Hi Zmn!

While you have some interesting thoughts about the implementation of
min-cost flow based routing, I feel that there are at least two grave
misunderstandings here:

First, the problem with the base-fee or any similar constant is that
they make the fee function concave at the point where the flow
increases from 0 to 1. That is, there is a jump from 0 to
base-fee+1*prop_fee or similar, and the function does increase slower
from there. This is the case for any function that includes a constant
per edge that is only paid IF there is non-zero flow along that edge.

Second, while there might be additional problems in the disect-phase
(which I am sure can be handled by a plethora of techniques), the fact
remains that the min-cost flow problem itself is NP-hard when the
cost-function is of this form.

I realise that this is a fairly theoretical term, so let me shortly
describe what it means: We know (there is a simple mathematical proof
for this, see reference in our paper) that if we could find an
algorithm that solves the min-cost flow problem ON ALL GRAPHS for all
cost-functions of this form (basically, a constant per edge) in
polynomial time, we could then use this algorithm to solve ANY problem
in NP in polynomial time. This means, we would have an algorithm that
can find a solution to any problem in polynomial time whose solutions
can be VERIFIED in polynomial time. Because there are thousands of
problems of this kind that the smartest people in the world have tried
to solve for decaded without success (and just one success would be
enough to solve all of them!), it is highly doubtful that such an
algorithm exists. That is why we can be fairly certain that we cannot
in general solve the min-cost flow problem for concave functions in
polynomial time.

HOWEVER, this does not mean that in practice it isn't possible to find
good solutions for the subset of problems we are likely to encounter
in our application. It might be that the LN-graph is of a certain
structure that makes finding a solution easier. It is certain that we
do not need the optimal solution at all and approximations will be
easier to find. All we know is that we do know how to solve the
problem generally for convex functions, and that we are very confident
that we cannot in general solve it optimally for concave functions.


Am Mo., 30. Aug. 2021 um 12:58 Uhr schrieb ZmnSCPxj via Lightning-dev
> Good morning list,
> It seems to me that the cost function defined in Pickhard-Richter
> can in fact include a base fee:
>     -log(success_probability) + fudging_factor * amount * prop_fee + 
> fudging_factor * base_fee
> (Rant: why do mathists prefer single-symbol var names, in software
> engineering there is a strong injunction **against** single-char
> var names, with exceptions for simple numeric iteration loop vars,
> I mean keeping track of the meaning of each single-symbol var name
> takes up working memory space, which is fairly limited on a wetware
> CPU, which is precisely why in software engineering there is an
> injunction against single-char var names, srsly.)
> It seems to me that the above is "convex", as the `fudging_factor`
> for a run will be constant, and the `base_fee` for the channel
> would also be constant, and since it seems to me, naively, that
> the paper defines "convex" as "the second derivative >= 0 for all
> `amount`", and since the derivative of a constant is 0, the above
> would still remain convex.
> (I am not a mathist and this conjecture might be completely asinine.)
> So, it seems to me that the *real* reason for `#zerobasefee` is
> not that the **mincostflow** algorithm cannot handle non-zero `base_fee`,
> but rather, the **disect** phase afterwards cannot handle non-zero
> `base_fee`.
> For example, suppose the minflowcost algorithm were instructed to
> deliver 3,000msat from `S` to `D`, and returned the following flow:
>     S -->3000--> A ->1000-> B
>                  |          |
>                  |        1000
>                  |          v
>                  +-->2000-> C ->3000-> D
> In the "disect" phase afterwards, the above flow solution would have
> to be split into two sub-payments, a 1000 sub-payment `S-A-B-C-D`
> and a 2000 sub-payment `S-A-C-D`.
> However, this does mean that the base cost for `C->D` and `S->A` in
> the above flow will pay *twice* the base cost than what the above
> cost function would have computed, because in reality two independent
> payments pass through those channels.
> Thus, the current Pickhardt-Richter scheme cannot work with non-zero
> base fees if it were modified to consider fees, but not due to the
> mincostflow algorithm, rather because due to the disect algorithm.
> ### Converting Non-Zero Base Fees To Proportional Fees
> An alternative solution would be to optimize for a *maximum* fee
> rather than optimize for the *exact* fee.
> We can do this by virtually splitting up the entire payment into
> smaller bunches of value, and asking mincostflow to solve individual
> bunches rather than individual msats.
> For example, we can decide that the smallest practical HTLC would
> be 1000 msat.
> Let us call this the "payment unit", and the mincostflow algo
> solves for paying in these payment units instead of 1msat units.
> (Actual implementations would need to have some heuristic or
> reasoned rule-of-thumb assumption on what the smallest practical
> HTLC would be.)
> In our example, suppose we need to send 2001msat from `S` to `D`,
> with the payment unit being 1000 msat.
> Then we would actually have the mincostflow algorithm work to
> deliver 3 payment units (3 * 1000msat) from `S` to `D`.
> Let us suppose that the mincostflow algorithm returns the following
> flow:
>     S -->3-----> A ->1----> B
>                  |          |
>                  |        1 |
>                  |          v
>                  +-->2----> C ->3----> D
> Actually, what the mincostflow algorithm uses as a cost function
> would be:
>     -log(success_probability) + fudging_factor * amount * prop_fee * 
> payment_unit + fudging_factor * base_fee * amount
>     == -log(success_probability) + fudging_factor * amount * (prop_fee * 
> payment_unit + base_fee)
>     where: amount is in units of 1000msat, our smallest practical HTLC.
>            payment_unit is the unit, i.e. 1000msat.
> What the above means is that the mincostflow algorithm *allocates*
> `3 * base_fee` for `C->D`, since the `amount` flowing through
> `C->D` would be 3.
> However, when we pass the above flow to the disect algorithm, it
> would actually only split this into 2 sub-payments, so the
> actual payment plan would only pay `2 * base_fee` for the
> `C->D` leg on both sub-payments.
> In short, this effectively converts the base fee to a proportional
> fee, removing the zerobasfee requirement imposed by the disect
> algorithm.
> That is, the cost computed by the mincostflow algorithm is really a
> maximum cost budget that the subsequent disect algorithm could later
> spend.
> In effect, this converts the base fee to a proportional fee.
> This may be acceptable in practice.
> This approximation has a bias against non-zerobasefee --- it would
> treat those channels as being far more expensive than they actually
> would end up being in an *actual* payment attempt --- but at least
> does not *require* zerobasefee.
> It would be able to still use non-zerobasefee channels if those are
> absolutely required to reach the destination.
> This should at least help create a practical payment algorithm that
> handles current LN with nonzerobasefee, and provide an impetus
> towards making zerobasefee a reality.
> Note that this approximation would be non-optimal --- there may
> be solutions that provide lower costs than the output of this
> approximation would provide.
> It may be practical to have the payment unit be adjustable by a
> higher-level (but still automated) process.
> For example, it may be considered that having more than 500 splits
> would be heuristically determined to be impractical, in which
> case the payment unit could be the actual amount divided by 500
> rounded up.
> Then if that results in extreme overestimation of base costs
> (i.e. the subsequent disect stage results in far fewer than 500
> splits) the payment unit could be adjusted upwards.
> Similarly, if that results in a payment plan that has high
> base fees, the payment unit could be adjusted downwrds.
> --
> Similarly, having particular split amounts is helpful to reduce
> the ability of intermediate nodes snooping on payment amounts,
> so having a particular fixed payment unit may also be useful
> nevertheless, even if this approximation is otherwise undesirable.
> That is, in the above example, we just send 3 sub-payments
> instead of the minimum 2, so that every sub-payment is always
> uniformly 1000msat.
> This leaks the least amount of information to intermediate nodes,
> especially after payment decorrelation is widely deployed.
> (this is the same principle for why a uniform unit amount is
> needed in practical CoinJoin implementations; it increases the
> anonymity set of individual sub-payments.)
> This would be helped greatly by payment decorrelation;
> intermediate nodes would remain uncertain that multiple
> HTLCs with the same amount and the same in-channel and
> out-channel are from the same overall payment or not.
> Assuming many software uses the same payment unit consistently,
> less information can be extracted by surveillors, and the
> increased cost may be considered justifiable in paying for
> additional privacy.
> --
> ### Whole-flow Payments
> Suppose we were to instead modify the LN protocol such that,
> instead of MPP having individual sub-payments that split at the
> source and join at the destination, we allow the source to
> instruct arbitrary intermediate nodes to join and/or split
> payments.
> That is, suppose LN allowed the source to indicate to arbitrary
> forwarding nodes to split payments and join them.
> (This would require nearly all nodes to upgrade, which may be
> impractical, but let us consider its implications first without
> committing to actually doing this.)
> A node that is a join node would be instructed to wait for all
> incoming HTLCs to arrive before they create any outgoing
> HTLCs.
> A node that is a split node would make more than one outgoing
> HTLCs.
> Forwarding nodes can be both join and split nodes.
> This would allow the output of the mincostflow algorithm to be
> used directly, without the problematic disect stage.
> Nodes that have more than one in-flow would be join nodes,
> while nodes with more than one out-flow would be split nodes.
> Returning to the example:
>     S -->3000--> A ->1000-> B
>                  |          |
>                  |        1000
>                  |          v
>                  +-->2000-> C ->3000-> D
> In the above, `S` sends a *single* payment out on channel
> `S->A`, and instructs `A` to split the payment.
> Then `S` also instructs `C` to join two payments, resulting
> in a single HTLC out on `C->D` if `C` receives on both
> `A->C` and `B->C`.
> The effect is that, for a single payment plan (presented as a
> flow, outputted by a mincostflow algorithm) for each channel
> with nonzero flow, there would only be one HTLC that gets paid
> for using a single `base_fee`.
> Then, the base fee computed by the cost function would be exact
> and there would be no underestimation of the base fee for
> non-zerobasefee channels.
> Let us then consider the practicality of this scheme.
> If a hop were to fail to deliver, then some join nodes would
> be left hanging, waiting for an incoming HTLC that will not
> arrive.
> The failure would have to be handled by a split node, when it
> receives a `update_fail_htlc`.
> That split node cannot propagate the failure backwards to the
> source, since it would still have other outgoing HTLCs in
> play, which it cannot safely ignore; it can only propagate
> the failure backwards if all the other outgoing HTLCs also
> fail.
> To support this, a split node could give a request-to-fail
> signal to its other outgoing HTLCs.
> Then, any other join nodes still waiting for an incoming
> HTLC (and would not yet have created any outgoing HTLCs,
> or for the destination, would not have claimed any incoming
> HTLCs) would fail all its incoming HTLCs.
> Otherwise, an intermediate node receiving a request-to-fail
> signal from any incoming HTLCs would simply propagate
> this request-to-fail outwards to all its outgoing HTLCs.
> This implies that for this scheme, if *any* hop fails, the
> entire payment fails and has to be restarted "from the top".
> This is in contrast with the current scheme of splitting
> only at the source and joining only at the destination.
> In the current LN, if *any* hop of an MPP fails, only the
> sub-payment(s) going through that hop will need to be
> retried.
> Other sub-payment(s) can keep being "in play", and the
> source, having all the information needed, only needs to
> recompute the failing outgoing payments.
> On the other hand, the same information (particular hops are
> failing) would be propagated back to the source, and the
> source having to recompute the entire flow rather than just
> some subset of the payments is not much worse computationally
> (work is still dominated by `O((m*m+m*n)*log n)` where `m` is
> the channels of the *entire network* and `n` is the nodes
> of the *entire network*, amount is just a piddling little
> `O(log(u))` term).
> It also removes the need to virtually reduce the capacity of
> intermediate hops while other sub-payments are in play
> (however, a practical implementation would need to consider
> the possibility of multiple *simultaneous* outgoing payments
> anyway and would still retain this reduction of capacity in
> case a new parallel outgoing payment is triggered while one
> is still ongoing).
> This leads us to some fairly interesting choices:
> * [ ] Whole-flow payment is a brilliant idea and we should
>   totally do it.  `#deletethezerobasefeedebate`
> * [ ] Whole-flow payment is a horrible idea and we should
>   totally not do it.
>   * [ ] As we can see, the LN community needs to get together
>     for better payment success, and supporting whole-flow
>     payments is worse than setting `basefee=0`. `#zerobasefee`
>   * [ ] As we can see, the current LN provides a vital service
>     (partial failure of MPP) and intermediate nodes need to be
>     compensated for this service.  `#nonzerobasefee`
> Regards,
> ZmnSCPxj
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