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The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

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Abstract

We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points. Focusing on constant dimension d, we show that any such k-quantifier, d-dimensional problem is solvable in \(O(n^{k-1} \log ^{d-1} n)\) time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, \((3k-3)\)-dimensional problem in this class that requires time \(\Omega (n^{k-1-o(1)})\). Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time \(O(n\log ^{d-1} n)\), and k-quantifier problems with \(k>3\) reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination \(\mathsf {VCND}_d\) (Given three sets of vectors X, Y and Z of dimension d, d and 2d, respectively, is there an \(x \in X\) and a \(y \in Y\) so that their concatenation \(x \circ y\) is not dominated by any \(z \in Z\), where vector u is dominated by vector v if \(u_i \le v_i\) for each coordinate \(1 \le i \le d\)), and determine it as the “unique” candidate to be complete for this class (under fine-grained assumptions).

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Notes

  1. The given expression could model the following feature: if a person is of working age (\(x_1\in [\ell _1,u_1]\)), use criterion \(x_2 \in [\ell _2,u_2]\), otherwise use \((x_3,\dots ,x_d)\in [\ell _3,u_3]\times \cdots \times [\ell _d,u_d]\).

  2. Recall that a set X is Pareto optimal if there are no distinct points \(x,x' \in X\) such that x is coordinate-wise at least as large as \(x'\).

  3. For even more involved types of algorithmic tasks beyond k-quantifier first-order properties, see, e.g., [20] (All-Pairs Shortest Paths) or [25] (NP-hard problems).

  4. Strictly speaking, we identify the following 3-dimensional problem (which is linear-time equivalent to triangle detection in sparse graphs) as complete for \(PTO_{\exists \exists \exists , d}\): \(\exists x,y,z: x_1=z_1 \wedge x_2 = y_2 \wedge y_3=z_3\).

  5. Strong Exponential Time Hypothesis (\(\mathsf {SETH}\)) for CNF-SAT: For all \(\epsilon > 0\), there exists a k so that k-CNF-SAT cannot be solved in time \(O(2^{n(1-\epsilon )})\) [34].

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Acknowledgements

We would like to thank Rex Lei, Jiawei Gao, and Victor Vianu for helpful comments and discussion.

Funding

Russell Impagliazzo: Work supported by the Simons Foundation and NSF grant CCF-1909634. Marvin Künnemann: Research supported by Dr. Max Rössler, by the Walter Haefner Foundation, and by the ETH Zürich Foundation. Part of this research was performed while the author was employed at MPI Informatics.

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Authors and Affiliations

  1. University of Maryland, College Park, Maryland, USA

    Haozhe An

  2. Cornell University, Ithaca, USA

    Mohit Gurumukhani

  3. UC San Diego, San Diego, USA

    Russell Impagliazzo & Maria Paula Parga Nina

  4. University of Texas at Austin, Austin, USA

    Michael Jaber

  5. Institute for Theoretical Studies, ETH Zurich, Switzerland

    Marvin Künnemann

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  1. Haozhe An
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Haozhe An, Mohit Gurumukhani, Michael Jaber and Maria Paula Parga Nina: This research was conducted while the authors were undergraduates at UC San Diego.

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An, H., Gurumukhani, M., Impagliazzo, R. et al. The Fine-Grained Complexity of Multi-Dimensional Ordering Properties. Algorithmica 84, 3156–3191 (2022). https://doi.org/10.1007/s00453-022-01014-x

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