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An Analysis of the Correctness and Computational Complexity of Path Planning in Payment Channel Networks

Padraig Corcoran, Rhyd Lewis

TL;DR

This work analyzes path planning in payment channel networks, focusing on a Dijkstra-like algorithm whose correctness and complexity hinge on a consistency property of the fee function. It proves that when the fee map is consistent, the problem is solvable in polynomial time, and verifies that the Lightning Network satisfies this condition; without consistency, the problem becomes NP-hard. A bidirectional variant is proposed, maintaining asymptotic efficiency while delivering empirical performance gains on LN-like topologies. Experiments on a Lightning Network snapshot demonstrate substantial reductions in explored nodes and wall-clock time for the bidirectional approach, supporting its practical value. The results inform the design of PCN routing and fee structures, while acknowledging remaining challenges in privacy, security, and network topology.

Abstract

Payment Channel Networks (PCNs) are a method for improving the scaling and latency of cryptocurrency transactions. For a payment to be made between two peers in a PCN, a feasible low-fee path in the network must be planned. Many PCN path planning algorithms use a search algorithm that is a variant of Dijkstra's algorithm. In this article, we prove the correctness and computational complexity of this algorithm. Specifically, we show that, if the PCN satisfies a consistency property relating to the fees charged by payment channels, the algorithm is correct and has polynomial computational complexity. However, in the general case, the algorithm is not correct and the path planning problem is NP-hard. These newly developed results can be used to inform the development of new or existing PCNs amenable to path planning. For example, we show that the Lightning Network, which is the most widely used PCN and is built on the Bitcoin cryptocurrency, currently satisfies the above consistency property. As a second contribution, we demonstrate that a small modification to the above path planning algorithm which, although having the same asymptotic computational complexity, empirically shows better performance. This modification involves the use of a bidirectional search and is empirically evaluated by simulating transactions on the Lightning Network.

An Analysis of the Correctness and Computational Complexity of Path Planning in Payment Channel Networks

TL;DR

This work analyzes path planning in payment channel networks, focusing on a Dijkstra-like algorithm whose correctness and complexity hinge on a consistency property of the fee function. It proves that when the fee map is consistent, the problem is solvable in polynomial time, and verifies that the Lightning Network satisfies this condition; without consistency, the problem becomes NP-hard. A bidirectional variant is proposed, maintaining asymptotic efficiency while delivering empirical performance gains on LN-like topologies. Experiments on a Lightning Network snapshot demonstrate substantial reductions in explored nodes and wall-clock time for the bidirectional approach, supporting its practical value. The results inform the design of PCN routing and fee structures, while acknowledging remaining challenges in privacy, security, and network topology.

Abstract

Payment Channel Networks (PCNs) are a method for improving the scaling and latency of cryptocurrency transactions. For a payment to be made between two peers in a PCN, a feasible low-fee path in the network must be planned. Many PCN path planning algorithms use a search algorithm that is a variant of Dijkstra's algorithm. In this article, we prove the correctness and computational complexity of this algorithm. Specifically, we show that, if the PCN satisfies a consistency property relating to the fees charged by payment channels, the algorithm is correct and has polynomial computational complexity. However, in the general case, the algorithm is not correct and the path planning problem is NP-hard. These newly developed results can be used to inform the development of new or existing PCNs amenable to path planning. For example, we show that the Lightning Network, which is the most widely used PCN and is built on the Bitcoin cryptocurrency, currently satisfies the above consistency property. As a second contribution, we demonstrate that a small modification to the above path planning algorithm which, although having the same asymptotic computational complexity, empirically shows better performance. This modification involves the use of a bidirectional search and is empirically evaluated by simulating transactions on the Lightning Network.
Paper Structure (9 sections, 11 theorems, 12 equations, 7 figures, 2 algorithms)

This paper contains 9 sections, 11 theorems, 12 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Problem Definition
  4. Variant of Dijkstra’s algorithm
  5. Correctness and Computational Complexity Analysis
  6. Bidirectional Variant of Dijkstra’s algorithm
  7. Experimental Results
  8. Conclusions
  9. Recurrence Relation

Key Result

Theorem 1

Let $G$ be a graph representation of the LN and let $G^T$ be its transpose. Let $(v_1, v_2)$, $(v_2, v_3), \dots, (v_{n-1}, v_n)$ be a lowest-fee path in $G^T$ from $t$ to $s$, where the amount transferred to $t$ is $a$. The transpose path $(v_n, v_{n-1}), (v_{n-1}, v_{n-2}), \dots, (v_2, v_1)$ in $

Figures (7)

  • Figure 1: An example LN containing two vertices and a single arc is displayed. The arc is labelled with a tuple containing the corresponding base fee, fee rate, and balance respectively.
  • Figure 2: An example LN containing three vertices and two arcs is displayed. Each arc is labelled with a tuple containing the corresponding base fee, fee rate and balance.
  • Figure 3: An example LN is displayed in (a) and the corresponding transpose, where arc directions are reversed, is displayed in (b).
  • Figure 4: An LN graph $G$ and corresponding transpose graph $G^T$ displayed in (a) and (b) respectively. The fee map $f_{br}$ corresponding to $G^T$ has the properties $f_{br}((t,i), 100)=10$, $f_{br}((i,j), 110)=10$, $f_{br}((j,s), 120)=5$, $f_{br}((t,j), 100)=10$ and $f_{br}((j,s), 110)=20$.
  • Figure 5: An example LN with a hub-and-spoke topology is displayed.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Definition 2
  • ...and 14 more