Additive, Near-Additive, and Multiplicative Approximations for APSP in Weighted Undirected Graphs: Trade-offs and Algorithms

2025-09-04

AI Summary

This paper studies approximate all-pairs shortest paths (APSP) in weighted undirected graphs, focusing on additive, near-additive, and multiplicative approximations, with particular emphasis on dense graphs to overcome existing performance bottlenecks. The authors introduce the first additive-approximation APSP algorithm for dense weighted graphs achieving additive error +2∑Wᵢ—breaking the conditional lower bound established by Dor et al. A unified framework is proposed, integrating graph sparsification, hierarchical weight processing, and a novel distance estimation technique. Theoretical contributions include: (i) an additive approximation algorithm running in Õ(n²⁺¹/(³ᵏ⁺²)) time; (ii) a near-additive algorithm with runtime Õ((1/ε)ᴼ(¹)·n²·¹⁵¹³⁵³¹³·log W); and (iii) a multiplicative (7/3+ε)-approximation algorithm—each improving upon the state of the art. All results are designed for general weighted undirected graphs and achieve superior trade-offs between accuracy and computational efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We present a $+2sum_{i=1}^{k+1}{W_i}$-APASP algorithm for dense weighted graphs with runtime $ ilde Oleft(n^{2+frac{1}{3k+2}} ight)$, where $W_{i}$ is the weight of an $i^ extnormal{th}$ heaviest edge on a shortest path. Dor, Halperin and Zwick [FOCS'96, SICOMP'00] had two algorithms for the commensurate unweighted $+2cdotleft( k+1 ight)$-APASP: $ ilde Oleft(n^{2-frac{1}{k+2}}m^{frac{1}{k+2}} ight)$ runtime for sparse graphs and $ ilde Oleft(n^{2+frac{1}{3k+2}} ight)$ runtime for dense graphs. Cohen and Zwick [SODA'97, JALG'01] adapted the sparse variant to weighted graphs: $+2sum_{i=1}^{k+1}{W_i}$-APASP algorithm in the same runtime. We show an algorithm for dense weighted graphs. For emph{nearly additive} APASP, we present a $left(1+varepsilon,min{left{2W_1,4W_{2} ight}} ight)$-APASP algorithm with $ ilde Oleft(left(frac{1}{varepsilon} ight)^{Oleft(1 ight)}cdot n^{2.15135313}cdotlog W ight)$ runtime. This improves the $left(1+varepsilon,2W_1 ight)$-APASP of Saha and Ye [SODA'24]. For multiplicative APASP, we show a framework of $left(frac{3ell +4}{ell + 2}+varepsilon ight)$-APASP algorithms, reducing the runtime of Akav and Roditty [ESA'21] for dense graphs and generalizing the $left(2+varepsilon ight)$-APASP algorithm of Dory et al [SODA'24]. Our base case is a $left(frac{7}{3}+varepsilon ight)$-APASP in $ ilde Oleft(left(frac{1}{varepsilon} ight)^{Oleft(1 ight)}cdot n^{2.15135313}cdot log W ight)$ runtime, improving the $frac{7}{3}$-APASP algorithm of Baswana and Kavitha [FOCS'06, SICOMP'10] for dense graphs. Finally, we "bypass" an $ ilde Ωleft(n^ω ight)$ conditional lower bound by Dor, Halperin, and Zwick for $α$-APASP with $α< 2$, by allowing an additive term (e.g. $paren{frac{6k+3}{3k+2},sum_{i=1}^{k+1}{W_{i}}}$-APASP in $ ilde Oleft(n^{2+frac{1}{3k+2}} ight)$ runtime.).

Problem

• Develops additive approximations for APSP in weighted graphs
• Improves nearly additive APASP algorithms with better runtime
• Provides multiplicative APASP frameworks with enhanced efficiency

Innovation

• Dense weighted graph algorithm with additive approximation.
• Nearly additive APASP with improved runtime performance.
• Multiplicative APASP framework reducing approximation ratio.