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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? 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Advances in Algorithmic Meta Theorems
Sebastian Siebertz, Alexandre Vigny · 2024-11-23 · via math.CO updates on arXiv.org

Tractability results for the model checking problem of logics yield powerful algorithmic meta theorems of the form: Every computational problem expressible in a logic $L$ can be solved efficiently on every class $\mathscr{C}$ of structures satisfying certain conditions. The most prominent logics studied in the field are (counting) monadic second-order logic (C)MSO, and first-order logic FO and its extensions. The complexity of CMSO model checking in general and of FO model checking on monotone graph classes is very well understood. In recent years there has been a rapid and exciting development of new algorithmic meta theorems. On the one hand there has been major progress for FO model checking on hereditary graph classes. This progress was driven by the development of a combinatorial structure theory for the logically defined monadically stable and monadically dependent graph classes, as well as by the advent of the new width measure twinwidth. On the other hand, new algorithmic meta theorems for new logics with expressive power between FO and CMSO offer a new unifying view on methods like the irrelevant vertex technique and recursive understanding. In this paper we overview the recent advances in algorithmic meta theorems and provide rough sketches for the methods to prove them.