A deterministic near-linear time approximation scheme for geometric transportation
Given a set of points P=\left(P^{+} \sqcup P^{-}\right) \subset \mathbb{R}^{d} for some constant d and a supply function \mu: P \rightarrow \mathbb{R} such that \mu(p)\gt 0 \forall p \in P^{+}, \mu(p)\lt 0 \forall p \in P^{-}, and \sum_{p \in P} \mu(p)=0, the geometric transportation problem asks on...
Uloženo v:
| Vydáno v: | Proceedings / annual Symposium on Foundations of Computer Science s. 1301 - 1315 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Konferenční příspěvek |
| Jazyk: | angličtina |
| Vydáno: |
IEEE
06.11.2023
|
| Témata: | |
| ISSN: | 2575-8454 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Shrnutí: | Given a set of points P=\left(P^{+} \sqcup P^{-}\right) \subset \mathbb{R}^{d} for some constant d and a supply function \mu: P \rightarrow \mathbb{R} such that \mu(p)\gt 0 \forall p \in P^{+}, \mu(p)\lt 0 \forall p \in P^{-}, and \sum_{p \in P} \mu(p)=0, the geometric transportation problem asks one to find a transportation map \tau: P^{+} \times P^{-} \rightarrow \mathbb{R}_{\geq 0} such that \sum_{q \in P^{-}} \tau(p, q)=\mu(p) \forall p \in P^{+}, \sum_{p \in P^{+}} \tau(p, q)=-\mu(q) \forall q \in P^{-}, and the weighted sum of Euclidean distances for the pairs \sum_{(p, q) \in P^{+} \times P^{-}} \tau(p, q) \cdot\|q-p\|_{2} is minimized. We present the first deterministic algorithm that computes, in near-linear time, a transportation map whose cost is within a (1+\varepsilon) factor of optimal. More precisely, our algorithm runs in O\left(n \varepsilon^{-(d+2)} \log ^{5} n \log \log n\right) time for any constant \varepsilon>0. While a randomized n \varepsilon^{-O(d)} \log ^{O(d)} n time algorithm for this problem was discovered in the last few years, all previously known deterministic (1+\varepsilon)-approximation algorithms run in \Omega\left(n^{3 / 2}\right) time. A similar situation existed for geometric bipartite matching, the special case of geometric transportation where all supplies are unit, until a deterministic n \varepsilon^{-O(d)} \log ^{O(d)} n time (1+\varepsilon)-approximation algorithm was presented at STOC 2022. Surprisingly, our result is not only a generalization of the bipartite matching one to arbitrary instances of geometric transportation, but it also reduces the running time for all previously known (1+\varepsilon)-approximation algorithms, randomized or deterministic, even for geometric bipartite matching. In particular, we give the first (1+\varepsilon)-approximate deterministic algorithm for geometric bipartite matching and the first (1+\varepsilon) approximate deterministic or randomized algorithm for geometric transportation with no dependence on d in the exponent of the running time's polylog. As an additional application of our main ideas, we also give the first randomized near-linear O\left(\varepsilon^{-2} m \log ^{O(1)} n\right) time (1+\varepsilon)-approximation algorithm for the uncapacitated minimum cost flow (transshipment) problem in undirected graphs with arbitrary real edge costs. |
|---|---|
| ISSN: | 2575-8454 |
| DOI: | 10.1109/FOCS57990.2023.00078 |