derive fine-grained lower bounds

This work investigates, under the assumption that P ≠ NP, whether NP-complete problems admit algorithms substantially faster than naïve brute-force search and whether current best-known algorithms are already optimal. By integrating fine-grained complexity theory, algebraic techniques, extremal and additive combinatorics, cryptography, and conditional hypotheses such as the Strong Exponential Time Hypothesis (SETH), the project establishes a unified framework for deriving conditional time lower bounds for NP-complete problems. Through a systematic synthesis of classical and recent results, and by leveraging reductions and combinatorial analyses, the study provides strong evidence for the hardness of improving existing algorithms for several canonical NP-complete problems, thereby advancing our understanding of the fine-grained structure of computational complexity.

This work addresses the longstanding barrier in complexity theory concerning non-uniform lower bound proofs, focusing on three fundamental measures: monotone circuit size, matrix rigidity, and tensor rank. Leveraging the Nondeterministic Strong Exponential Time Hypothesis (NSETH) and its variants, we establish the first unified framework that systematically translates co-nondeterministic time lower bounds—such as the impossibility of solving k-SAT or MAX-3-SAT in O(2^{(1/2+ε)n}) co-nondeterministic time—into constructive combinatorial lower bounds. Our approach integrates pseudorandomness, circuit complexity, algebraic analysis, and explicit construction techniques. Key results include: assuming no O(2^{(1/2+ε)n}) co-nondeterministic algorithm for k-SAT, there exists an explicit family of Boolean functions requiring monotone circuits of size 2^{Ω(n/log n)}; moreover, we obtain explicit families of highly rigid matrices and high-rank 3D tensors. This reveals a “win-win” lower-bound structure, simultaneously advancing all three central complexity measures.

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 establishes the first polynomial sensitivity lower bounds for randomized approximation algorithms solving constraint satisfaction problems (CSPs), filling a key theoretical gap. To overcome the limitation of classical lower-bound techniques—which fail to preserve sensitivity—the authors innovatively adapt the PCP framework into a sensitivity-preserving variant, integrating Hamming distance metrics and analysis within the LOCAL model of distributed computing. The results yield tight polynomial sensitivity lower bounds for fundamental problems including Maximum Clique, Minimum Vertex Cover, and Maximum Cut. Concurrently, they imply tight round-complexity lower bounds for these problems in the LOCAL model. This is the first systematic demonstration of a deep connection between algorithmic sensitivity and distributed computational complexity, laying the foundation for a unified theory bridging the robustness of approximation algorithms and the scalability of distributed computation.

Constructing exponential lower bounds on proof length in intuitionistic implication logic has been notoriously complex and difficult to comprehend. Method: Building upon Gordeev and Haeusler’s directed acyclic graph (DAG)-based natural deduction system, we introduce a significantly simplified proof technique. By carefully designing a family of propositional formulas and their combinatorial encoding into DAG structures, coupled with refined reduction analysis, we establish a tight exponential lower bound of $2^{Omega(n)}$. Contribution/Results: Our approach preserves rigor and generality while markedly enhancing conceptual clarity and technical reusability. It reduces the structural complexity of prior constructions and yields a more transparent, broadly applicable paradigm for proof complexity analysis in intuitionistic logic—offering a concise, scalable framework that facilitates further theoretical development and cross-system adaptation.

This study addresses the problem of verifying correctness preservation under natural reductions in parameterized concurrent programs: given a program template and (semi-)commutativity relations, it determines whether the reduction maintains correctness. The work proposes the first systematic framework that characterizes the semantics of natural reductions by introducing atomic blocks and global convergence points to simplify verification. The main contributions include a polynomial-time complete decision algorithm for the synchronization-free setting, and a proof that the problem becomes coNP-hard in the presence of synchronization primitives such as locks. Furthermore, the paper establishes general complexity lower bounds dependent on the synchronization mechanism, revealing that even simple forms of synchronization induce substantial computational hardness.

This work addresses the limited accuracy of upper bound estimation for the Maximum Clique Problem (MCP) by proposing a novel framework that integrates upper bounding functions with graph reduction techniques. The approach introduces adjustable reduction rules—namely $(k,\omega^u)$-core, $(k,\omega^u)$-truss, and the more general $(k,d,\omega^u)$-truss—by embedding upper bound tests into core and truss decompositions for the first time, and further enhances pruning power through struction operations while maintaining computational efficiency. Experimental results demonstrate that the method significantly improves upon multiple classical upper bounds across 73 benchmark graphs. Notably, it reduces certified integer upper bounds to 73, 115, and 168 on the challenging DIMACS instances C500.9, C1000.9, and C2000.9, respectively, and achieves comparable accuracy faster than direct semidefinite programming (SDP) on sparse graphs.

This work investigates conditional lower bounds on the running time of the Global Label Minimum Cut problem under the Exponential Time Hypothesis (ETH). By constructing a deterministic reduction, the paper strengthens the previously known lower bound from $(np)^{o(\log n / (\log \log n)^2)}$ to $(np)^{o(\log n / \log \log n)}$. This improvement significantly sharpens the characterization of the problem’s computational hardness, demonstrating that it cannot be solved substantially faster than the current best-known algorithms. Consequently, the result provides a stronger conditional hardness guarantee within the framework of complexity theory, reinforcing the belief that near-optimal algorithms for this problem are unlikely to exist under standard complexity assumptions.

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.

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.