The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

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-1log ^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 Ω (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(nlog ^d-1 n) , and k -quantifier problems with k>3 reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination 𝖵𝖢𝖭𝖣_d (Given three sets of vectors X , Y and Z of dimension d , d and 2 d , respectively, is there an x ∈ X and a y ∈ Y so that their concatenation x ∘ y is not dominated by any z ∈ Z , where vector u is dominated by vector v if u_i ≤ v_i for each coordinate 1 ≤ i ≤ d ), and determine it as the “unique” candidate to be complete for this class (under fine-grained assumptions).