Exponential Inapproximability of Selecting a Maximum Volume Sub-matrix

On selecting a maximum volume sub-matrix of a matrix and related problems

Given a matrix A ∈ Rm×n (n vectors in m dimensions), we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible. This notion turned out to be important in low-rank approximations to matrices and rank revealing QR factorizations which have been investigated in the linear algebra community and can be quantified in a few different wa...

Finding Maximum Volume Sub-matrices of a Matrix

Given a matrix A ∈ Rm×n (n vectors in m dimensions), we consider the problem of selecting a submatrix (subset of the columns) with maximum volume. The motivation to study such a problem is that if A can be approximately reconstructed from a small number k of its columns (A has “numerical” rank k), then any set of k independent columns of A should suffice to reconstruct A. However, numerical sta...

Braess's Paradox, Fibonacci Numbers, and Exponential Inapproximability

We give the first analyses in multicommodity networks of both the worst-case severity of Braess’s Paradox and the price of anarchy of selfish routing with respect to the maximum latency. Our first main result is a construction of an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both of these quantities grow exponentially with the size of the network. This...

On the inapproximability of maximum intersection problems

Inapproximability of Matrix $p \rightarrow q$ Norms

This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = ∞, q = 1), and has been widely studied in various regimes. When p ≥ q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 ∈ [q, p], and the problem is hard to approximate within almost polynomial factors when 2 / ∈ [q, p]. The regime when p < q, known as hy...