hardness amplification

This paper addresses the optimization of $n$-ary functions over integer boxes. We propose a reducibility-based compression method grounded in *equivalent small-domain substitution*: leveraging integer lattice theory and domain reduction, we construct an equivalent function with a significantly smaller domain, thereby upgrading originally weakly polynomial-time algorithms to strongly polynomial-time guarantees. This work establishes the first unified framework for systematically strengthening a broad class of weakly polynomial algorithms to strong polynomial complexity, while simultaneously simplifying both the analysis and construction of reducibility bounds. By decoupling computational efficiency from input size—particularly the bit-length of coefficients—the method overcomes a fundamental scalability bottleneck. It yields a qualitative leap in computational efficiency for integer programming and discrete optimization, offering a novel paradigm for large-scale combinatorial optimization.

This paper resolves Yao’s (1979) open problem: proving that computing the deterministic communication complexity $ D(f) $ of a Boolean function $ f $ is NP-hard. It establishes this hardness for constant-round (i.e., constantly many alternations) communication protocols—surpassing prior results requiring unbounded rounds. The core techniques are: (1) constructing a self-similar gadget enabling recursive embedding, yielding a reusable, modular lower-bound tool; (2) introducing a relaxed interleaving lemma and a polynomial-time reduction framework; and (3) proving, under the Exponential Time Hypothesis (ETH), an unbounded additive inapproximability gap for $ D(f) $. These results not only confirm the inherent computational hardness of determining $ D(f) $, but also provide a new paradigm and structural foundation for subsequent approximation algorithms and lower-bound investigations in communication complexity.

The Strong Exponential Time Hypothesis (SETH) faces growing skepticism regarding its plausibility. Method: We propose a systematic framework for weakening SETH by introducing five natural, hierarchically ordered variants grounded in circuit complexity, SAT backtracking gates, graph width parameters, and weighted satisfiability. Contribution/Results: We establish the first five-level equivalence hierarchy among heterogeneous problems—including pathwidth/treewidth/tree-depth modulators, strong 2-SAT/Horn backtracking gates, and shallow-circuit SAT—revealing their fine-grained computational equivalence. Using parameterized reductions, circuit modeling, and graph-structural modularization, we prove that breaking brute-force algorithms for diverse classical problems is computationally equivalent under these assumptions. This unifies explanations of SETH-dependent phenomena and provides a more credible, layer-wise verifiable foundation for fine-grained conditional lower bounds.

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 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.

This work investigates lower bounds on the query complexity of Boolean functions in the noisy query model, where each query is flipped independently with a fixed probability. By analyzing degree statistics of specific subgraphs of the Boolean hypercube and integrating tools from Boolean function analysis, hypercube graph theory, and probabilistic methods, the authors introduce the first general lower bound framework that surpasses the trivial random-query baseline. This framework not only unifies and simplifies existing results but also resolves an open problem concerning the relationship between total influence $I(f)$ and noisy query complexity $N_p(f)$, establishing that $N_p(f) = \Omega(I(f) \log I(f))$. Furthermore, it yields tight bounds for several new functions and recovers nearly all known lower bounds up to constant factors.

This work addresses #P-hard counting problems—such as counting independent sets in general graphs and #2-SAT—that are inapproximable in polynomial time and prohibitively expensive to solve exactly. The authors propose a novel framework based on bounded, unweighted self-reducibility, which recursively decomposes problem instances and aggregates upper bounds from subproblems at a square-root recursion depth. By integrating enumeration with a hybrid sampling estimator, the approach substantially reduces the base of the exponential time complexity. The method achieves improved runtimes of O*(1.1869ⁿ) for independent set counting and O*(1.2373ⁿ) for #2-SAT approximation, outperforming the best known exact algorithms. It further extends to counting maximum cliques, minimal separators, and perfect matchings in subcubic graphs, and admits black-box quantum speedup.

This study investigates the average-case complexity of refuting "clique-free" instances in random dense graphs, analyzed through the lenses of proof complexity and communication complexity. For binary-encoded clique formulas, it establishes the first exponential lower bounds on refutation length for Cutting Planes and bounded-depth Frege systems augmented with parity axioms in the average case. Concurrently, it demonstrates that the randomized communication complexity required to identify a violated clause remains polynomially bounded. These results reveal a stark separation between proof length and communication cost for such formulas, underscoring how structural restrictions inherent to the average-case setting fundamentally constrain the power of proof systems.

Since the introduction of the Ideal Proof System (IPS) by Grochow and Pitassi (J. ACM 2018), a substantial body of work has established size lower bounds for IPS and its fragments. In particular, Forbes, Shpilka, Tzameret, and Wigderson (Theory Comput. 2021) developed the main lower-bound frameworks for restricted IPS fragments, namely functional lower bounds and the hard multiples method, while Alekseev, Grigoriev, Hirsch, and Tzameret (SIAM J. Comput. 2024) gave a general template for conditional lower bounds for full IPS. Yet all these lower bounds apply only to purely algebraic formulas over a field, that is, non-Boolean formulas not directly expressible in propositional logic. Proving lower bounds for CNF formulas has therefore remained a central open problem in this line of work. The current work resolves this question for IPS over read-once oblivious algebraic branching programs (roABPs) by proving lower bounds for refutations of CNF formulas in this system. Our approach is a rank-based feasible interpolation argument, following the method of Pudlák and Sgall (Proof Complexity and Feasible Arithmetic 1996) for monotone span programs, in which decomposing a given roABP refutation along a variable partition yields a low-dimensional space of polynomials from which we construct a span-program interpolant. We extend their result from Nullstellensatz refutations measured by degree to Nullstellensatz refutations measured by roABP size (i.e., roABP-IPS$_\text{LIN}$).