consistent update framework

This work addresses the fully dynamic submodular maximization problem, which supports both insertions and deletions, by proposing the first general algorithmic framework that achieves constant-factor approximation guarantees with sublinear adaptive complexity. Built upon a dynamic streaming model, the framework integrates greedy strategies with a buffering mechanism and applies to both cardinality and rank-$k$ matroid constraints. Specifically, under a cardinality constraint, it attains a $(1/2 - O(\varepsilon))$-approximation with $O(1/\varepsilon^2)$ adaptivity; under a rank-$k$ matroid constraint, it achieves a $(1/4 - O(\varepsilon))$-approximation with $O(\log k / \varepsilon^2)$ adaptivity. This represents a significant advance over prior approaches, which were limited to insertion-only settings.

This work addresses the theoretical analysis of algorithmic stability, aiming to characterize the sensitivity of combinatorial algorithms to input perturbations and to elucidate the fundamental trade-off between stability and solution quality—particularly generalization error. We propose the first unified stability framework encompassing randomized, iterative, and distributed algorithms. Within this framework, we establish tight equivalence conditions linking stability to uniform convergence and generalization bounds. Leveraging probabilistic inequalities, empirical process theory, and Lipschitz sensitivity decomposition, we derive sharp stability bounds for canonical algorithms including stochastic gradient descent (SGD) and empirical risk minimization (ERM). These results yield improved generalization error upper bounds and provide rigorous theoretical foundations for noise robustness analysis and model selection.

This work breaks the long-standing deterministic lower bound on update time for dynamic maximum matching in dense graphs. Previously, all deterministic algorithms required Ω(n) amortized update time on n-vertex dense graphs. We present the first deterministic dynamic algorithm that maintains a maximal matching in Õ(n⁸⁄₉) amortized time—surpassing the linear barrier. Our approach introduces three key techniques: (1) repurposing the Edge Degree Constrained Subgraph (EDCS) to guarantee full matching of high-degree vertices, diverging from its conventional use for approximation; (2) integrating sublinear-time matching computation, random walks on directed expander graphs, and monotonic Even–Shiloach trees; and (3) designing a randomized algorithm achieving Õ(n³⁄₄) amortized time against an adaptive adversary. These advances collectively establish new state-of-the-art bounds for deterministic and randomized dynamic maximal matching in dense graphs.

Computing fixed points of non-monotonic operators—such as those arising from negation-as-failure or hypothetical updates—is notoriously challenging, as traditional monotonic methods do not apply and existing approximation techniques are either imprecise or computationally expensive. This work introduces, for the first time, controlled incompleteness into approximate fixed-point computation, integrating abstract interpretation, Approximation Fixpoint Theory (AFT), lattice theory, and partitioning-based optimization to devise a practical algorithm. The proposed method guarantees termination, polynomial-time complexity, and soundness over finite lattices while substantially improving approximation precision. Empirical evaluations demonstrate its effectiveness: deployed as an accelerating preprocessor in Answer Set Programming and applied to speculative program analysis, it significantly reduces rollback frequency, thereby validating both its efficiency and practical utility.

Formal theories of algorithms have long been confined to non-interactive settings, leaving interactive and nondeterministic algorithms without rigorous foundational treatment. Method: This work introduces a unified formal framework encompassing both non-interactive and interactive, deterministic and nondeterministic algorithms. It proposes the “prototype algorithm” as an abstract computational model and rigorously defines its behavioral semantics. Three equivalence relations—behavioral, implementation, and specification equivalence—are formally introduced; their relationships are established, and specification equivalence is proven to be the appropriate criterion for capturing essential algorithmic identity. Contribution: The framework breaks the traditional boundaries of algorithm definitions, providing the first formal foundation for interactive algorithms. It establishes a layered, extensible meta-theory of algorithms and delivers a rigorous logical basis for reasoning about algorithmic essence, correctness verification, and cross-model comparison—thereby unifying previously fragmented formal approaches under a coherent theoretical umbrella.

This work addresses the problem of efficiently maintaining the rank, a column basis, and a maximum-rank submatrix of a matrix under dynamic updates—either to individual entries or entire columns—and applies these techniques to the dynamic maximum matching problem in graphs. The paper presents the first dynamic algorithm whose update time depends on the current rank \( r \) rather than the matrix dimension \( n \). By integrating sparse update strategies with rank-sensitive complexity analysis, it achieves an amortized update time of \( \tilde{O}(r^{1.405}) \) for single-entry modifications and \( \tilde{O}(r^{1.528} + z) \) for column updates, where \( z \) denotes the number of changed entries. This approach is the first to simultaneously support dynamic maintenance of rank, basis, and maximum-rank submatrix, yielding an edge update time of \( \tilde{O}(|M|^{1.405}) \) for dynamic graph matching and significantly improving upon prior methods.

This work addresses the challenge of verifying graph-theoretic properties on graph classes with bounded treewidth or pathwidth by proposing a unified framework that integrates tree-decomposition-based dynamic programming with formal reductions of graph properties. The framework enables automatic verification of atomic properties and their Boolean combinations, achieving for the first time a modular composition of dynamic programming algorithms coupled with parameterized automated theorem proving in treewidth. The developed TreeWidzard engine automatically checks whether all graphs of treewidth at most \(k\) satisfy a given Boolean expression \(P\) over graph properties, significantly enhancing the scalability and automation of complex graph property verification.

This work addresses the efficient computation of fixed points for specific variables in systems of equations over Noetherian partially ordered sets with a bottom element—a problem commonly arising in program verification. The paper proposes a local fixed-point algorithm based on a dependency oracle that dynamically identifies variable dependencies and explores only the subsystem influencing the target variable. By leveraging the Noetherian structure, the method ensures sound termination guarantees. The designed dependency oracle supports customization, composition, and approximation, enabling flexible trade-offs between precision and performance while preserving correctness. Experimental evaluation demonstrates that a prototype implementation outperforms existing approaches across multiple scenarios, offering both superior efficiency and a clean, adaptable architecture suitable for diverse application domains.

This study addresses the problem of determining whether an observed execution trace of a multithreaded program admits a sequentially consistent interleaving under a constraint of at most π preemptions. By integrating computational complexity theory with parameterized analysis, the work establishes the first trichotomy in complexity based on the number of writers: the single-writer case is polynomial-time decidable; the two-writer case is NP-hard; and the three-writer case admits a conditional lower bound under the Exponential Time Hypothesis. Moreover, the problem becomes W[1]-hard when the number of preemptions is unbounded. This paper presents the first fixed-parameter intractability result parameterized by the number of preemptions, thereby delineating precise computational boundaries for reasoning about sequential consistency under limited preemption.

Existing approaches to program resource analysis struggle to simultaneously achieve the completeness of static analysis and the worst-case coverage afforded by dynamic analysis. To address this limitation, this work proposes a hybrid analysis method that integrates dynamic symbolic execution with mixed-integer linear programming to systematically enumerate execution paths within a bounded input space and derive empirically sound upper bounds on maximum resource consumption. This approach represents the first deep integration of dynamic symbolic execution and linear programming for inferring tight and effective worst-case resource bounds for functional programs. The prototype tool CompAS demonstrates both practical utility and theoretical guarantees in estimating resource usage on complex programs.