Heuristic Search for Path Finding with Refuelling

TL;DR

The paper tackles the Gas Station Problem (GSP), a fuel-constrained path-finding problem where edge costs are fuel use and vertices have refuelling prices, bounded by tank capacity $q_{max}$ and max stops $k_{max}$. It introduces Refuel $A^*$ (RF-$A^*$), a heuristic A*-style search that builds and prunes labeled partial paths using dominance rules, while precomputing reachable sets and an admissible heuristic to guide search. RF-$A^*$ is proven to be complete and optimal, and experiments show it substantially outperforms dynamic programming and mixed-integer programming baselines on large city graphs, achieving 2–8x speedups over DP and up to 64x when a cached heuristic is available. The approach enables scalable, exact planning for fuel-constrained robots in urban networks, with potential extensions to time-varying environments and multi-agent settings.

Abstract

This paper considers a generalization of the Path Finding (PF) problem with refuelling constraints referred to as the Gas Station Problem (GSP). Similar to PF, given a graph where vertices are gas stations with known fuel prices, and edge costs are the gas consumption between the two vertices, GSP seeks a minimum-cost path from the start to the goal vertex for a robot with a limited gas tank and a limited number of refuelling stops. While GSP is polynomial-time solvable, it remains a challenge to quickly compute an optimal solution in practice since it requires simultaneously determine the path, where to make the stops, and the amount to refuel at each stop. This paper develops a heuristic search algorithm called Refuel A$^*$ (RF-A$^*$) that iteratively constructs partial solution paths from the start to the goal guided by a heuristic while leveraging dominance rules for pruning during planning. RF-A$^*$ is guaranteed to find an optimal solution and often runs 2 to 8 times faster than the existing approaches in large city maps with several hundreds of gas stations.

Figures (6)

• Figure 1: An illustrative example of GSP. This graph consists of six vertices representing gas stations, each associated with a gas price, and each edge with its fuel expenditure. The objective is to find a minimum-cost path from start to goal, assuming the tank capacity is 5 and the refuelling stop limit is 3. The figure shows the minimum-cost path (ACEF), using green arrows, and the minimum fuel consumption path, (ABDF) using red arrows. Note that the minimum fuel consumption path does not incur the lowest fuel cost. Along the optimal solution ACEF, the cost of refuelling at each vertex is: $\$10$ at A, $\$14$ at C, $\$18$ at E.
• Figure 2: A toy example showing the $\text{Refuel A}^{*}$ with four vertices. The graph $G$ has four vertices $(o,a,b \text{ and } {t})$, with fuel consumption in black, fuel price in blue, $q_{max}=6$ and $k_{max}=2$. The i-th label associated with a vertex $v$ is $l_{i} = (v, g_i, q_i, k_i)$. The optimal path is shown to be $o \to b \to t$, with a cost of $15$.
• Figure 3: (\ref{['fig:small-graph']}) Visualization of a random map. (\ref{['fig:city-graph']}) Visualization of the map for Philadelphia, USA. It showcases the road network of the city, where the red dots represent the gas stations.
• Figure 4: Runtime on the synthetic dataset.
• Figure 5: (a) and (b) show runtime and memory cost of all methods. (c) shows the speed-up of $\text{RF-A}^*$ variants compare to DP.

Theorems & Definitions (12)

• Definition 1: Gas Station Problem (GSP) khullerFillNotFill2011
• Remark 1
• Definition 2
• Lemma 1: Optimal Refuelling Strategy
• Lemma 2: Admissible Heuristic
• Remark 2
• Theorem 1
• Lemma 3: Path Feasibility
• Lemma 4: Complete Expansion
• Lemma 5: Dominance Pruning