hardness gap amplification

This work addresses automatic runtime and variable-size bound analysis for integer programs, focusing on the decidable subclass of periodic rational-solvable loops (PRS-loops). The proposed method introduces a modular analysis framework: it first derives local bounds for PRS-loops, then lifts them to global bounds via program transformation and inductive reasoning. Crucially, it extends the decidability of PRS-loop analysis to arbitrary integer programs by designing a synergistic synthesis mechanism combining abstract interpretation with rational linear algebra. The approach is fully automated in the tool KoAT, supporting precise derivation of polynomial and exponential complexity bounds as well as variable growth bounds. Experimental evaluation demonstrates effectiveness on diverse nontrivial integer programs, significantly improving the completeness, precision, and practicality of automated complexity analysis.

This work investigates the robust satisfiability problem for Promise Constraint Satisfaction Problems (PCSPs), focusing on the expressive power and limitations of semidefinite programming (SDP) algorithms. For symmetric Boolean PCSPs, we establish a complete computational complexity classification. We prove that SDP achieves robust approximation precisely when the PCSP admits either a majority or an alternating threshold polymorphism; moreover, we provide the first algebraic necessary and sufficient condition—formulated via minion homomorphisms—for SDP feasibility to imply exact satisfiability. Innovatively, we introduce spherical Ramsey theory into PCSP analysis, revealing a deep connection between SDP integrality gaps and spherical coloring unsatisfiability. This yields the first algebraic-geometric method for proving SDP hardness gaps. Our results unify and extend robust satisfiability theory to the promise setting, delivering systematic criteria for SDP tractability in PCSPs and proposing a central conjecture governing its scope.

This work addresses the long-standing bottleneck in prophet inequality research—manual construction of worst-case instances for tight competitive ratio proofs—by proposing the first unified framework that directly models tightness analysis as a computationally tractable optimization problem. Methodologically, it formalizes worst-case instance search as a convex optimization and linear programming problem subject to extremal probability distribution constraints, integrating tools from random-order theory and extremal probability analysis, thereby replacing the traditional decoupled paradigm of “algorithm analysis + counterexample construction.” Contributions include: (i) the first automated computation of tight competitive ratios; (ii) a unified derivation of tight bounds for multiple prophet inequality variants, yielding several new results; and (iii) rigorous verification of the optimality of several classical bounds. The framework significantly enhances the systematicity, scalability, and reliability of tightness proofs in online stochastic optimization.

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 paper investigates the computational power and complexity boundaries of families of nonuniform polynomial-size nondeterministic finite automata (NFA-poly) with respect to partial counting functions, gap functions, and their associated promise decision problems. Using state-complexity analysis and counting-complexity frameworks, it provides the first systematic characterization of the complexity hierarchy for NFA-poly without relying on unproven hardness assumptions. Key contributions include: (i) a strict separation between the counting class #NFA-poly and the gap class GapNFA-poly; (ii) an exact simulation equivalence between these classes and polynomially stack-bounded pushdown automata (PDA); and (iii) identification of novel conditions under which complexity collapses occur for promise problems. These results unify the theoretical connections between counting-based finite automata and resource-restricted PDAs, yielding a foundational stratification of automata-based complexity classes.

Existing large language models generate verifiable code through a sequential pipeline that separates program synthesis from proof construction, often yielding code that is difficult to verify or contains subtle bugs, necessitating costly and iterative repairs. Inspired by Dijkstra’s principle that programs and their correctness arguments should be co-developed, this work proposes the P³ framework, which introduces, for the first time, joint planning of programs and proofs: a unified plan is first derived from formal specifications, followed by synchronized generation of both implementation and proof. The contributions include a novel joint planning mechanism, the creation of Lean4Commit0—the first Lean4 verification benchmark grounded in real-world software libraries—and the integration of LLM agents, formal verification, and API-level specifications. P³ achieves state-of-the-art solve rates across Verina, AlgoVeri, and Lean4Commit0, outperforming the strongest baseline by 4.6–11.2 percentage points while reducing API invocation costs by ~40% and runtime by 37%.

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.