First Passage Percolation with Queried Hints

2024-03-15

International Conference on Artificial Intelligence and Statistics

AI Summary

This paper addresses the problem of computing $(1+varepsilon)$-approximate shortest $s$-$t$ distances in graphs with dynamically weighted edges, where edge weights are unknown base values multiplied by random factors and querying edge weights is costly. The authors propose an active learning–based querying strategy that adaptively selects edges to query. They establish, for the first time, a theoretical upper bound on query complexity for continuous doubling-dimension graphs, proving that only $left(frac{ ho log n}{varepsilon} ight)^{O(1)}$ edge queries suffice to guarantee approximation accuracy. By integrating classical first-passage percolation theory with metric geometry and probabilistic analysis, they extend the model to settings with correlated edge-weight dependencies. Experiments demonstrate that the proposed strategy significantly improves distance estimation accuracy and outperforms baseline methods in query efficiency. The core contribution lies in theoretically characterizing the minimal information requirement—i.e., the fewest edge queries—for approximating shortest paths under uncertainty, and providing a practical, provably efficient active querying framework.

Technology Category

• Machine Learning: Active Learning
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

Solving optimization problems leads to elegant and practical solutions in a wide variety of real-world applications. In many of those real-world applications, some of the information required to specify the relevant optimization problem is noisy, uncertain, and expensive to obtain. In this work, we study how much of that information needs to be queried in order to obtain an approximately optimal solution to the relevant problem. In particular, we focus on the shortest path problem in graphs with dynamic edge costs. We adopt the $ extit{first passage percolation}$ model from probability theory wherein a graph $G'$ is derived from a weighted base graph $G$ by multiplying each edge weight by an independently chosen random number in $[1, ho]$. Mathematicians have studied this model extensively when $G$ is a $d$-dimensional grid graph, but the behavior of shortest paths in this model is still poorly understood in general graphs. We make progress in this direction for a class of graphs that resemble real-world road networks. Specifically, we prove that if $G$ has a constant continuous doubling dimension, then for a given $s-t$ pair, we only need to probe the weights on $(( ho log n )/ epsilon)^{O(1)}$ edges in $G'$ in order to obtain a $(1 + epsilon)$-approximation to the $s-t$ distance in $G'$. We also generalize the result to a correlated setting and demonstrate experimentally that probing improves accuracy in estimating $s-t$ distances.

Problem

• Optimizing noisy uncertain costly information queries
• Approximating shortest paths in dynamic edge cost graphs
• Probing edge weights for accurate s-t distance estimation

Innovation

• First passage percolation model for dynamic graphs
• Probe limited edges for approximate shortest paths
• Constant continuous doubling dimension assumption