Log In Sign Up

Efficient Determinant Maximization for All Matroids

by   Adam Brown, et al.

Determinant maximization provides an elegant generalization of problems in many areas, including convex geometry, statistics, machine learning, fair allocation of goods, and network design. In an instance of the determinant maximization problem, we are given a collection of vectors v_1,…, v_n ∈ℝ^d, and the goal is to pick a subset S⊆ [n] of given vectors to maximize the determinant of the matrix ∑_i ∈ S v_iv_i^⊤, where the picked set of vectors S must satisfy some combinatorial constraint such as cardinality constraint (|S| ≤ k) or matroid constraint (S is a basis of a matroid defined on [n]). In this work, we give a combinatorial algorithm for the determinant maximization problem under a matroid constraint that achieves O(d^O(d))-approximation for any matroid of rank r≥ d. This complements the recent result of <cit.> that achieves a similar bound for matroids of rank r≤ d, relying on a geometric interpretation of the determinant. Our result matches the best-known estimation algorithms <cit.> for the problem, which could estimate the objective value but could not give an approximate solution with a similar guarantee. Our work follows the framework developed by <cit.> of using matroid intersection based algorithms for determinant maximization. To overcome the lack of a simple geometric interpretation of the objective when r ≥ d, our approach combines ideas from combinatorial optimization with algebraic properties of the determinant. We also critically use the properties of a convex programming relaxation of the problem introduced by <cit.>.


page 1

page 2

page 3

page 4


Determinant Maximization via Matroid Intersection Algorithms

Determinant maximization problem gives a general framework that models p...

Maximizing Determinants under Matroid Constraints

Given vectors v_1,…,v_n∈ℝ^d and a matroid M=([n],I), we study the proble...

Maximizing Non-Monotone DR-Submodular Functions with Cardinality Constraints

We consider the problem of maximizing a non-monotone DR-submodular funct...

Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration

Several fundamental problems that arise in optimization and computer sci...

On the Complexity of Dynamic Submodular Maximization

We study dynamic algorithms for the problem of maximizing a monotone sub...

Optimal matroid bases with intersection constraints: Valuated matroids, M-convex functions, and their applications

For two matroids M_1 and M_2 with the same ground set V and two cost fun...

Circuit Lower Bounds for the p-Spin Optimization Problem

We consider the problem of finding a near ground state of a p-spin model...