Some Inapproximability Results of MAP Inference and Exponentiated Determinantal Point Processes
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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. 1. Unconstrained MAP inference for an n × n matrix is NP-hard to approximate within a factor of 2^β n, where β = 10^-10^13. This result improves upon a (9/8-ϵ)-factor inapproximability given by Kulesza and Taskar (2012). 2. 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. 3. 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^β pn. This gives a(nother) negative answer to open questions posed by Kulesza and Taskar (2012); Ohsaka and Matsuoka (2020).
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