gadget construction

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 addresses the lack of scalable, general-purpose tools for polynomial-time reductions among NP-hard optimization problems, which hinders flexible integration with quantum hardware, commercial solvers, or heuristic algorithms. The authors propose a "constraint engineering" framework that leverages AI-powered coding agents to automatically construct a comprehensive reduction library. Built in Rust, the system features type safety, multi-layer verification, and a fully automated pipeline for implementation, review, and integration, enabling composable, transitive reduction graphs. Within three months, the team developed over 170,000 lines of code, covering more than 100 NP-hard problems and 200 reduction rules. Once a new solver is registered, it immediately becomes available across the entire connected component of the reduction graph, significantly enhancing reusability and interoperability.

This work establishes lower bounds on certificate size for local certification of distributed graph properties. Focusing on classical properties—including connectivity, matching, and coloring—it introduces the first local hardness reduction framework. By constructing local graph gadgets and applying information-theoretic analysis within the local model, the framework systematically transfers certificate-size lower bounds from one class of properties to another, overcoming the limitations of property-specific proofs. This approach achieves the first transferable lower-bound results in local certification. It uniformly establishes polynomial-scale (Ω(n)) lower bounds for multiple fundamental properties, exposing their inherent local complexity. The framework provides a general tool for local certification theory and advances the understanding of the fundamental limits of distributed verification efficiency.

Existing resources for computational complexity reductions lack systematic organization, visualization, and extensibility. Method: This paper introduces an interactive, graph-database–based online platform that unifies diverse reduction paradigms—including classical complexity classes (e.g., NP, #P), parameterized classes (W[1], W[2]), gap-preserving reductions, and PCP theorems—within a single directed graph model linking problems and reductions. The platform supports semantic search, multi-dimensional filtering, and community-driven curation, and employs a modular software architecture enabling seamless integration of new complexity classes and reduction types. Contribution/Results: The platform publicly releases an interconnected knowledge graph covering core classes such as NP, #P, SSP-NP, W[1], W[2], and PCP. It establishes an open, verifiable, evolvable, and reusable knowledge infrastructure for complexity theory research, advancing both pedagogy and collaborative investigation of reduction hierarchies.

This paper investigates performance optimization of Local Computation Algorithms (LCAs) on random graphs under average-case analysis. Addressing the Erdős–Rényi and preferential attachment models, it pioneers the extension of the LCA framework to the average-case setting, designing sublinear-query local access schemes for combinatorial structures—including *k*-spanners and maximum independent sets. The main contributions are threefold: (1) It demonstrates that structural properties of random graphs enable circumvention of worst-case stretch–size trade-offs, yielding improved parameter balances; (2) It introduces a novel “joint generation” paradigm, simultaneously constructing the random graph instance and its target combinatorial structure; (3) It constructs the first efficient LCA that locally generates a maximum independent set on ER graphs with query complexity significantly below the worst-case bound, and achieves optimal stretch–size trade-offs for *k*-spanner access on both models.

This work addresses the efficient identification of constraints that are indispensable for any constant-factor approximation to Boolean Minimum Constraint Satisfaction Problems (MinCSP)—termed 𝒪(1)-essential constraints—with the aim of reducing the search space for subsequent fixed-parameter tractable (FPT) algorithms. By extending graph-theoretic preprocessing frameworks to Boolean MinCSP, we establish a dichotomy theorem for constraint languages ℱ, providing the first systematic characterization of Boolean constraint types that admit efficient detection of such essential constraints. Notably, for the bijunctive constraint class, we devise a polynomial-time algorithm that identifies these essential constraints, enabling effective instance preprocessing even under the Unique Games Conjecture (UGC), where constant-factor approximation is believed to be intractable.

This study investigates how reductions in NP search problems affect the accessibility of hidden witness information and the capacity for local inference. It introduces a novel perspective that views reductions, gadgets, and auxiliary constructions as mechanisms for redistributing information. By incorporating auxiliary variables and consistency constraints to extend problem representations, the approach enhances local inferential power while preserving the recoverability of the original witness. Through case studies—such as reductions from the structured P-matrix violation search problem to classic NP-complete problems like 3-SAT and Subset Sum—the analysis demonstrates that representation expansion substantially improves local inferability. The work further reveals that search algorithms essentially function as decoders of the original hidden witness, thereby establishing a unified conceptual framework for understanding the dynamics of information transformation in NP reductions.

We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem $\Pi$ (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any \emph{constant} approximation ratio $\alpha$, the task of finding an $\alpha$-approximation of $\Pi$ in directed cycles has one of the following complexities: 1. $O(1)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 2. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $O(1)$ rounds in randomized LOCAL, 3. $\Theta(\log^* n)$ rounds in deterministic LOCAL, $\Theta(\log^* n)$ rounds in randomized LOCAL, 4. $\Theta(n)$ rounds in deterministic LOCAL, $\Theta(n)$ rounds in randomized LOCAL. Moreover, for any given $\Pi$ and $\alpha$, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

This work investigates logical laddgets—graph-theoretic gadgets for Boolean function realization—under a single-anchor constraint, where only one vertex is fixed to a “false” color in 3-coloring. Method: We formalize the laddget framework, rigorously defining its primitives and structural constraints; combine graph-theoretic modeling, non-isomorphic graph enumeration, and exhaustive search over graphs with ≤10 vertices; and integrate Boolean function mapping with universal k-coloring embedding techniques. Contribution/Results: We establish the structural scarcity of Boolean expressibility under single-anchor constraints, identifying—*for the first time*—exactly two non-isomorphic minimal XNOR laddgets. Compared to conventional double-anchor models, the single-anchor setting substantially increases construction complexity and shifts the boundary of realizable logic. Our work proves completeness of minimal laddget constructions for 3-coloring and generalizes the framework to arbitrary k-coloring, thereby introducing a new paradigm for constraint-driven graph-based logic computation.

This work addresses the limitations of traditional complexity frameworks—such as PLS—in capturing the core computational challenges inherent in designing efficient pivoting rules for local search. The authors propose a novel framework that requires algorithms to output not only a locally optimal solution but also the complete improvement path leading to it. By integrating parameterized complexity theory with the entire trajectory of local search and focusing on improvement chains rather than individual steps, this approach more accurately models the computational hardness of pivoting rules. Using a new form of reduction, the study analyzes the fixed-parameter tractability of canonical problems—including Subset Weight Optimization and Weighted Circuit—under c-swap and flip neighborhoods. It shows that these problems are efficiently solvable when parameterized by the number of weights, yet become intractable when parameterized by the distance to an optimal solution.