The worst case of greedy volume maximization is Hadamard's maximal determinant
Year: 2026
Type: preprint
Abstract: The greedy algorithm for volume maximization picks k of a set of given vectors one at a time, each time taking the vector farthest from the span of those already picked. It is greedy MAP inference for determinantal point processes, and it makes the same choices as pivoted Cholesky factorization and column-pivoted QR. Çivril and Magdon-Ismail proved that greedy keeps at least a 1/k! fraction of the optimal volume and conjectured that the worst-case fraction is 1/2^(k-1). We show that it is 1/D(k) (approached, not attained), where D(k) is the largest determinant of a k×k matrix with entries ±1; D(k) = k^(k/2) whenever a Hadamard matrix of order k exists. After rescaling by greedy's distances, greedy's vectors form a unit triangular matrix and the optimal vectors form a matrix with entries in [-1,1], whose determinant is at most D(k); geometrically separated scales approach this bound. Since D(4) = 16 > 8, the conjecture fails for every k ≥ 4, even for vectors of nearly equal length, while it holds for k ≤ 3. The record includes the paper, its LaTeX source, and the code and certificates that reproduce every computed result.
Authors: Baliga, Kundan, Xie, Aiden, Byju, Aditya
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225468479