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Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics [Puk06], convex geometry [Kha96], fair allocations [AGSS16], combinatorics [AGV18], spectral graph theory [NST19a], network design, and random processes [KT12]. In an instance of a determinant maximization problem, we are given a collection of vectors U = {v1, . . . , vn} ⊂ Rd , and a goal is to pick a subset S ⊆ U of given vectors to maximize the determinant of the matrix ∑i∈S vivi^T. Often, the set S of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint (|S| ≤ k) or matroid constraint (S is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a r O(r)-approximation for any matroid of rank r ≤ d. This improves previous results that give e O(r^2)-approximation algorithms relying on e^O(r)-approximate estimation algorithms [NS16, AG17,AGV18, MNST20] for any r ≤ d. All previous results use convex relaxations and their relationship to stable polynomials and strongly log-concave polynomials. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the det(.) function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.
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This content will become publicly available on January 1, 2026
Web invariants for flamingo Specht modules
Webs yield an especially important realization of certain Specht modules, irreducible representations of symmetric groups, as they provide a pictorial basis with a convenient diagrammatic calculus. In recent work, the last three authors associated polynomials to noncrossing partitions without singleton blocks, so that the corresponding polynomials form a web basis of the pennant Specht module S(d,d,1n−2d). These polynomials were interpreted as global sections of a line bundle on a 2-step partial flag variety. Here, we both simplify and extend this construction. On the one hand, we show that these polynomials can alternatively be situated in the homogeneous coordinate ring of a Grassmannian, instead of a 2-step partial flag variety, and can be realized as tensor invariants of classical (but highly nonplanar) tensor diagrams. On the other hand, we extend these ideas from the pennant Specht module S(d,d,1n−2d) to more general flamingo Specht modules S(dr,1n−rd). In the hook case r=1, we obtain a spanning set that can be restricted to a basis in various ways. In the case r>2, we obtain a basis of a well-behaved subspace of S(dr,1n−rd), but not of the entire module.
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- Award ID(s):
- 2247089
- PAR ID:
- 10580795
- Publisher / Repository:
- Algebraic Combinatorics
- Date Published:
- Journal Name:
- Algebraic Combinatorics
- Volume:
- 8
- Issue:
- 1
- ISSN:
- 2589-5486
- Page Range / eLocation ID:
- 235 to 266
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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