DRCC: Dimensionality Reduction as Complexity Collapse A Structural Framework for Controlled Reconstruction in Complex Search Spaces

Abstract

Dimensionality Reduction as Complexity Collapse (DRCC) is proposed as a structural framework for reducing the effective complexity of large search spaces through controlled reconstruction. Instead of focusing solely on computational resources such as time or memory, DRCC studies how structural information can be used to collapse classes of equivalent continuations and thereby reduce the dimensionality of a problem. The central idea is that many combinatorial and computational systems contain hidden structural regularities that allow large families of candidate solutions to be represented by a significantly smaller set of continuation classes. By exploiting these classes, reconstruction can be performed on a reduced representation while preserving essential problem structure. The framework introduces the concepts of fragments, continuation classes, reconstruction paths, and controlled collapse operators. These notions provide a unified language for analyzing how information propagates through a search process and how ambiguity can be reduced without exhaustive enumeration of all possibilities. DRCC is not presented as a replacement for classical complexity theory but as a complementary perspective that focuses on structural reduction. The framework is applicable to combinatorial optimization, search problems, reconstruction tasks, routing systems, and other domains in which the effective search space can be reduced through structural constraints. The paper develops the theoretical foundations of DRCC, formal definitions, illustrative examples, and discusses its relationship to dimensionality reduction, reconstruction theory, and computational complexity. Keywords: DRCC, Computational Complexity, Dimensionality Reduction, Reconstruction Theory, Search Spaces, Combinatorics.

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