Document Open Access Logo

On the Parameterized Intractability of Determinant Maximization

Author Naoto Ohsaka

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2022.46.pdf
  • Filesize: 0.88 MB
  • 16 pages

Document Identifiers

Author Details

Naoto Ohsaka
  • CyberAgent, Inc., Tokyo, Japan

Cite AsGet BibTex

Naoto Ohsaka. On the Parameterized Intractability of Determinant Maximization. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ISAAC.2022.46

Abstract

In the Determinant Maximization problem, given an n × n positive semi-definite matrix A in ℚ^{n × n} and an integer k, we are required to find a k × k principal submatrix of A having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (2006); i.e., a f(k)n^𝒪(1)-time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, we show that Determinant Maximization is solvable in polynomial time on tridiagonal matrices. Thereafter, we demonstrate the W[1]-hardness with respect to the rank r of an input matrix. Our result is stronger than Koutis' result in the sense that any k × k principal submatrix is singular whenever k > r. We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of 2^{-c√k} for some universal constant c > 0. Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov, Ramanujan, Saurab, and Zehavi (2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an ε-additive approximation algorithm that runs in ε^{-r²}⋅r^𝒪(r³)⋅n^𝒪(1) time for the rank r of an input matrix, provided that the diagonal entries are bounded.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Determinant maximization
  • Parameterized complexity
  • Approximability

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Amir Abboud, Kevin Lewi, and Ryan Williams. Losing weight by gaining edges. In ESA, pages 1-12, 2014. Google Scholar
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. Google Scholar
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. Google Scholar
  4. Rajendra Bhatia. Perturbation Bounds for Matrix Eigenvalues. SIAM, 2007. Google Scholar
  5. Erdem Bıyık, Kenneth Wang, Nima Anari, and Dorsa Sadigh. Batch active learning using determinantal point processes. CoRR, abs/1906.07975, 2019. URL: http://arxiv.org/abs/1906.07975.
  6. Alexei Borodin and Eric M. Rains. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys., 121(3-4):291-317, 2005. Google Scholar
  7. L. Elisa Celis, Vijay Keswani, Damian Straszak, Amit Deshpande, Tarun Kathuria, and Nisheeth K. Vishnoi. Fair and diverse DPP-based data summarization. In ICML, pages 715-724, 2018. Google Scholar
  8. Wei-Lun Chao, Boqing Gong, Kristen Grauman, and Fei Sha. Large-margin determinantal point processes. In UAI, pages 191-200, 2015. Google Scholar
  9. Diego Cifuentes and Pablo A. Parrilo. An efficient tree decomposition method for permanents and mixed discriminants. Linear Algebra Appl., 493:45-81, 2016. Google Scholar
  10. Ali Çivril and Malik Magdon-Ismail. On selecting a maximum volume sub-matrix of a matrix and related problems. Theor. Comput. Sci., 410(47-49):4801-4811, 2009. Google Scholar
  11. Ali Çivril and Malik Magdon-Ismail. Exponential inapproximability of selecting a maximum volume sub-matrix. Algorithmica, 65(1):159-176, 2013. Google Scholar
  12. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Appl. Math., 108(1-2):23-52, 2001. Google Scholar
  13. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  14. Marco Di Summa, Friedrich Eisenbrand, Yuri Faenza, and Carsten Moldenhauer. On largest volume simplices and sub-determinants. In SODA, pages 315-323, 2014. Google Scholar
  15. Rod G. Downey and Michael R. Fellows. Fixed-parameter tractability and completeness II: On completeness for W[1]. Theor. Comput. Sci., 141(1-2):109-131, 1995. Google Scholar
  16. Rodney G. Downey and Michael R. Fellows. Parameterized Complexity. Springer, 2012. Google Scholar
  17. Andreas Emil Feldmann, C. S. Karthik, Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. Google Scholar
  18. Leonardo Pisano Fibonacci. The book of squares, 1225. Google Scholar
  19. Jörg Flum and Martin Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  20. Fedor V. Fomin and Dieter Kratsch. Exact Exponential Algorithms. An EATCS Series: Texts in Theoretical Computer Science. Springer, 2010. Google Scholar
  21. Peter Gritzmann, Victor Klee, and David G. Larman. Largest j-simplices in n-polytopes. Discrete Comput. Geom., 13:477-515, 1995. Google Scholar
  22. Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In STOC, pages 291-300, 2004. Google Scholar
  23. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. Google Scholar
  24. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. Google Scholar
  25. Chun-Wa Ko, Jon Lee, and Maurice Queyranne. An exact algorithm for maximum entropy sampling. Oper. Res., 43(4):684-691, 1995. Google Scholar
  26. Ioannis Koutis. Parameterized complexity and improved inapproximability for computing the largest j-simplex in a V-polytope. Inf. Process. Lett., 100(1):8-13, 2006. Google Scholar
  27. Alex Kulesza and Ben Taskar. k-DPPs: Fixed-size determinantal point processes. In ICML, pages 1193-1200, 2011. Google Scholar
  28. Alex Kulesza and Ben Taskar. Determinantal point processes for machine learning. Found. Trends Mach. Learn., 5(2-3):123-286, 2012. Google Scholar
  29. Donghoon Lee, Geonho Cha, Ming-Hsuan Yang, and Songhwai Oh. Individualness and determinantal point processes for pedestrian detection. In ECCV, pages 330-346, 2016. Google Scholar
  30. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurab, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In SODA, pages 2181-2200, 2020. Google Scholar
  31. Odile Macchi. The coincidence approach to stochastic point processes. Adv. Appl. Probab., 7(1):83-122, 1975. Google Scholar
  32. Vivek Madan, Aleksandar Nikolov, Mohit Singh, and Uthaipon Tantipongpipat. Maximizing determinants under matroid constraints. In FOCS, pages 565-576, 2020. Google Scholar
  33. Dániel Marx. On the optimality of planar and geometric approximation schemes. In FOCS, pages 338-348, 2007. Google Scholar
  34. Dániel Marx. Parameterized complexity and approximation algorithms. Comput. J., 51(1):60-78, 2008. Google Scholar
  35. Dániel Marx. A tight lower bound for planar multiway cut with fixed number of terminals. In ICALP, pages 677-688, 2012. Google Scholar
  36. Aleksandar Nikolov. Randomized rounding for the largest simplex problem. In STOC, pages 861-870, 2015. Google Scholar
  37. Aleksandar Nikolov and Mohit Singh. Maximizing determinants under partition constraints. In STOC, pages 192-201, 2016. Google Scholar
  38. Naoto Ohsaka. On the parameterized intractability of determinant maximization. CoRR, abs/2209.12519, 2022. URL: http://arxiv.org/abs/2209.12519.
  39. Naoto Ohsaka. Some inapproximability results of MAP inference and exponentiated determinantal point processes. J. Artif. Intell. Res., 73:709-735, 2022. Google Scholar
  40. Naoto Ohsaka and Tatsuya Matsuoka. On the (in)tractability of computing normalizing constants for the product of determinantal point processes. In ICML, pages 7414-7423, 2020. Google Scholar
  41. Mihai Pătraşcu and Ryan Williams. On the possibility of faster SAT algorithms. In SODA, pages 1065-1075, 2010. Google Scholar
  42. Mark Wilhelm, Ajith Ramanathan, Alexander Bonomo, Sagar Jain, Ed H. Chi, and Jennifer Gillenwater. Practical diversified recommendations on YouTube with determinantal point processes. In CIKM, pages 2165-2173, 2018. Google Scholar
  43. Jin-ge Yao, Feifan Fan, Wayne Xin Zhao, Xiaojun Wan, Edward Y. Chang, and Jianguo Xiao. Tweet timeline generation with determinantal point processes. In AAAI, pages 3080-3086, 2016. Google Scholar
  44. Martin J. Zhang and Zhijian Ou. Block-wise MAP inference for determinantal point processes with application to change-point detection. In SSP, pages 1-5, 2016. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact