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This content will become publicly available on October 31, 2023

Title: Determinant Maximization via Matroid Intersection Algorithms
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 more » 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. « less
Authors:
; ; ; ;
Award ID(s):
1910423 2106444
Publication Date:
NSF-PAR ID:
10362731
Journal Name:
Annual Symposium on Foundations of Computer Science
ISSN:
0272-5428
Sponsoring Org:
National Science Foundation
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