algorithm design

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.

This work addresses the satisfiability problem for constraint satisfaction problems (CSPs) over homogeneous infinite hypergraphs, circumventing the fundamental limitation that finite-domain reduction techniques do not directly extend to infinite domains. We devise the first symmetry-driven algorithm for this setting—relying on an external linear order—and integrate tools from homogeneous structure model theory, first-order definability reductions, and group action analysis to systematically classify the computational complexity of broad classes of infinite hypergraph CSPs. Our main contribution is the establishment of a dichotomy theorem—classifying each CSP as either in P or NP-complete—for a wide family of infinite hypergraphs. This result confirms the Bodirsky–Pinsker conjecture in full generality and significantly extends prior graph-CSP dichotomies to higher-arity hypergraph structures. The framework provides a unified classification theory and foundational technical machinery for infinite-domain CSPs.

This paper investigates the computational complexity and efficient solvability of high-multiplicity identical parallel-machine scheduling $P||C_{max}$. Addressing the severe dependence of prior algorithms on the optimal makespan $C_{max}$—with runtime $(log C_{max})^{2^{O(d)}}$—and their inefficiency under large job-type multiplicities, we introduce three key techniques: problem-tailored preprocessing, Frank–Tardos compression encoding, and a novel upper-bound analysis based on the number of vertices of the integer convex hull. We establish, for the first time, an FPT-equivalence between $P||C_{max}$ and $Q||C_{max}$ parameterized by the number $d$ of job and machine types. Our algorithm achieves runtime $(log p_{max})^{2^{O(d)}}$, drastically weakening dependence on $C_{max}$. Furthermore, we provide a tight parameterized lower bound, fully resolving the central open question posed by Mnich and van Bevern regarding FPT status of $d$-type scheduling.

This paper investigates the parameterized complexity of “almost satisfying all constraints” for finite Boolean constraint languages Γ, formalized as Min SAT(Γ) (minimize the number k of unsatisfied constraints) and its weighted variant Weighted Min SAT(Γ) (ensure total violation weight ≤ W). Employing a novel synthesis of directed flow augmentation, algebraic classification of constraint languages, weight-sensitive kernelization, and structural analysis of graph cuts, we establish, for the first time, a unified complexity dichotomy for both unweighted and weighted cases. For every Γ, we completely characterize fixed-parameter tractability: either (i) Weighted Min SAT(Γ) is FPT while Min SAT(Γ) is W[1]-hard, (ii) both are W[1]-hard, or (iii) Weighted Min SAT(Γ) is FPT (implying Min SAT(Γ) is also FPT). Our framework overcomes prior limitations in modeling implication constraints (u → v), and systematically generalizes and unifies landmark results including Almost 2-SAT, ℓ-Chain SAT, and Coupled Min-Cut.

This work investigates the finite tractability of Promise Constraint Satisfaction Problems (PCSPs): specifically, when a PCSP admits a polynomial-time reduction to an efficiently solvable CSP over some finite domain (unless P = NP). Employing algebraic methods—including clone theory, polymorphism analysis, and categorical tools—alongside structural properties of templates such as symmetry and negation closure, we establish the first necessary and sufficient condition for finite tractability within the class of symmetric Boolean PCSP templates. This resolves a fundamental gap left open by prior work of Barto and Brakensiek–Guruswami. Our characterization shows that most classical approximation problems—including 1-in-3-SAT, Not-All-Equal SAT, and various approximate graph coloring variants—are not finitely tractable. Moreover, we provide a complete classification boundary delineating precisely which symmetric Boolean PCSPs are finitely tractable.

This work addresses the high sensitivity of constraint programming solver performance to hyperparameter configurations and the prohibitive cost of manual tuning. The authors propose a resource-aware, two-phase auto-tuning framework that, within a limited time budget, first explores promising configurations and then solves the target problem using the best identified configuration. Innovatively integrating Bayesian optimization with Hamming distance-based search within a unified framework, the approach is implemented using CPMpy. Experimental evaluation on 114 combinatorial optimization instances demonstrates that the method outperforms the default configurations on 25.4% and 38.6% of instances for the ACE and Choco solvers, respectively, significantly surpassing either search strategy in isolation.

This work addresses the lack of machine-verifiable formalizations of line search methods in nonlinear optimization, which has hindered algorithmic reliability. Within the Lean 4 theorem prover, it presents the first systematic formalization of several classical line search criteria—including Armijo, Goldstein, Wolfe, and their nonmonotone variants—alongside rigorous definitions of gradient descent, descent directions, and backtracking step-size selection. The study fully verifies the Zoutendijk convergence theorem within this framework, thereby establishing a comprehensive formal foundation for line search theory. This contribution significantly enhances the verifiability and trustworthiness of nonlinear optimization algorithms through mechanized mathematical reasoning.

This work addresses the Partial Constraint Satisfaction Problem (PCSP) on program control flow graphs, where variable assignments are sought to minimize the total penalty incurred by partially violating constraints—a formulation well-suited for compiler optimizations such as register allocation and redundancy elimination. Leveraging the Series-Parallel-Loop (SPL) decomposition structure of control flow graphs, the paper presents the first unified and efficient algorithm for general PCSP on SPL graphs. By integrating dynamic programming with the graph’s sparse structural properties, the approach achieves linear-time solvability for fixed-size domains. Experimental evaluation on optimal bank selection tasks demonstrates a fourfold speedup over the state-of-the-art method, confirming both the efficiency and generality of the proposed technique.

This work addresses the critical challenge that modern GPU-accelerated linear programming solvers—such as cuPDLP, which is based on the primal-dual hybrid gradient (PDHG) algorithm—exhibit performance highly sensitive to hyperparameters, yet lack tuning methods with provable generalization guarantees. For the first time, this study establishes structural relationships between hyperparameters and solution trajectories for multiple adaptive techniques in complex first-order LP solvers, including preconditioning, restart strategies, and smoothed weight updates. By integrating convergence analysis of PDHG with a model of structural sensitivity, the authors propose a data-driven hyperparameter learning framework that offers theoretical generalization guarantees under polynomial sample complexity. Experimental results demonstrate that the framework significantly enhances solver efficiency across diverse problem instances.

This study addresses the computational bottlenecks in scientific computing arising from the infeasibility of exact algorithms for large-scale problems. Through a systematic evaluation of approximation methods across 118 core algorithmic problems—integrating complexity analysis, taxonomies of approximation algorithms, and historical context—the work presents the first large-scale empirical evidence demonstrating that only approximately 20% of these problems derive substantial benefit from approximation. Notably, one-quarter of exponential-time-hard problems admit polynomial-time approximation schemes, and the adoption of approximation strategies increases the proportion of linear-time solvable problems by 23%. By quantifying the trade-offs between accuracy and efficiency, this research offers theoretical insights to guide the design of AI-driven and high-performance algorithms.