Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks

Diarmaid Conaty, Denis D. Maua, Cassio P. de Campos
Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence, PMLR R15:111-120, 2017.

Abstract

We discuss the computational complexity of approximating maximum a posteriori infer- ence in sum-product networks. We first show NP-hardness in trees of height two by a reduc- tion from maximum independent set; this im- plies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to ap- proximate the problem within a factor 2f(n) for any sublinear function f of the size of the input n. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor 2c\cdotn for some constant c < 1. Last, we present a sim- ple algorithm, and show that it provably pro- duces solutions at least as good as, and poten- tially much better than, the max-product algo- rithm. We empirically analyze the proposed algorithm against max-product using synthetic and real-world data.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR15-conaty17a, title = {Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks}, author = {Conaty, Diarmaid and Maua, Denis D. and de Campos, Cassio P.}, booktitle = {Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence}, pages = {111--120}, year = {2017}, editor = {Elidan, Gal and Kersting, Kristian}, volume = {R15}, series = {Proceedings of Machine Learning Research}, month = {11--15 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r15/main/assets/conaty17a/conaty17a.pdf}, url = {https://proceedings.mlr.press/r15/conaty17a.html}, abstract = {We discuss the computational complexity of approximating maximum a posteriori infer- ence in sum-product networks. We first show NP-hardness in trees of height two by a reduc- tion from maximum independent set; this im- plies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to ap- proximate the problem within a factor 2f(n) for any sublinear function f of the size of the input n. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor 2c\cdotn for some constant c < 1. Last, we present a sim- ple algorithm, and show that it provably pro- duces solutions at least as good as, and poten- tially much better than, the max-product algo- rithm. We empirically analyze the proposed algorithm against max-product using synthetic and real-world data.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks %A Diarmaid Conaty %A Denis D. Maua %A Cassio P. de Campos %B Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2017 %E Gal Elidan %E Kristian Kersting %F pmlr-vR15-conaty17a %I PMLR %P 111--120 %U https://proceedings.mlr.press/r15/conaty17a.html %V R15 %X We discuss the computational complexity of approximating maximum a posteriori infer- ence in sum-product networks. We first show NP-hardness in trees of height two by a reduc- tion from maximum independent set; this im- plies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to ap- proximate the problem within a factor 2f(n) for any sublinear function f of the size of the input n. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor 2c\cdotn for some constant c < 1. Last, we present a sim- ple algorithm, and show that it provably pro- duces solutions at least as good as, and poten- tially much better than, the max-product algo- rithm. We empirically analyze the proposed algorithm against max-product using synthetic and real-world data. %Z Reissued by PMLR on 04 October 2026.
APA
Conaty, D., Maua, D.D. & de Campos, C.P.. (2017). Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks. Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R15:111-120 Available from https://proceedings.mlr.press/r15/conaty17a.html. Reissued by PMLR on 04 October 2026.

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