, axis concatenation, random products, distributive recomposition (including output stripping and pair fusion), and downward projection, the study systematically recombines and extends known algorithms, yielding numerous new low-rank schemes—including ternary integer algorithms—and automatically generates DIS09 comparison tables categorized by number field and commutativity.

This work addresses the limitation of existing exact solvers for large-scale Maximum k-Cut problems (k > 2), which stems from the absence of effective preprocessing techniques. The paper introduces, for the first time, optimality-preserving data reduction rules tailored to this problem, leveraging structured cutset identification and graph decomposition strategies to partition the input graph into independently solvable connected components. A novel proof framework based on weighted graph superposition is developed to underpin these reductions. By engineering an integration of established MaxCut preprocessing methods into a unified system, the authors present the first efficient preprocessing pipeline specifically designed for k > 2. Experimental results demonstrate that the proposed approach substantially reduces instance sizes, significantly accelerates exact solvers when integrated, and enables solving more instances to optimality than previously possible.