Bit-Parallel Maximum Clique via 1024-bit Virtual Registers

Abstract

Stochastic Bit-Parallel Maximum Clique Solver (1024-bit Virtual Register) We introduce a stochastic bit-parallel solver for the Maximum Clique Problem (MCP) based on a 1024-bit virtual register architecture implemented as 16 contiguous uint64_t words in standard C++17, ensuring full portability across 64-bit platforms (x86-64, ARM, RISC-V). Core operations—candidate intersection, population count, and leading-zero detection—execute in exactly 16 instructions per 1024-bit operation. The solver integrates three key components: (i) a co-neighborhood heuristic that identifies high-coreness nodes via O(N²) pairwise popcount over 1024-bit adjacency rows; (ii) a stochastic swarm of independent worker threads; and (iii) greedy clique expansion through iterative bitwise intersection. Exact branch-and-bound solvers (MaxCliqueDyn, MCQ) become computationally intractable on dense random graphs such as G(1024, 0.5), where chromatic coloring bounds lose effectiveness and the search tree grows exponentially, requiring hours of computation on commodity hardware. The proposed method operates specifically within this hard regime, achieving 100% recovery of all 28 planted clique vertices in 153 milliseconds—a setting where exact state-of-the-art methods cannot remain competitive regardless of hardware scaling. Experimental validation was performed on a Qualcomm Snapdragon 8 Gen 2 (8-core ARM) and independently reproduced on Linux x86-64 server hardware. The solver requires no cloud infrastructure and no GPU acceleration. STATEMENT OF PRIOR ART AND LICENSE TERMS (PolyForm Noncommercial Framework) 1. Statement of Prior Art This document constitutes a public disclosure of the stochastic bit-parallel Maximum Clique methodology, including its virtual register architecture, heuristic structure, and execution model.The mathematical and algorithmic concepts are released solely to establish Prior Art and prevent third-party patent claims under 35 U.S.C. § 102 and international equivalents. 2. Software License While the conceptual methods are disclosed defensively, all source code, implementations, binaries, and hardware realizations are not in the public domain and are licensed under the PolyForm Noncommercial License 1.0.0. Permitted (Non-Commercial)• Academic research and experimentation• Peer review and independent verification• Educational and non-profit use• Non-commercial open-source research implementations Condition: Publications must cite the canonical DOI or primary reference. 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The following activities are permitted on a non-commercial basis: • Research, benchmarking, and experimental validation• Publication of scientific analyses, comparisons, or extensions• Use in university courses, laboratories, and academic projects• Inclusion in research solver portfolios• Independent theoretical or empirical study All academic use must include proper citation to the original work.