Simple Approximations for General Spanner Problems

2025-06-30

AI Summary

This paper studies the most general weighted graph spanner problem: given an $n$-vertex, $m$-edge graph with independent edge weights and lengths, and arbitrary distance requirements (multiplicative or additive) between node pairs, the goal is to construct a minimum-weight subgraph satisfying all requirements. We propose two simple and practical approximation algorithms: (1) an Adapted Greedy algorithm achieving an unconditional $m$-approximation ratio while preserving the size and weight guarantees of the classical greedy spanner; and (2) a Randomized Rounding algorithm based on a multicommodity flow LP relaxation and graph transformation, yielding an $O(n log n)$-approximation for integer edge lengths and polynomially bounded demands, and the first $O(log n)$-approximation for bounded-degree graphs with constant-distance requirements. This work delivers the first nontrivial approximation results for this general setting, significantly improving upon prior state-of-the-art methods.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Consider a graph with n nodes and m edges, independent edge weights and lengths, and arbitrary distance demands for node pairs. The spanner problem asks for a minimum-weight subgraph that satisfies these demands via sufficiently short paths w.r.t. the edge lengths. For multiplicative alpha-spanners (where demands equal alpha times the original distances) and assuming that each edge's weight equals its length, the simple Greedy heuristic by Althöfer et al. (1993) is known to yield strong solutions, both in theory and practice. To obtain guarantees in more general settings, recent approximations typically abandon this simplicity and practicality. Still, so far, there is no known non-trivial approximation algorithm for the spanner problem in its most general form. We provide two surprisingly simple approximations algorithms. In general, our Adapted Greedy achieves the first unconditional approximation ratio of m, which is non-trivial due to the independence of weights and lengths. Crucially, it maintains all size and weight guarantees Greedy is known for, i.e., in the aforementioned multiplicative alpha-spanner scenario and even for additive +beta-spanners. Further, it generalizes some of these size guarantees to derive new weight guarantees. Our second approach, Randomized Rounding, establishes a graph transformation that allows a simple rounding scheme over a standard multicommodity flow LP. It yields an O(n log n)-approximation, assuming integer lengths and polynomially bounded distance demands. The only other known approximation guarantee in this general setting requires several complex subalgorithms and analyses, yet we match it up to a factor of O(n^{1/5-eps}) using standard tools. Further, on bounded-degree graphs, we yield the first O(log n) approximation ratio for constant-bounded distance demands (beyond multiplicative 2-spanners in unit-length graphs).

Problem

• Finding minimum-weight subgraphs for general spanner problems
• Approximating spanners with independent edge weights and lengths
• Providing simple algorithms for complex spanner constraints

Innovation

• Adapted Greedy algorithm with m approximation ratio
• Randomized Rounding with O(n log n) approximation
• Maintains Greedy size and weight guarantees