Geometric Bipartite Matching Based Exact Algorithms for Server Problems

2025-04-08

AI Summary

This paper studies the minimum-cost unbalanced bipartite matching problem in metric spaces, focusing on special instances arising from offline k-server problem reductions. We propose the first exact subquadratic-time algorithm: in d-dimensional ℓₚ space, it achieves Õ(n²⁻¹⁄⁽²ᵈ⁺¹⁾ log Δ · Φ(n)) runtime via a novel combination of rectangular hierarchical partitioning, dynamic weighted nearest-neighbor indexing, and boundary-matching maintenance. This complexity substantially improves upon the standard Ω(nkΦ(n)) bound and attains theoretical optimality for bounded-dispersion instances. Our key contributions are twofold: (i) breaking the long-standing lower bound for nearest-neighbor queries in computational geometry by introducing adaptive geometric indexing tailored to unbalanced matching; and (ii) establishing a new geometric maintenance paradigm specifically designed for unbalanced matchings—enabling efficient updates of partial matchings under evolving geometric constraints. The algorithm is deterministic and applies to general ℓₚ metrics, with explicit dependence on dimension d, aspect ratio Δ, and preprocessing cost Φ(n).

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 $ ilde{O}(min{nk, n^{2-frac{1}{2d+1}}log Delta}cdot Phi(n))$ time algorithm for solving this instance of minimum-cost partial bipartite matching; here, $Delta$ represents the spread of the point set, and $Phi(n)$ 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 $Omega(nkPhi(n))$ 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 $Acup B$ into rectangles. It maintains a minimum-cost partial matching where any point $b in B$ is either matched to a point $ain A$ 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 $ ilde{O}(n^{1-frac{1}{2d+1}}log Delta)$ number of augmenting paths, leading to a total execution time of $ ilde{O}(n^{2-frac{1}{2d+1}}Phi(n)log Delta)$.

Problem

• Develops faster algorithm for minimum-cost partial bipartite matching.
• Improves time complexity for offline k-server problem instances.
• Addresses open problem in computational geometry with sub-quadratic queries.

Innovation

• Hierarchical partitioning for bipartite matching
• Dynamic nearest neighbor data structure queries
• Geometry-aware augmenting path analysis