Given two point sets A and B in ℝd of size n each, for some constant dimension d≥ 1, and a parameter ε>0, we present a deterministic algorithm that computes, in n·(ε−1 logn)O(d) time, a perfect matching between A and B whose cost is within a (1+ε) factor of the optimal matching under any ℓp-norm. Although a Monte-Carlo algorithm with a similar running time is proposed by Raghvendra and Agarwal [J. ACM 2020], the best-known deterministic ε-approximation algorithm takes Ω(n3/2) time. Our algorithm constructs a (refinement of a) tree cover of ℝd, and we develop several new tools to apply a tree-cover based approach to compute an ε-approximate perfect matching.

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our article are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in ℝ 1 and ℝ 2 . The first framework uses bootstrapping and gives a (1 + ε)-approximate data structure for dynamic interval set cover in ℝ 1 with O ( n α / ε) amortized update time for any constant α > 0; in ℝ 2 , this method gives O (1)-approximate data structures for unit-square set cover and hitting set with O ( n 1/2+α ) amortized update time. The second framework uses local modification and leads to a (1 + ε)-approximate data structure for dynamic interval hitting set in ℝ 1 with Õ(1/ε) amortized update time; in ℝ 2 , it gives O (1)-approximate data structures for unit-square set cover and hitting set in the partially dynamic settings with Õ(1) amortized update time.

We investigate dynamic versions of geometric set cover and hitting set where points and ranges may be inserted or deleted, and we want to efficiently maintain an (approximately) optimal solution for the current problem instance. While their static versions have been extensively studied in the past, surprisingly little is known about dynamic geometric set cover and hitting set. For instance, even for the most basic case of one-dimensional interval set cover and hitting set, no nontrivial results were known. The main contribution of our paper are two frameworks that lead to efficient data structures for dynamically maintaining set covers and hitting sets in $\mathbb{R}^1$ and $\mathbb{R}^2$. The first framework uses bootstrapping and gives a $(1+\varepsilon)$-approximate data structure for dynamic interval set cover in $\mathbb{R}^1$ with $O(n^α/\varepsilon)$ amortized update time for any constant $α> 0$; in $\mathbb{R}^2$, this method gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set with $O(n^{1/2+α})$ amortized update time. The second framework uses local modification, and leads to a $(1+\varepsilon)$-approximate data structure for dynamic interval hitting set in $\mathbb{R}^1$ with $\widetilde{O}(1/\varepsilon)$ amortized update time; in $\mathbb{R}^2$, it gives $O(1)$-approximate data structures for unit-square (and quadrant) set cover and hitting set in the \textit{partially} dynamic settings with $\widetilde{O}(1)$ amortized update time.

Let $A$ and $B$ be two point sets in the plane of sizes $r$ and $n$ respectively (assume $r \leq n$), and let $k$ be a parameter. A matching between $A$ and $B$ is a family of pairs in $A \times B$ so that any point of $A \cup B$ appears in at most one pair. Given two positive integers $p$ and $q$, we define the cost of matching $M$ to be $c(M) = \sum_{(a, b) \in M}\|{a-b}\|_p^q$ where $\|{\cdot}\|_p$ is the $L_p$-norm. The geometric partial matching problem asks to find the minimum-cost size-$k$ matching between $A$ and $B$. We present efficient algorithms for geometric partial matching problem that work for any powers of $L_p$-norm matching objective: An exact algorithm that runs in $O((n + k^2) {\mathop{\mathrm{polylog}}} n)$ time, and a $(1 + \varepsilon)$-approximation algorithm that runs in $O((n + k\sqrt{k}) {\mathop{\mathrm{polylog}}} n \cdot \log\varepsilon^{-1})$ time. Both algorithms are based on the primal-dual flow augmentation scheme; the main improvements involve using dynamic data structures to achieve efficient flow augmentations. With similar techniques, we give an exact algorithm for the planar transportation problem running in $O(\min\{n^2, rn^{3/2}\} {\mathop{\mathrm{polylog}}} n)$ time.

Our goal is to compare two planar point sets by finding subsets of a given size such that a minimum-weight matching between them has the smallest weight. This can be done by a translation of one set that minimizes the weight of the matching. We give efficient algorithms (a) for finding approximately optimal matchings, when the cost of a matching is the L_p-norm of the tuple of the Euclidean distances between the pairs of matched points, for any p in [1,infty], and (b) for constructing small-size approximate minimization (or matching) diagrams: partitions of the translation space into regions, together with an approximate optimal matching for each region.

Let R, B be a set of n points in R^d, for constant d, where the points of R have integer supplies, points of B have integer demands, and the sum of supply is equal to the sum of demand. Let d(.,.) be a suitable distance function such as the L_p distance. The transportation problem asks to find a map tau : R x B --> N such that sum_{b in B}tau(r,b) = supply(r), sum_{r in R}tau(r,b) = demand(b), and sum_{r in R, b in B} tau(r,b) d(r,b) is minimized. We present three new results for the transportation problem when d(.,.) is any L_p metric: * For any constant epsilon > 0, an O(n^{1+epsilon}) expected time randomized algorithm that returns a transportation map with expected cost O(log^2(1/epsilon)) times the optimal cost. * For any epsilon > 0, a (1+epsilon)-approximation in O(n^{3/2}epsilon^{-d}polylog(U)polylog(n)) time, where U is the maximum supply or demand of any point. * An exact strongly polynomial O(n^2 polylog n) time algorithm, for d = 2.