Tightest Admissible Shortest Path

TL;DR

This work presents a complete algorithm for solving TASP; a path with the tightest suboptimality bound on the optimal cost, generalization of the shortest path problem to bounded uncertainty, where edge-weight uncertainty can be traded for computational cost.

Abstract

The shortest path problem in graphs is fundamental to AI. Nearly all variants of the problem and relevant algorithms that solve them ignore edge-weight computation time and its common relation to weight uncertainty. This implies that taking these factors into consideration can potentially lead to a performance boost in relevant applications. Recently, a generalized framework for weighted directed graphs was suggested, where edge-weight can be computed (estimated) multiple times, at increasing accuracy and run-time expense. We build on this framework to introduce the problem of finding the tightest admissible shortest path (TASP); a path with the tightest suboptimality bound on the optimal cost. This is a generalization of the shortest path problem to bounded uncertainty, where edge-weight uncertainty can be traded for computational cost. We present a complete algorithm for solving TASP, with guarantees on solution quality. Empirical evaluation supports the effectiveness of this approach.

Figures (2)

• Figure 1: Left: Digraph $G$. Right: costs and estimates of $G$.
• Figure 2: Histograms of $1-(\theta^{max}$($\mathsf{BEAST}$$(\infty)$)$/\theta^{max}$($\mathsf{EI}$-$\mathsf{UCS}$)) [top left], $1-(\theta^{max}$($\mathsf{BEAST}$$(u(\pi_{SLB}))$)$/\theta^{max}$($\mathsf{BEAST}$$(\infty)$)) [top right], pruned nodes percentage for $\mathsf{BEAST}$$(u(\pi_{SLB}))$ [bottom left], $\mathcal{B}^*$ [bottom right], based on all domains.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Theorem 1: Generality
• proof
• Theorem 2: $\mathcal{B}^*=U^*/L^*$
• proof
• Corollary 1
• Example 1