factor-2 approximation algorithms

This work proposes an improved threshold rounding algorithm for Boolean MAX 2-CSP instances where each variable appears in at most $d$ constraints. By integrating semidefinite programming with a refined analysis based on graph degree constraints, the algorithm achieves—for the first time—a $\widetilde{\Omega}(1/d^4)$ improvement in approximation ratio across a broad class of such problems. Notably, for MAX 2-SAT, it attains an approximation factor of $(\beta^* + \widetilde{\Omega}(1/d^2))$, where $\beta^*$ denotes the optimal constant-factor approximation achievable without degree constraints. This result generalizes the bounded-degree MAX CUT guarantee of Hsieh and Kothari and provides a theoretical foundation for improved approximation algorithms for related problems such as MAX DI-CUT and MAX 2-AND.

This work addresses the long-standing challenge of achieving super-polynomial speedups for weighted NP-hard problems—such as the Traveling Salesman Problem (TSP), weighted Max-Cut, and edge-weighted k-Clique—whereas their unweighted counterparts have seen significant algorithmic advances. The authors introduce the doubling constant of the weight set as a key parameter and leverage a constructive Freiman theorem to compress arbitrary weights into polynomially bounded integers. By combining dynamic programming over (min, +) or (max, +) semirings with polynomial embedding techniques, they effectively reduce weighted instances to near-unweighted complexity regimes. When the weight set exhibits a small doubling constant, this approach yields substantially improved running times, breaking free from traditional pseudo-polynomial dependencies and establishing an efficient bridge between weighted and unweighted problem formulations.

This work addresses the approximability of NP-hard optimization problems by introducing a novel approximation notion that characterizes the intermediate approximability of problems defined via binary functions—situating them between polynomial-time solvability and the existence of a fully polynomial-time approximation scheme (FPTAS). Under the standard assumption that P ≠ NP, the authors rigorously establish through formal reductions and complexity-theoretic analysis that this newly defined approximation hierarchy is strictly stronger than FPTAS yet strictly weaker than polynomial-time solvability. This result fills a critical gap in the theory of approximation algorithms and offers a refined perspective on the boundary of approximability for NP-hard problems.

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.

This work addresses the approximability of Max-CSP, establishing for the first time an algorithmic hardness tightness framework applicable to **all satisfiable k-CSP instances**, thereby overcoming Raghavendra’s restriction to nearly satisfiable instances. We introduce the **mixed invariance principle**, which systematically links third-order correlations in discrete domains to expectations over hybrid Gaussian/Abelian group spaces—a novel connection. Our method combines Gaussian elimination with semidefinite programming to yield a hybrid approximation algorithm and constructs a perfectly complete “dictator vs. pseudorandom” test. For a broad class of predicates, we achieve optimal approximation ratios: the algorithm’s performance exactly matches the Unique Games Conjecture (UGC)-based hardness lower bounds, yielding tightness. This resolves the approximability threshold for these CSPs under UGC, unifying algorithm design and hardness analysis across the full spectrum of satisfiable instances.

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.

This work addresses the Boolean Max-k-CSP problem, which seeks to maximize the number of satisfied constraints in a Boolean constraint satisfaction instance where each constraint involves exactly k variables. We propose a polynomial-time algorithm that improves the best-known approximation ratio from 0.626612·k/2^k to k/2^k. This advancement is achieved by establishing a novel Gaussian comparison inequality and integrating techniques from coding theory originally developed to resolve the weak simplex conjecture. Under the Unique Games Conjecture, our result nearly matches the known NP-hardness threshold, thereby substantially advancing both the theoretical understanding and algorithmic performance for this fundamental optimization problem.

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.