Geometric Bipartite Matching Based Exact Algorithms for Server Problems

TL;DR

This work studies offline $k$-server problems in geometric spaces by reducing them to minimum-cost bipartite matching and then solving the resulting instances with a novel geometric primal-dual framework. It introduces a hierarchical partitioning scheme and a constrained dual formulation, enabling augmentation paths to be localized within cells and merged efficiently, which yields sub-quadratic DWNN-query time. The authors obtain deterministic runtimes of $\tilde{O}(\min\{nk, n^{2-\frac{1}{2d+1}}\log \Delta\}\cdot\Phi(n))$ for the $k$-SP and $k$-SPI problems in $d$ dimensions, with a refined $2$-dimensional exponent of $1.8$ in practice, and provide fast Hungarian-implementation variants and random-color analyses to bolster the main results. The results have practical implications for offline metric matching and online metric matching reductions, offering near-optimal geometric algorithms with provable sub-quadratic performance under DWNN models. Open questions include removing the $\log \Delta$ factor and translating these techniques to tighter online competitive ratios for the $k$-server setting.

Abstract

For any given metric space, obtaining an offline optimal solution to the classical $k$-server problem can be reduced to solving a minimum-cost partial bipartite matching between two point sets $A$ and $B$ within that metric space. For $d$-dimensional $\ell_p$ metric space, we present an $\tilde{O}(\min\{nk, n^{2-\frac{1}{2d+1}}\log Δ\}\cdot Φ(n))$ time algorithm for solving this instance of minimum-cost partial bipartite matching; here, $Δ$ represents the spread of the point set, and $Φ(n)$ is the query/update time of a $d$-dimensional dynamic weighted nearest neighbor data structure. Our algorithm improves upon prior algorithms that require at least $Ω(nkΦ(n))$ time. The design of minimum-cost (partial) bipartite matching algorithms that make sub-quadratic queries to a weighted nearest-neighbor data structure, even for bounded spread instances, is a major open problem in computational geometry. We resolve this problem at least for the instances that are generated by the offline version of the $k$-server problem. Our algorithm employs a hierarchical partitioning approach, dividing the points of $A\cup B$ into rectangles. It maintains a minimum-cost partial matching where any point $b \in B$ is either matched to a point $a\in A$ or to the boundary of the rectangle it is located in. The algorithm involves iteratively merging pairs of rectangles by erasing the shared boundary between them and recomputing the minimum-cost partial matching. This continues until all boundaries are erased and we obtain the desired minimum-cost partial matching of $A$ and $B$. We exploit geometry in our analysis to show that each point participates in only $\tilde{O}(n^{1-\frac{1}{2d+1}}\log Δ)$ number of augmenting paths, leading to a total execution time of $\tilde{O}(n^{2-\frac{1}{2d+1}}Φ(n)\log Δ)$.

Figures (12)

• Figure 1: The graph ${\mathcal{G}}_\sigma$ constructed for $\sigma=\langle r_1, r_2, \ldots, r_6\rangle$. The vertex set $A$ (red disks) and $B$ (blue squares) represent the exit and entry gates of each request, and the purple dashed lines show an $(n-2)$-partial matching on ${\mathcal{G}}_\sigma$ representing a $2$-partitioning $\langle r_1, r_2, r_4, r_5\rangle$ and $\langle r_3, r_6\rangle$.
• Figure 2: Illustration of the GRS algorithm.
• Figure 3: (left) A sub-problem of the $k$-SP problem, where the optimal solution has a high cost, and (right) there exists a low-cost high-cardinality matching inside the sub-problems.
• Figure 4: We Partition a rectangle into two children by picking a divider (the purple dashed vertical line) with the minimum number of points close to it (gray shaded area) within the middle-third of its longer side (the part between the two vertical dotted lines).
• Figure 5: (left) The low-cost high-cardinality matching constructed inside a cell $\Box$ of ${\mathcal{H}}$, and (right) a ${\mathcal{C}}$-extended $20$-matching with $9$ matched points (blue discs), $11$ boundary-matched points (blue circles), and $8$ free points (blue crosses). The matching cost is the total length of the solid and dashed lines. The boundary-matched point $b$ is matched to the boundary $\Gamma$.

Theorems & Definitions (77)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4