identify equivalence classes

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This paper addresses the logical definability of structured graph decompositions—specifically modular, split, and biconnected decompositions—and their generalizations to weak partition systems and weak bipartition systems. Methodologically, it introduces a unified logical characterization framework based on Counting Monadic Second-Order logic (CMSO) and its two-sorted extension C2MSO, integrating weak partition system theory with the algebraic properties of graph decompositions to devise multiple efficient CMSO translation schemes. Crucially, it achieves CMSO definability for all three classical decompositions without relying on order-invariant MSO—a first—and extends Courcelle’s seminal result to weaker combinatorial systems. The main contributions are: (i) the first unified CMSO-definability framework for these decompositions; (ii) elimination of dependence on stronger logics such as MSO with built-in order; (iii) significantly improved logical expressiveness and translation efficiency; and (iv) a more concise and broadly applicable logical foundation for the metatheory of graph algorithms.

本文通过引入程序路径和转换系统来分析BSS RAMs在初等结构上的决策过程，并提供算法枚举有限的程序路径。

This work addresses structural characterization and efficient sparsification for graph classes of bounded tree rank. Prior approaches face challenges including intractable decomposition, non-computable inverse explanations, and the absence of a unifying sparsification framework. To overcome these, we introduce a novel paradigm grounded in graph logic and structural decomposition. First, we provide a complete, interpretable characterization of graphs with tree rank at most two. Second, we design a computable decomposition framework that uniformly captures both bounded-tree-rank and bounded-expansion graph classes, supporting polynomial-time invertible explanation recovery. Third, we present the first polynomial-time sparsification algorithm for the class of interpretable graphs with tree rank ≤ 2. Finally, we generalize the seminal result of Gajarský et al. — originally restricted to bounded-degree graphs — to the broader class of bounded-expansion graphs, substantially enhancing both applicability and constructiveness.

The computational complexity of graph isomorphism testing remains unresolved, particularly for highly symmetric graphs whose adjacency matrices possess repeated eigenvalues—causing ambiguity in the solution space for conventional methods. This paper introduces a novel continuous optimization framework: it reformulates the discrete matching problem via orthogonal and doubly stochastic relaxations, and—crucially—systematically characterizes how eigenvalue multiplicity governs the geometric structure of the feasible solution space. Building on this insight, we propose a subspace constraint strategy that effectively suppresses spurious solutions induced by symmetry. Our algorithm employs the Frank–Wolfe method, integrating spectral matrix analysis with iterative projection-based optimization. Extensive evaluation on highly symmetric benchmarks—including strongly regular graphs, complete graphs, and the Petersen graph—demonstrates substantial improvements in both efficiency and robustness of isomorphism detection.

This work addresses the challenge of unifying the representation of conditional independence structures induced by feedback, latent variables, and selection mechanisms. It proposes a class of separable graphical models based on mixed graphs containing directed, undirected, and bidirected edges, where the absence of an edge corresponds to a separating set between its endpoints. By introducing separable graphs and their essential forms, the framework subsumes several existing graphical models. The study establishes an equivalence among graph structure, separation properties, and canonical parametrization, and leverages this correspondence to design an algorithm for identifying equivalence classes. Under mild assumptions, the algorithm consistently recovers the separation-equivalence class of a separable graph, thereby providing both theoretical foundations and computational tools for modeling and learning complex dependency structures.

This study addresses the lack of systematic analysis regarding the normativity and minimality of weak simulation equivalence and coupled similarity—two simulation-induced behavioral equivalences. For the first time, the theory of canonical and minimal quotients is extended to these equivalences. The work proposes an abstraction construction based on labeled transition systems (LTSs), integrating simulation preorders with complexity-theoretic analysis to establish the existence of unique canonical representatives and achieve simultaneous minimization of states and transitions. Furthermore, it proves that the associated minimization problems are NP-complete, thereby filling a significant theoretical gap in the field.

This study addresses the problem of efficiently recovering a concrete linear isometry from a decision oracle for linear code equivalence. The proposed method integrates polynomial-time algorithms, oracle query techniques, and orbit membership testing from linear algebra and group theory to progressively reconstruct the permutation and diagonal components via successive queries to a decision oracle. This work establishes the first proof that the search variant of the linear code equivalence problem admits a polynomial-time reduction to its decision counterpart, accompanied by an explicit reconstruction procedure. Specifically, the deterministic algorithm requires only O(n²) oracle calls to successfully recover the monomial equivalence between matrix representations, thereby achieving an efficient transition from decision to search.

This work addresses the challenge of efficiently reducing the search version to the decision version for Linear Code Equivalence (LCE) and Generalized Code Equivalence (GCE). Inspired by recent advances in permutation code equivalence reductions, it proposes the first deterministic polynomial-time search-to-decision reduction framework applicable to both LCE and GCE. The approach leverages a decision oracle to recover the permutation component and employs the Engel–Schneider algorithm to efficiently reconstruct the diagonal scaling and field automorphism components, thereby fully recovering the equivalence transformation. This contribution not only extends the applicability of classical reduction techniques in coding theory but also significantly enhances the computational efficiency of solving LCE and GCE problems.

This study addresses the lack of polyhedral theory for the multi-separator problem in image segmentation by formulating the feasible solution polytope via integer linear programming. Leveraging graph-theoretic conditions, we characterize its facets and strengthen associated inequalities. The work provides a complete description of canonical facets, establishes a totally dual integral formulation for path instances, and reveals geometric projection relationships with Boolean quadric and lifted multicut polytopes. These contributions fill critical theoretical gaps by achieving a comprehensive characterization of multi-separator polytope facets and inequality strengthening. Ultimately, this research offers robust theoretical foundations and novel perspectives for combinatorial optimization in image segmentation, advancing the mathematical understanding of partition-based vision problems through rigorous polyhedral analysis.