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1. Towards Nearly-linear Time Algorithms for Submodular Maximization with a Matroid Constraint

   Ene, Alina; Nguyen, Huy L. (July 2019, International Colloquium on Automata, Languages, and Programming)

   We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a $$1 - 1/e - \epsilon$$ approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, $$1 - 1/e - \epsilon$$ approximation, using a nearly-linear number of function evaluations and arithmetic operations. 

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2. Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint

   https://doi.org/10.4230/LIPIcs.ICALP.2019.54  

   Ene, Alina; Nguyen, Huy L. (January 2019, Leibniz international proceedings in informatics)

   We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations. 

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3. Purity and Separation for Oriented Matroids

   https://doi.org/10.1090/memo/1439  

   Galashin, Pavel; Postnikov, Alexander (September 2023, Memoirs of the American Mathematical Society)

   Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions are closely related to the theory of cluster algebras, to the combinatorics of the double Bruhat cells, and to the totally positive Grassmannian. A key feature, called the purity phenomenon, is that every maximal by inclusion strongly (resp., weakly) separated collection of subsets in $[n]$ has the same cardinality. In this paper, we extend these notions and define $$\mathcal{M}$$-separated collections for any oriented matroid $$\mathcal{M}$$. We show that maximal by size $$\mathcal{M}$$-separated collections are in bijection with fine zonotopal tilings (if $$\mathcal{M}$$ is a realizable oriented matroid), or with one-element liftings of $$\mathcal{M}$$ in general position (for an arbitrary oriented matroid). We introduce the class of pure oriented matroids for which the purity phenomenon holds: an oriented matroid $$\mathcal{M}$$ is pure if $$\mathcal{M}$$-separated collections form a pure simplicial complex, i.e., any maximal by inclusion $$\mathcal{M}$$-separated collection is also maximal by size. We pay closer attention to several special classes of oriented matroids: oriented matroids of rank $$3$$, graphical oriented matroids, and uniform oriented matroids. We classify pure oriented matroids in these cases. An oriented matroid of rank $$3$$ is pure if and only if it is a positroid (up to reorienting and relabeling its ground set). A graphical oriented matroid is pure if and only if its underlying graph is an outerplanar graph, that is, a subgraph of a triangulation of an $$n$$-gon. We give a simple conjectural characterization of pure oriented matroids by forbidden minors and prove it for the above classes of matroids (rank $$3$$, graphical, uniform). 

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4. Ehrhart theory of paving and panhandle matroids

   https://doi.org/10.1515/advgeom-2023-0020  

   Hanely, Derek; Martin, Jeremy L.; McGinnis, Daniel; Miyata, Dane; Nasr, George D.; Vindas-Meléndez, Andrés R.; Yin, Mei (October 2023, Advances in Geometry)

   Abstract We show that the base polytopeP_Mof any paving matroidMcan be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial ofP_M, starting with Katzman’s formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni’s work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation ofstressed-hyperplane relaxationintroduced by Ferroni, Nasr and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent. 

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5. Stellahedral geometry of matroids

   https://doi.org/10.1017/fmp.2023.24  

   Eur, Christopher; Huh, June; Larson, Matt (January 2023, Forum of Mathematics, Pi)

   Abstract We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety. 

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