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-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 logd-1 n), and k-quantifier problems with k > 3 reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination VCNDd (Given three sets of vectors X, Y and Z of dimension d, d and 2d, respectively, is there an x is an element of X and a y is an element of Y so that their concatenation x is an element of y is not dominated by any z is an element of 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).