The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

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).