complexity analysis

This work investigates the evolutionary trajectory of algorithmic space complexity across 118 core problems in computer science, encompassing over 800 algorithms. Method: Leveraging a large-scale literature survey, historical complexity data analysis, and theoretical evaluation, the study quantifies trends in memory efficiency improvements relative to hardware advances. Contribution/Results: It is the first to empirically demonstrate that, in 20% of cases, algorithmic space optimization outpaces DRAM latency reduction. The paper introduces the “time–space trade-off Pareto frontier” framework to characterize optimal algorithmic trade-offs over time. Findings confirm that memory efficiency has emerged as a critical constraint in modern algorithm design. To support reproducible research and engineering practice, the authors release an open-source algorithm knowledge base (https://algorithm-wiki.csail.mit.edu), providing standardized benchmarks and decision-support tools for both theoretical analysis and system implementation.

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.

Evaluating learning-augmented online algorithms under uncertainty remains challenging, as conventional metrics focus narrowly on worst-case prediction errors, neglecting both prediction accuracy and risk sensitivity. Method: We propose a dual-track evaluation framework grounded in decision theory, jointly incorporating distance-based prediction error quantification (deterministic aspect) and risk-sensitive modeling (stochastic aspect). By embedding decision-theoretic loss functions into online algorithm analysis, we integrate prediction error modeling with risk-controllable optimization, designing novel learning-augmented algorithms for contract scheduling and 1-max search. Contribution/Results: Our approach achieves provable robustness to prediction errors, performance guarantees with tight bounds, and explicit risk controllability. It is the first to unify prediction accuracy, worst-case robustness, and risk preference within a single theoretical framework—establishing a systematic evaluation paradigm and design principle for learning-augmented online algorithms.

Conventional algorithm analysis treats basic operations as equally costly, ignoring substantial disparities in execution time, energy consumption, carbon emissions, and monetary cost across modern processor architectures. Method: We propose a multidimensional weighted operation complexity model that unifies computational cost, energy usage, carbon footprint, and financial expense—enabling architecture-aware, sustainability-oriented algorithm evaluation. Our approach integrates instruction-level fine-grained cost modeling, automated source-code analysis, and empirical measurement tooling, supporting user-defined weight configurations for diverse optimization objectives. Contribution/Results: Experiments demonstrate strong correlation with ground-truth measurements (Spearman ρ > 0.9) and significantly higher prediction accuracy for runtime and energy than baseline methods—including Big-O, ICE, and EVM gas metrics. The model establishes a novel, interpretable, cross-architectural paradigm for algorithmic efficiency assessment in green computing and resource-constrained environments.

This paper investigates whether the branch-and-bound (B&B) algorithm exhibits polynomial-time approximation scheme (PTAS) behavior for NP-hard combinatorial optimization problems, including the knapsack and scheduling problems. Through structural problem analysis, novel truncation strategy design, and rigorous convergence analysis, we establish—for the first time—theoretical guarantees that the standard B&B framework asymptotically generates (1−ε)-approximate solutions within polynomial time. This result fundamentally challenges the conventional view that B&B only ensures eventual optimality, and instead bridges B&B with approximation algorithms by formally extending its theoretical applicability to polynomial-time approximation. Extensive experiments on benchmark instances confirm that the proposed approach achieves arbitrary approximation accuracy ε > 0 in polynomial time, matching or surpassing the performance of specialized PTASs and state-of-the-art heuristic methods.

Segmented linear approximation (PLA) in learned indexes suffers from suboptimal storage efficiency, and no information-theoretic space lower bound exists for PLA under both compression and indexing constraints. Method: We establish the first information-theoretic space lower bound for PLA in these dual settings, then design a novel, minimalist data structure that achieves theoretically optimal compact representation for 2D monotonic point sequences under a given error bound. The structure supports O(log n)-time x-value lookup and segment evaluation. Contribution/Results: Our approach unifies the modeling of PLA’s compressibility and queryability—yielding the first systematic lower-bound analysis, constructive guarantee, and efficient implementation for PLA-based learned indexes. The space usage is asymptotically tight to the lower bound, achieving succinctness on most practical distributions. This work bridges a critical theoretical and engineering gap in learned indexing research.

This work addresses the computational intractability of large-scale combinatorial optimization problems arising from their exponentially sized search spaces by proposing a structure-aware parallel decomposition framework. The approach constructs a constrained maximum-cut model based on variable interaction structures, reformulates it as a QUBO problem, and leverages an Ising machine to efficiently cluster variables for automatic problem decomposition. The resulting subproblems are then solved in parallel using mathematical optimization solvers. This method uniquely integrates structure-aware clustering with Ising-based computation, substantially reducing the effective problem size. Experimental results on the capacitated vehicle routing problem demonstrate up to a 95.32% reduction in variable count, achieving within one minute the solution quality that conventional methods require thirty minutes to attain, while significantly improving the rate of feasible solutions.

This work investigates the fundamental theoretical limits of learned indexes in terms of the trade-off between query time and space overhead, revealing intrinsic constraints on their algorithmic complexity. By developing a unified analytical framework, it introduces tools from approximation theory—specifically quantization theory and Kolmogorov widths—into the study of learned indexing for the first time. The analysis systematically characterizes the optimal space-time trade-offs under piecewise-constant and piecewise-linear models by integrating probabilistic sampling control with approximation of cumulative distribution functions. Rigorous lower bounds on query time are derived across diverse data distributions and modeling assumptions, clarifying the performance bottlenecks inherent in current learned index methods and establishing a new theoretical foundation for both understanding and designing such algorithms.

This work investigates the trade-off between memory and computation under a fixed online computational budget, specifically quantifying the amount of offline storage required to achieve a target accuracy. The authors propose a semi-amortized parameter optimization framework that stores a limited set of pre-solved instances offline and, at test time, retrieves the most relevant one to warm-start K steps of projected gradient descent. For smooth convex problems, they establish matching upper and lower bounds on memory complexity in the strongly convex setting, revealing a phase transition phenomenon: under β-growth conditions, the benefit of additional memory vanishes once K exceeds a certain threshold. A general framework is provided to quantify the memory cost necessary for acceleration. The theoretical findings are validated through parametric ridge regression experiments, accurately characterizing the interplay among memory, computation, and solution accuracy.

This work proposes a reservoir computing–based mechanism for cross-task dynamic programming reuse to reduce the overall computational cost of solving multiple combinatorial optimization problems. By encoding intermediate states generated during dynamic programming into feature representations and feeding them into a lightweight linear regression model, the method automatically learns transferable computational patterns across distinct problems, enabling efficient sharing and reuse of intermediate results. As the first approach to leverage machine learning for automatically identifying and exploiting cross-task computational redundancy, this study demonstrates its effectiveness on the Traveling Salesman Problem and the Subset Sum Problem. Experimental results show that, compared to solving each problem independently, the proposed framework significantly improves the accuracy of approximate solutions while substantially reducing computation time, highlighting the potential of this novel computational paradigm.

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.