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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? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$ Weisfeiler-Leman Is Incomplete on Simple Spectrum Graphs, so Canonicalize Them A superlinear improvement on line-free sets in $\mathbb{F}_p^3$ Central limit theorems for high dimensional lattice polytopes: cosmological polytopes Finding blowups one vertex at a time Maximum Probability of Independence in Transitive Matroids Higher Rank-Support Weights and q-Polymatroids Localized Turán-type inequalities for $Q$-index Functional codes arising from rank $n$ Hermitian varieties and hypersurfaces in low dimensions Generalizing OOOOOOB Complex Representations of Groups and Involutions of its Automorphisms Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs The Closure of LCD-to-GI Reductions via Generalized Inner Products Sets of large values of polynomial multi-correlation functions Polylogarithmic Full-Chord Buffon Discrepancy Star-Shaped Integral Cartan-Type Matrices and an Egyptian-Fraction Classification of Affine Weighted Trees Symplectic lattice counting and zeta functions of higher Heisenberg groups Concise and elegant proofs of three formulas for complete Bell polynomials On Reed-Muller subcodes, Grassmannian partitions and sum-free functions Holographic functions and neural networks An Ore-type condition for $H$-tilings in graphs From Halin's Edge Removability to Matching Removability in $k$-Connected Graphs Reciprocals of Partition Polynomials A Comprehensive Study of Clique Graphs and Clique Regular Graphs Targeting Clause Type Distributions: a Picklock for Random Satisfiability Problems Hypercube geodesics with few colour changes Every signed planar graph is $5$-choosable: A short proof and refinements The Symmetries of Three-Layer ReLU Networks The Geometry of Cooperative Game Solutions: Stratified Egalitarian Shapley Values An Axiomatic Theory of Tie-Breaking: Impossibility, Characterization, and Decomposition Every Minimal Counterexample to the Erdős-Gyárfás Conjecture is Predominantly Cubic TriVAL: A Tri-Validation Framework for Faithful Automatic Optimization Modeling Learning Minimally Rigid Graphs with High Realization Counts Geometry of Rényi Entropy on the Majorization Lattice On $\ell$-weakly cross $t$-intersecting families for sets and vector spaces Invariant-Based Diagnostics for Graph Benchmarks Thresholds for Tic-Tac-Toe on Finite Affine Spaces Almost-Orthogonality in Lp Spaces: A Case Study with Grok On the Architectural Complexity of Neural Networks Most ReLU Networks Admit Identifiable Parameters A Partition-Based Generating Function for Row-Convex Polyominoes Word-Representability of Shift Graphs Trees and Graphs with Non Log-concave Dominating Set Sequence via AI Tools The Banach-Butterfly Invariant: Influence-Adaptive Walsh Geometry for Ternary Polynomial Threshold Functions New Bounds for Zarankiewicz Numbers via Reinforced LLM Evolutionary Search Order-Sensitive Sequential Interventions on Ideal Lattices Mathematical Foundations for Peer-to-Peer Lattice Computation Doubly Saturated Ramsey Graphs: A Case Study in Computer-Assisted Mathematical Discovery A weighted angle distance on strings Achieving the Kesten-Stigum bound in the non-uniform hypergraph stochastic block model A Complete Symmetry Classification of Shallow ReLU Networks Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks Adaptive Threshold-Driven Continuous Greedy Method for Scalable Submodular Optimization A semicontinuous relaxation of Saito's criterion and freeness as angular minimization Representing Higher-Order Networks: A Survey of Graph-Based Frameworks Reinforced Generation of Combinatorial Structures: Ramsey Numbers Two Dimensional Subtraction -- Transfer Games Almost All Vectorial Functions Have Trivial Extended-Affine Stabilizers Split-Twin Extensions Preserving Seymour Vertices Three formulas for CSM classes of open quiver loci Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs The existence of unexpected automorphisms in direct product graphs When are Hopf algebras determined by integer sequences? Self-affine quadrangles Pointwise order of generalized Hofstadter functions G, H and beyond Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler Characteristic A Mathematical Reconstruction of Endothelial Cell Networks Progressive and Rushed Dyck Paths Symmetric noncrossing partitions of an annulus with double points Improved Bounds for Multicovering Hypergraphs The pre-Pieri rules
Finding a Shortest Even Hole in Polynomial Time
Hou-Teng Cheong, Hsueh-I Lu · 2020-08-15 · via math.CO updates on arXiv.org

An even (respectively, odd) hole in a graph is an induced cycle with even (respectively, odd) length that is at least four. Bienstock [DM 1991 and 1992] proved that detecting an even (respectively, odd) hole containing a given vertex is NP-complete. Conforti, Chornuéjols, Kappor, and Vušković [FOCS 1997] gave the first known polynomial-time algorithm to determine whether a graph contains even holes. Chudnovsky, Kawarabayashi, and Seymour [JGT 2005] estimated that Conforti et al.'s algorithm runs in $O(n^{40})$ time on an $n$-vertex graph and reduced the required time to $O(n^{31})$. Subsequently, da~Silva and Vušković~[JCTB 2013], Chang and Lu [JCTB 2017], and Lai, Lu, and Thorup [STOC 2020] improved the time to $O(n^{19})$, $O(n^{11})$, and $O(n^9)$, respectively. The tractability of determining whether a graph contains odd holes has been open for decades until the algorithm of Chudnovsky, Scott, Seymour, and Spirkl [JACM 2020] that runs in $O(n^9)$ time, which Lai et al. also reduced to $O(n^8)$. By extending Chudnovsky et al.'s techniques for detecting odd holes, Chudnovsky, Scott, and Seymour [Combinatorica 2020 to appear] (respectively, [arXiv 2020]) ensured the tractability of finding a long (respectively, shortest) odd hole. They also ensured the NP-hardness of finding a longest odd hole, whose reduction also works for finding a longest even hole. Recently, Cook and Seymour ensured the tractability of finding a long even hole. An intriguing missing piece is the tractability of finding a shortest even hole, left open for at least 15 years by, e.g., Chudnovsky et al. [JGT 2005] and Johnson [TALG 2005]. We resolve this long-standing open problem by giving the first known polynomial-time algorithm, running in $O(n^{31})$ time, for finding a shortest even hole in an $n$-vertex graph that contains even holes.