Unconstrained MAP Inference, Exponentiated Determinantal Point Processes, and Exponential Inapproximability

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 unconstrained MAP inference and the normalizing constant for E-DPPs. Unconstrained MAP inference for an n x n matrix is NP-hard to approximate within a 2(beta n)-factor, where beta = 10(-1013). This result improves upon a (9/8 - epsilon)-factor inapproximability given by Kulesza and Taskar (2012). The normalizing constant for E-DPPs of any (fixed) constant exponent p >= beta(-1) = 10(1013) is NP-hard to approximate within a 2(beta pn)-factor. This gives a(nother) negative answer to open questions posed by Kulesza and Taskar (2012); Ohsaka and Matsuoka (2020).