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REVIEW 5 major objections 5 minor 41 references

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

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A proof shows that the Lightning Network's Dijkstra-style routing algorithm is correct and near-linear exactly when channel fees are consistent, and that arbitrary fee maps make the problem NP-hard.

desk verdict The paper's core mapping is solid, but the printed Algorithm 1 computes fees on the wrong amount, so the correctness proof doesn't cover the stated algorithm; fix that and it's a worthwhile contribution. read the letter →

arxiv 2501.11419 v1 pith:WGUKDO34 submitted 2025-01-20 cs.DM cs.CE

Padraig Corcoran , Rhyd Lewis This is my paper · ORCID
classification cs.DMcs.CE MSC 68Q1768R1005C85
keywords paymentchannelnetworksLightningNetworkpathplanningDijkstra'salgorithmtime-dependentNP-hardnessfeeconsistencybidirectionalsearch
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Payment channel networks such as the Lightning Network route payments by finding a low-fee path through a network of channels, each charging a fee that depends on the amount sent to it. This paper proves that the Dijkstra-style algorithm actually used by Lightning implementations is correct and runs in time $O(|E|+|V|\log|V|)$ exactly when the fee map is 'consistent': sending a larger amount through a channel never results in a smaller amount arriving on the other side. The paper further shows that Lightning's linear base-fee-plus-rate fee formula satisfies this condition, and that if fees are allowed to be arbitrary, the same path planning problem becomes NP-hard. A small modification to the algorithm, a partial bidirectional search, is shown empirically to explore fewer vertices on a real Lightning Network snapshot.

What carries the argument

The load-bearing mechanism is the consistency (or monotonicity) property of the fee map: $a+f(e,a)\le a'+f(e,a')$ whenever $a\le a'$. This condition plays the role of a FIFO / non-decreasing-horizon requirement in time-dependent networks; it guarantees that a greedy, label-setting search never needs to revisit a vertex with a larger accumulated fee. The paper reduces Lightning path planning to a minimum-time problem on a time-dependent network, transfers the classical correctness result for that setting, and then shows that Algorithm 1 prunes the search space of Algorithm 1' to feasible paths, yielding the same near-linear bound. The closed-form solution of the amount recurrence $a_{i-1}=a_i+f_{br}(e_i,a_i)$ supplies the formula for the amount that must be sent at each hop.

What would settle it

A concrete check: on the paper's own Figure 3 example, compare the fee returned by the pseudocode (which evaluates the arc fee at the constant amount $a$ on line 9) with the fee computed by Equation (4) on the accumulated amount. If the two disagree, the correctness theorem applies to the intended algorithm, not to the literal pseudocode.

Watch

Extended reading notes

Core claim

The central claim is that the Lightning Network's path planning algorithm, a variant of Dijkstra's algorithm that reverses the search direction and accumulates fees, is correct and has complexity $O(|E|+|V|\log|V|)$ precisely when the channel fee map $f_{br}$ is consistent in the sense of Definition 1: for every arc $e$ and amounts $a \le a'$, $a + f_{br}(e,a) \le a' + f_{br}(e,a')$. The proof works by equating the payment routing problem with a lowest-time path problem in a time-dependent network whose travel times are the fee amounts, then applying the known correctness and complexity results for Dijkstra-style search in such networks. Under this condition the algorithm returns the true lowest-fee feasible path; if the fee map is arbitrary and inconsistent, the paper proves that the path planning problem is NP-hard. It also proves that the Lightning Network's fee formula $f_{br}(e,a)=f_b(e)+f_r(e)\,a$ with nonnegative rate is always consistent, and introduces a partial bidirectional search variant whose correctness is established by the same argument and whose practical speedup is demonstrated by simulation.

Load-bearing premise

The load-bearing assumption is that channel fees are consistent: sending a larger amount through a channel never leaves a smaller amount arriving out the other side. The proof of correctness and efficiency collapses if this condition fails.

Editorial extensions

If this is right

  • New payment channel networks that want efficient routing should design their fee schedules to satisfy $a+f(e,a)\le a'+f(e,a')$ for all $a\le a'$.
  • Lightning's existing linear fee formula already meets this condition, giving a theoretical explanation for why its routing search is usually fast.
  • If a network operator uses a non-consistent fee map, no efficient exact algorithm can be guaranteed; the paper's NP-hardness result applies to the general case.
  • The partial bidirectional search variant can be dropped into existing implementations without changing asymptotic complexity and offers a measurable reduction in explored vertices on hub-and-spoke topologies.
  • The correctness guarantee carries over to any payment channel network whose fee map is consistent, not just to Lightning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extending the paper's logic
  • consistency should become a first-class design requirement for any new PCN fee schedule
  • so that non-linear or state-dependent fee functions are checked against the condition before deployment. The closed-form recurrence in the appendix could be used to compute the required onward amount exactly without iterating each hop
  • reducing per-hop arithmetic in future routing engines. Because correctness depends on the amount used at each hop
  • implementations should verify that their fee computation matches Equation (4) on the accumulated amount
  • a discrepancy between the published pseudocode and the intended formula is testable on real node software.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper analyzes a Dijkstra-style algorithm (Algorithm 1) for path planning in payment channel networks (PCNs), specifically the Lightning Network (LN). It claims that if the fee map is 'consistent' (a FIFO-type condition), the algorithm correctly finds a lowest-fee path satisfying balance constraints and runs in O(|E| + |V| log |V|). For arbitrary inconsistent fee maps, it claims the problem is NP-hard. A second contribution is a 'partial bidirectional' variant (Algorithm 2) with the same asymptotic complexity, plus an empirical evaluation on an LN snapshot. The paper also solves a recurrence for accumulated fees along a path and proves that the LN's linear fee function is consistent.

Significance. If the results were fully correct, this would be a valuable formal analysis of a heuristic widely used in LN implementations (LND, Core Lightning), giving the first correctness and complexity guarantees under a realistic consistency condition, plus a hardness result for the general case. The bidirectional variant is a small but potentially useful practical improvement, and the consistency observation about the LN's fee structure is nontrivial context for protocol design. The paper also provides a closed-form solution to the fee-accumulation recurrence and ships Python code to verify it, which is a strength. However, the central correctness claims do not hold for the algorithms as printed, and several proof steps are asserted rather than demonstrated; these issues must be resolved before the contribution can be accepted.

major comments (5)
  1. [Algorithm 1, line 9; Definition 2; Equation (4)] The relaxation step computes cv' = c(v) + fbr((v,v'), a), but the fee on the transposed arc must be evaluated at the accumulated amount a + c(v), and on the reverse original arc (v',v), not on (v,v'). Equation (4) and the example in Section 4 require this: after reaching i with c(i)=3 in Figure 3, the next fee is fbr((s,i), 13)=4.6, so c(s)=7.6, whereas line 9 yields fbr((s,i),10)=4 and c(s)=7. The same error propagates through Algorithm 1' (Definition 2) and Algorithm 2 (line 11). Because Lemma 5's appeal to Kaufman et al.'s time-dependent Dijkstra requires evaluating the travel-time function at the departure time, the correctness proofs do not apply to the printed algorithms. This is a load-bearing error for Theorem 7 and Theorem 10.
  2. [Algorithm 1, line 10; Algorithm 2, line 8] The balance condition c(v)+a <= b(v,v') checks the balance of the wrong arc and uses the amount before the fee is added. The amount that must fit in the channel is a + c(v') (the amount at the head of the GT arc), which equals a + c(v) + fbr((v',v), a+c(v)) under the correct relaxation, and the relevant balance is that of the original arc (v',v). As printed, the algorithm can declare a path feasible even when the true required amount exceeds the channel balance, so the feasibility part of Theorem 7's correctness claim is unsupported.
  3. [Theorem 7, proof] The pruning argument is asserted, not proven. The proof states that Algorithm 1 prunes precisely the infeasible paths and that the optimal feasible path therefore remains in the pruned search space, but no invariant is established showing that the tentative costs c(v) correctly track the accumulated amount a+c(v) for partially explored paths, nor that the balance checks preserve an optimal feasible path under the order of vertex processing. The boundedness assumption on fbr is not used to bridge this gap. A rigorous proof must relate the pruned search to the (corrected) Algorithm 1' and show the optimal feasible path is not eliminated.
  4. [Theorem 9, proof] The NP-hardness proof is a two-sentence appeal to Lemma 4 and Zeitz [33], and it is not sufficient. Lemma 4's equivalence is not formalized for balance constraints, and the 'forbidden waiting' problem's structure is not shown to be encodable by LN fee maps of the form fb(e)+fr(e)*a with the balance constraints. The proof needs at least a precise reduction or a rigorous statement of how arbitrary non-consistent fee maps simulate the hardness source. As written, Theorem 9 is not established.
  5. [Theorem 10, proof and Algorithm 2, lines 6-9] Algorithm 2's early termination condition is not justified. The proof cites a nonexistent 'Theorem 5' (intended Lemma 5) and does not prove that the first popped neighbor v of s in GT yields a lowest-fee path to s. The text preceding the algorithm relies on a transformation that sets the fees of arcs leaving s to zero, but Algorithm 2 does not apply that transformation, and the proof does not use the consistency property to rule out better paths through vertices with larger c(v). Additionally, the balance check at line 8 uses the pre-fee amount and the wrong balance quantity, as in Algorithm 1. Hence the correctness of the bidirectional variant is not demonstrated.
minor comments (5)
  1. [Section 6, Theorem 10 proof] The proof refers to 'Theorem 5' but no Theorem 5 exists; the intended reference appears to be Lemma 5. This should be corrected.
  2. [Section 3, Equation (2) discussion] The sentence 'to transfer an amount a from vi to vj, we must first transfer a plus the fee in question to vi' is correct for the model, but the definition of ei in the recurrence (2) could be stated more explicitly (which of the two arcs is meant) to avoid confusion with the transposed graph used later.
  3. [Section 7, experimental setup] The experimental results are reported without implementation code or details of the priority-queue and graph representations. Since the printed algorithms contain the fee/balance bugs, it is unclear what implementation was actually benchmarked; the reported vertex counts and wall-clock reductions are therefore hard to reproduce or interpret.
  4. [Appendix A, Listing 1] The comment in the code says 'directly using Equation (6)' but the cited equation is (7) in the theorem. Also, the index notation in the proof of Theorem 11 could be simplified; the product limits in equations (10)-(13) are error-prone as written.
  5. [General] Papers should be checked for typos such as 'Inter i9-14900K' (Intel) and the use of 'Theorem 5' instead of 'Lemma 5'. These are minor but contribute to an impression of insufficient proofreading.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's results are derived from external benchmarks and direct algebra, not from fitted inputs or self-citations.

full rationale

The derivation chain is self-contained with respect to circularity concerns. The correctness theorem for the unidirectional variant (Lemma 5, Theorem 7) is imported from Kaufman et al. 'Fastest paths in time-dependent networks' [29], an external refereed result about FIFO time-dependent Dijkstra; the paper's own contribution is the reduction (Lemma 4) from LN path planning to TDN path planning and the direct verification that the LN fee map fbr(e,a)=fb(e)+fr(e)a satisfies the consistency condition (Theorem 2) because fr(e) >= 0. The NP-hardness claim (Theorem 9) likewise rests on Zeitz's external NP-hardness proof for non-FIFO time-dependent networks, not on a self-citation. No fitted parameter is renamed as a prediction: LN consistency is derived from fee-rate non-negativity, not estimated from data, and the empirical section compares two concrete algorithms on a real LN snapshot and simulated payments, which is externally falsifiable. The substantive concern visible in the manuscript is a correctness/proof gap, not a circularity: Algorithm 1 line 9 relaxes using fbr((v,v'),a) with the original amount a rather than a+c(v) as required by Equation (4), so the printed algorithm may not be the one covered by the Kaufman-based lemmas; in addition, Theorem 7's claim that Algorithm 1's search space is exactly the feasible payment paths is asserted rather than proved. But a missing proof or an incorrect application of an external theorem is not a circular step unless the target claim is presupposed in the definitions, which it is not.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theory introduces no free parameters and no invented entities. The main premises are the non-negativity of LN fees, the equivalence to time-dependent networks, and two imported external theorems. The pseudocode indexing convention is an ad hoc assumption that is inconsistent with the paper's own equations.

assumptions (6)
  • domain assumption LN fee rates and base fees are non-negative, making fbr consistent (Definition 1 and Theorem 2).
    The Lightning Network protocol defines fee rate fr as a non-negative rational, so fbr(e,a) = fb(e)+fr(e)*a satisfies a+fbr(e,a) <= a'+fbr(e,a') for a <= a'. This is asserted in Section 5 (Theorem 2) and is true for the real LN, but it is a property of the protocol, not a mathematical necessity.
  • domain assumption Path planning in a PCN with fee map f∞br is equivalent to a shortest path problem in a time-dependent network with travel time map fd = f∞br (Lemma 4).
    This equivalence identifies the payment amount with departure time and the fee with travel time; it is the bridge that imports TDN results. It is plausible but stated without formal isomorphism proof, and the capacity/balance constraint is encoded as infinite travel times.
  • standard math Kaufman et al. [29], Theorem 4: Dijkstra's algorithm is correct and has the same complexity for TDNs with consistent (FIFO) travel time maps.
    External result cited in proofs of Lemmas 5 and 6; the paper does not reprove it.
  • standard math Zeitz [33]: the 'forbidden waiting' shortest path problem in non-FIFO TDNs is NP-hard.
    External result used in Theorem 9's NP-hardness proof.
  • domain assumption Payment amounts and the range of fbr are bounded (used in Theorems 7 and 8).
    Assumed so that feasible payment paths have finite fee and Algorithm 1 searches a bounded space; in the LN, amounts are bounded by channel capacities, so this is reasonable, but it is stated without proof.
  • ad hoc to paper In Algorithm 1, the fee for transposed arc (v,v') is computed by fbr((v,v'), a) (line 9), which implicitly assumes the fee on the reverse original arc at amount a equals this value; the paper's own Equation (4) instead requires the fee at amount a+c(v) on the opposite arc.
    This is the paper's implicit indexing convention, but it conflicts with the recurrence and the worked example, making the pseudocode ambiguous.

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Cite this review

Pith. "Pith review of An Analysis of the Correctness and Computational Complexity of Path Planning in Payment Channel Networks." pith.science (2026). https://pith.science/paper/WGUKDO34

@misc{pith2026250111419,
  author       = {Pith},
  title        = {Pith review of: An Analysis of the Correctness and Computational Complexity of Path Planning in Payment Channel Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WGUKDO34}},
  note         = {Machine review of arXiv:2501.11419}
}
read the original 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.

Figures

Figures reproduced from arXiv: 2501.11419 by the authors.

Figure 1
Figure 1. An example LN containing two vertices and a single arc is displayed. The arc is [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. An example LN is displayed in (a) and the corresponding transpose, where arc [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. An LN graph G and corresponding transpose graph GT displayed in (a) and (b) respectively. The fee map fbr corresponding to GT has the properties fbr((t, i), 100) = 10, fbr((i, j), 110) = 10, fbr((j, s), 120) = 5, fbr((t, j), 100) = 10 and fbr((j, s), 110) = 20. the lowest fee path from t to s is in fact (t, i),(i, j),(j, s), which has a fee of fbr((t, i), 100) +fbr((i, j), 110) +fbr((j, s), 120) = 10 + 10 + 5 = 25. … view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: An example LN with a hub-and-spoke topology is displayed. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: An example LN is displayed. vertex t to the vertex s or a vertex in the set v ∈ {u : (u, s) ∈ E}. When either of these events occurs, the algorithm, in turn, computes a lowest-fee path from t to s in GT . Hence, Algorithm 2 is correct. Algorithm 2 is a variant on bidir…
Figure 7
Figure 7. Figure 7: For two sets of simulated payments, the corresponding histograms of the percentage [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.