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Posted by O3marchnative 14 hours ago

Missile defense is NP-complete(smu160.github.io)
266 points | 283 commentspage 2
Havoc 5 hours ago|
There is also a big real world experience element that is hard to account for mathematically.

There are stories of ukrainian operators expressing bewilderment and BCC countries sending 8x interceptors at millions a pop at a 20k shaheed. The world doesn't seem to have acclimatized to well...how the world works now.

There is a very fundamental disconnect at play here and I fear it'll get us all into trouble

mattmaroon 5 hours ago|
Well, it’s less about the value of the interceptor and more about the value of what the drones are heading for. If it’s a desalination plant that makes 20% of your freshwater, eight missiles are probably well worth it because a lot of people can die.

Middle Eastern countries have much more condensed critical infrastructure and economic targets than we do.

Iran has expected a war like this for decades and been continuously preparing, most of the other nations they have not.

Havoc 2 hours ago||
That logic right there is why the US is going to lose. That math only works in isolation

The Ukrainians don’t think about today’s tradeoff but also about tomorrows. They learned that when when a three day special operation turned into four years.

You have to match your best defenses against the best expected incoming attack- across time- even if it means taking a hit. Yes desalination plant included

The us doctrine of defend all the things all the time against everything has failed in light of modern drone warefare

angelgonzales 1 hour ago||
The US already has “shoot the archer” doctrine which strategizes to target the site(s) launching cheap drones rather than the drones themselves. With US air superiority this seems feasible.
blueblisters 10 hours ago||
Gaming this out for peer adversaries is mostly moot, right? The post-Cold War strategic balance has mostly hung on MAD. And Russia, in particular, has responded to any attempt at building missile shields with more capable missiles.

It's likely more relevant for asymmetric conflicts that involve conventional weapons, and would enable an otherwise less resourced adversary to become a near peer.

Dennis Bushnell from NASA presented this deck in 2001, and is quite prescient about UAVs and distributed warfare.

https://alachuacounty.us/Depts/epd/EPAC/Future%20Strategic%2...

CamperBob2 4 hours ago|
Eh, he threw so much random stuff at the wall that some of it is bound to stick. An early slide in his presentation says there will be "no pixie dust," but that's 90% of what follows.
maxglute 12 hours ago||
>Hence, for one warhead, a defender can launch 4 interceptors and have a 96% chance of successfully intercepting the incoming warhead. >Unfortunately, those numbers are optimistic.

This part worth stressing, ceiling for more performant missiles, i.e. faster, terminal maneuvering, decoys are geometrically harder to intercept. Past mach ~10 terminal and functionally impossible because intercept kinematics will break interceptor airframes apart.

AFAIK there hasn't been tests (i.e. FTM series) done on anything but staged/choreographed "icbm representative" targets. Iran arsenal charitably pretty shit, including high end. Hypothetical high end missile with 10%-20% single shot probability of kill requires 20-40 interceptors for 98% confidence, before decoys, i.e. 40x6=240 interceptors for 1 missile with 5 credible decoys.

The math / economics breaks HARD with offensive missile improvements.

Voultapher 12 hours ago|
Lasers. No really, near-future laser systems with adaptive optics and good spotting - for example distributed SAR satellites - dramatically shift that balance [0].

[0] https://toughsf.blogspot.com/2025/05/the-laser-revolution-pa...

gpderetta 12 hours ago|||
I doubt a MW laser can reliably intercept a reentry vehicle. A lot of energy is lost through the atmosphere when intercepting a warhead in space, from a land based laser. Once it reenters the atmosphere there might not be enough time. You also need to burn through the heatshield that the warhead is equipped with for reentry.

Even if you can can deliver enough energy for long enough, there is no fuel to burn and it might not be easy to detonate or disable the warhead.

For ICBMs, one idea was to use orbital, nuclear powered lasers to hit the missile on the boost phase.

But that's very much not near-future.

Lasers might still be useful for rockets, drones and cruise missiles of course.

O3marchnative 11 hours ago|||
> For ICBMs, one idea was to use orbital, nuclear powered lasers to hit the missile on the boost phase.

Author here. Thank you for your insight.

I took some time to read about the recently proposed "Golden Dome" defense system, and what you laid out seems to be the end goal [0]. It's difficult to tell how realistic this actually is. The size of the constellation of satellites needed seems prohibitive, to say the least.

[0] https://armscontrolcenter.org/fact-sheet-golden-dome/

[1] https://www.defenseone.com/ideas/2026/02/space-based-interce...

Voultapher 8 hours ago|||
I get the impression you didn't read the linked post. It goes into the details, atmospheric absorption for different wavelength, weather conditions, tracking time, interception time based on warhead hardness ratings and many more details. It's paper based, so in practice it will be more complicated and there are things it will have missed, and things we don't even know yet as being operational challenges for such systems. At the same time, it does present a compelling narrative and I'd much rather discuss individual assumptions or sources than dismiss it entirely based on a gut feeling.
maxglute 11 hours ago||||
Maybe for subsonic, high end missiles I'm extremely skeptical. Need 5-10MW to get useful dwell power on high end hypersonic inherently shielded against reentry thermals. Speculative laser defense are infra size defense, not mobile trailer size. Factor in duty cycles (i.e. shots per minute) and it seems dead end. Half of economics of missile defense is mobility - building density relative to threats by moving platforms. Last 2 parts real constraints, high-end adversaries coordinate salvos to arrive in time. Interceptor magazine depth limited = still throw 100s of interceptors to engage multiple targets if required. Lasers = serial visual range engagement. Figure out dwell time + duty cycle to saturate. Hypersonic can go from over horizonal to hit target in 10 seconds, a laser couldn't engage more than 1-2 missiles in that time. Technically 1, because by the time you fried 1st target the 2nd is so close the shrapnel will hit on momentum.
consumer451 11 hours ago||||
Lasers are not "all weather" weapons as far as I am aware. Clouds, snow, fog, rain, and just humidity all degrade their performance greatly.
O3marchnative 11 hours ago|||
The recently announced "Golden Dome" project intends to get around this issue by putting a vast constellation of satellites into orbit. Each satellite would likely need a serious source of power in order to use its laser. Assuming that's just an engineering problem, then the issue becomes coverage. That is, depending on the adversary's capabilities, you'd need an absolutely massive constellation in orbit [0].

[0] https://en.wikipedia.org/wiki/Golden_Dome_(missile_defense_s...

consumer451 11 hours ago|||
This is such an insane plan, and I don't mean that in a good way.

For one thing, it can do little to nothing about low flying nuclear tipped cruise missiles, especially in less than ideal weather. These already exist, so the Golden Dome system is already inadequate on day one.

gpderetta 10 hours ago|||
The idea has been around for a while: https://en.wikipedia.org/wiki/Brilliant_Pebbles

> https://en.wikipedia.org/wiki/Golden_Dome_(missile_defense_s...

> President Donald Trump signed an executive order directing the United States Armed Forces to construct the [...] Golden Dome

Well, that's just taking the piss!

Voultapher 8 hours ago|||
The linked article covers that in depth, it's not implausible to punch a hole through a storm with pulsed laser of that class. Honestly we don't know enough about these systems to know their operational limits but we know weather will play a role.
apt-apt-apt-apt 11 hours ago|||
Try that on my spinning, mirror-coated missile!
contravariant 12 hours ago||
> Note that a more complete model would multiply each term by P(track)_j — the common-mode detection-tracking-classification factor developed in the previous section — but the standard WTA formulation assumes perfect tracking.

I'm not sure that is a useful model, or more complete. I don't think you can assign interceptors to undetected missiles, so considering their effect on the value is rather pointless. It's effectively a sunk cost.

Multiplying with the probability also makes no sense from an optimisation point of view. Why would you assign lower value to a target about to be hit simply because you were unlikely to detect the missile?

The tracking probability only shows up in the meta game described at the end, where one side is trying to optimise their ability to hit valuable targets and the other is trying to optimise their ability to prevent that from happening.

jcul 12 hours ago||
Really interesting.

Forgive my ignorance, but I thought Israel's "iron dome" offered a very effective defense.

Is this just from short distance missiles from neighbouring countries?

This article seems to indicate it's very difficult to achieve a high success rate against multiple missiles.

Admittedly I probably need to read up on this more.

icegreentea2 10 hours ago|
There are multiple tiers of missile (and ballistic missile defense).

Especially with ballistic missiles, the longer the range, the faster the inbound warhead will be in the terminal phase (roughly). So longer range ~= faster meaning more difficult to intercept.

"Iron Dome" is the name generally used to describe Israel's lowest tier set of defenses. Very roughly Iron Dome is designed to defend against stuff that you could plausibly fire from the back of a truck, and have a max range of around ~50km.

Very roughly, these were intended to take on something like GMLRS (realistically, massed volleys of unguided rockets) - these are rockets that one or two people could conceivably manhandle, and are traveling in the neighborhood of Mach 2-3. One of the key innovations of Iron Dome is its ability to quickly ascertain and design on which rockets were unlikely to strike valuable areas, and only engage the actually threatening ones.

The next tier up is David's Sling, and then Israel's wider set of high performance anti-ballistic missile systems. Returning the the range <-> speed thing, we'd need something like a medium range ballistic missile to get from Iran to Israel. For something like the Shahab-3, that's like ~Mach 7 during re-entry.

If we step up to IRBMs (so something that China might use to strike at Guam), we're probably talking like Mach 10.

jcul 7 hours ago||
Interesting stuff, thanks for the long and detailed response.
enaaem 5 hours ago||
Yeah this is really bad news for Israel. America does not have the resolve for a major ground invasion, and Israel cannot intercept missiles indefinitely.
u_sama 13 hours ago||
Great nerd title, the maths made me nostalgic as I haven't seen a Sigma/Pi in a few years
captainswirly 12 hours ago||
This is rocket defense, not missile defense.

Pretty much nothing can stop those ICBMs - those aren't rockets.

If you dig deeper than mainstream news - Iran is lighting Israel up with those ICBMs, but they don't use them too often.

nerfbatplz 12 hours ago|
Technically Iran fires SRBMs and MRBMs not ICBMs. They intentionally gimp their missiles to avoid advertising ICBM range as a way of placating Europe.
captainswirly 12 hours ago||
Cope harder. I saw the videos that Israel tried to ban.

Whatever you want to call them, they are "hypersonic" traveling over 15k mph.

They definitely look fast too in the videos where they smash into Israeli housing destroying everything in an instant.

Maken 11 hours ago|||
ICBM means the missile is able to be launched into other continent (more than 5K kilometers in range), not about its effective warhead. Iranian missiles are fast and efficient, but their effective range is essentially being able to target Israel and the Gulf States, which means they are not ICBMs. Also, the one country in the Middle East who does have ICBM missiles and can target Europe is Israel, not Iran.
echoangle 9 hours ago|||
The IC in ICBM mean Intercontinental and specifies the range, not how impressive it looks in videos.
energy123 13 hours ago||
What is the steady state? Assume you have two competent superpowers, both researching missile offense and defense, over the next 1000 years. What are the asymptotics of the interception rate from 0 to 1000?
bob1029 12 hours ago||
The steady state would look like a sinusoidal signal. This is more of a cycle than a hill climbing thing.
energy123 12 hours ago||
Are you sure there isn't a structural advantage to either offense or defense that will reveal itself with more iterations, and we won't converge to either 0% or 100%?
adampunk 12 hours ago||
https://en.wikipedia.org/wiki/The_bomber_will_always_get_thr...

That’s the steady state. Interceptors are expensive; missiles are (relatively) cheap. There’s no sine wave or cat and mouse game. If you’re trying to defend against a peer, missile defense loses.

energy123 12 hours ago||
I'm talking about interception probability, not the relative cost. I get that interceptors will probably be more expensive indefinitely (unless we start putting lasers into orbit to get around atmosphere, or something unexpected like that).
hedora 12 hours ago|||
OK, assume infinite resources, but the attacker only has one missile, and the defender only has one interceptor.

Optimal strategy for the attacker: Figure out how fast the interceptor can reach your missile, and have it split into a dozen warheads on different trajectories a mile before that. Include the blast radius of the interceptor in the calculation in case the defender decides to set of high-altitude nukes to defend itself against your missile.

The non-proliferation treaties we just pulled out of banned multi-warhead ICBMs decades ago because there's no feasible counter-move. That's bad for the missile business.

Back in reality, the attacker just builds 100,000 conventional drones, and 1 identical looking one with a nuke in it. Eventually, the defender runs out of interceptors, so the intercept probability trends to 0. At that point, the attacker sends the nuke without varying the behavior of the conventional drones.

adampunk 12 hours ago|||
Well there the cost plays a part! It’s not independent. If I can build 1000 missiles for every 1 interceptor, the probability of interception hardly matters.

This actually was why we planned to put lasers in space: the economics of one nuclear-pumped laser reflected through Unobtanium were better than any other interceptor. And even that if the effect worked (it didn’t, they could not prove lasing and fired an engineer who blew the whistle on that), the system could be defeated by a staggered salvo.

femiagbabiaka 13 hours ago|
could use some investigation of the ukranians techniques -- the number of interceptors the U.S. used within the first four days of the war eclipsed the total amount Ukranians have had for the war
OrangePilled 12 hours ago||
"Lessons US and Gulf could learn from Ukraine’s air defence warriors"

https://www.thetimes.com/world/russia-ukraine-war/article/us...

"Ukraine’s low-cost Shahed killers draw US and Gulf interest, but a wartime ban blocks sales"

https://apnews.com/article/iran-ukraine-shahed-russia-drone-...

"Ukraine Helps U.S. Bases in the Mideast With Stopping Drones"

https://www.nytimes.com/2026/03/09/world/middleeast/ukraine-...

"Ukraine deploys units to five Middle East countries to intercept drones"

https://www.reuters.com/world/europe/ukraine-deploys-units-i...

Maken 11 hours ago||
What the USA learned from Ukraine is that the Shaheds are amazing and they want them too [1], apparently.

[1] https://en.wikipedia.org/wiki/Low-cost_Uncrewed_Combat_Attac...

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