The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

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