reductions to communication complexity

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.

This paper addresses the fundamental open question of P vs NP. Method: It introduces the novel “computation-as-game” paradigm, modeling any computational problem as a two-player zero-sum game between an algorithm and nature: the algorithm asymptotically refines solutions within a Scott domain, while nature imposes penalties based on bias; correctness is defined as Nash equilibrium in the limit. Complexity classes—including P and NP—are newly characterized via the existence of Nash equilibria under specific informational and temporal constraints; P = NP is thus recast as the equivalence of equilibrium existence under these two constraint regimes. Contribution/Results: The framework unifies domain theory, game theory, and computational complexity theory, yielding a game-theoretic, semantic, and structurally principled characterization of complexity classes—providing a novel analytical pathway for understanding the nature of computation and the P vs NP problem.

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 study investigates the intrinsic connection between finite-time reachability in dynamical systems and computational complexity. By introducing a family of decision problems termed “telic problems,” which formalize reachability under coarse-grained state-space representations, the work establishes—via polynomial-time reductions and topological entropy analysis—the first direct link between algorithmic time lower bounds (e.g., exponential time) and positive topological entropy in dynamical systems. The paper proposes a novel paradigm for classifying dynamical systems based on algorithmic complexity: if the telic problem associated with a system is solvable only by exponential-time algorithms, then the system necessarily exhibits positive topological entropy. This result provides a computationally grounded characterization of complexity in dynamical systems, bridging theoretical computer science and dynamical systems theory.

This paper investigates the computational complexity of Arc-Kayles and its “non-disconnecting” variant—where each move must preserve graph connectivity after removing adjacent vertices. Using combinatorial game theory, structural graph analysis, and PSPACE-completeness reductions, it systematically delineates complexity boundaries across graph classes. The authors establish the first polynomial-time algorithms for non-disconnecting Arc-Kayles on cycles, clique trees, and several subclasses of threshold graphs. Conversely, they prove PSPACE-completeness on split graphs and all bipartite graphs with even girth. Furthermore, they characterize second-player winnability in standard Arc-Kayles via a precise graph isomorphism condition, yielding a tight equivalence between game solvability and the graph isomorphism problem. Collectively, these results unify and extend the tractability theory of impartial games on structured graph families.

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.