Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes

Abstract

We study the computational complexity of two hard problems on determinantal point processes (DPPs). One is maximum a posteriori (MAP) inference, i.e., to find a principal submatrix having the maximum determinant. The other is probabilistic inference on exponentiated DPPs (E-DPPs), which can sharpen or weaken the diversity preference of DPPs with an exponent parameter p. We prove the following complexity-theoretic hardness results that explain the difficulty in approximating MAP inference and the normalizing constant for E-DPPs.• Unconstrained MAP inference for an n × n matrix is NP-hard to approximate within a factor of 2 βn , where β = 10 −10 13 . This result improves upon a ( 9 8 − )-factor inapproximability given by Kulesza and Taskar (2012).• Log-determinant maximization is NP-hard to approximate within a factor of 5 4 for the unconstrained case and within a factor of 1 + 10 −10 13 for the size-constrained monotone case.• The normalizing constant for E-DPPs of any (fixed) constant exponent p ≥ β −1 = 10 10 13 is NP-hard to approximate within a factor of 2 βpn . This gives a(nother) negative answer to open questions posed by ; Ohsaka and Matsuoka (2020).