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New algorithm can dramatically streamline solutions to the ‘max flow’ problem
- piqufoh 13y agoThis sounds really cool. Are there any links to the paper / presentation anywhere? I would love to learn more.
- stonesixone 13y agoKelner's web site has the link (and confirms that it corresponds to what will be discussed at the ACM-SIAM symposium): http://math.mit.edu/~kelner/publications.html http://math.mit.edu/~kelner/publications.html http://math.mit.edu/~kelner/Publications/Docs/1304.2338v2.pdf http://math.mit.edu/~kelner/Publications/Docs/1304.2338v2.pd...
- jamra 13y agoCould it be? http://arxiv.org/abs/1304.2338 http://arxiv.org/abs/1304.2338
- X4 13y agoYes. Downloaded that too. Here is more: http://math.mit.edu/~kelner/publications.html http://math.mit.edu/~kelner/publications.html I cant wait to see the C99 code for that! (maybe I should try that myself)
- carterschonwald 13y agoi'm kinda hoping to have some time to hack out a haskell implementation of the SDD solver next month (not sure if i'll have the time then, but I hope to)
- X4 13y agothat's cool! I've not seen any use of a SDD, what do you intent to use this for, or could you tell how it could be used?
- carterschonwald 13y agoexcellent question! Basically you could use the SDD solver http://math.mit.edu/~kelner/Publications/Docs/1301.6628v1.pdf http://math.mit.edu/~kelner/Publications/Docs/1301.6628v1.pd... in any context where you'd use an SVD (pseudo inverse / least squares) solver, and your matrix is symmetric and diagonally dominant. The key bit however, is that the SDD solver as above, has really nice asymptotics for sparse matrices. theres a few fancy algorithms that need to be engineered along the way, and theres also the question about how the constant factors work out in practice!
- Shish2k 13y agoFor those (like me) who clicked because algorithms are fun but don't know the max flow problem specifically: http://en.wikipedia.org/wiki/Maximum_flow_problem http://en.wikipedia.org/wiki/Maximum_flow_problem TL;DR: what's the fastest way to transport a large amount of data over a mesh of many small pipes As someone who knows little about this problem or the laws of physics, I wonder if it could be solved using physics -- set up a series of physical pipes, pump water from A to B, measure the flow at each point? If that would work, how complicated would the mesh need to be that solving it with physics is faster than brute-force computation?
- seabee 13y agoSome people, when confronted with a problem, think "I know, I'll use a physical model." Now they have two problems. (The new problem is dealing with fluid dynamics.) Water-based computation has already been done of course: http://en.wikipedia.org/wiki/MONIAC_Computer http://en.wikipedia.org/wiki/MONIAC_Computer
- deleted 13y ago[deleted]
- pmahoney 13y agoThe Soap Film: An Analogue Computer http://www.americanscientist.org/issues/feature/the-soap-film-an-analogue-computer- http://www.americanscientist.org/issues/feature/the-soap-fil...
- leephillips 13y agoThis would be an example of an analogue simulation. You first need to prove that the behavior of the physical analogue is also a solution to the original problem. The classic example is solving Poisson's equation, which gives (among other things) the electric field in a space containing charges, by measuring the deformation of an elastic sheet.
- codehero 13y agoNetwork routing is directed flow; water flow is undirected, so backflow is possible. The dangers of analogy....
- yread 13y agoThe abstract talks about ϵ-approximation for m edges in O(m^(1+o(1))ϵ^−2). If this algorithm is practical that would be quite a speed up!
- murbard2 13y agoThis could be quite useful for Ripple, which is basically solving a max-flow min-cost problem over a credit lines graph.
- jplewicke 13y agoRipple is actually a generalized network flow problem, in which the flow through an edge can be multiplied by an exchange rate when converting between currencies. As a result, regular network flow algorithms don't apply. There's also another issue with using a minimum cost criterion, since the costs are in different currencies and it's not clear how to put them on the same footing.
- murbard2 13y agoIf you make cost the log of the conversion rate plus the log of the %age fee, and treat each node as a distinct node for each currency, you do get max flow min cost.
- mrdmnd 13y agoKelner was my advisor as an undergraduate - I loved listening to him talk about graph problems. He always seemed to have a novel way of looking at something like this. I'm glad to see him get some excellent new results!
- bidev 13y agoJust a thought that , could use of Drones for transportation make this problem less useful ? I admire solution presented here however just saying.
- cliveowen 13y agoOptimizing the distribution of goods through a network of highways is just one of the, literally, hundreds of applications where this could (and probably will) be useful.
- tubbzor 13y agoYou are very correct. Max-flow/min-cut reduces to tons of other problems where this algorithm will be optimized. Here's a nice little introduction I found with some reductions on the first couple slides: http://www.cs.princeton.edu/courses/archive/spr04/cos226/lectures/maxflow.4up.pdf http://www.cs.princeton.edu/courses/archive/spr04/cos226/lec... I just finished my algorithms undergrad class last semester; one of the questions on the final was 'you are a terrorist looking to cut off supplies to your enemy, which 3 nodes must you destroy to cut off the supply network?' And there was a complicated graph with probably 10 nodes and twice as many edges (very difficult to brute force in the allotted time), and we had to find the min-cut which would severe the flow. Some really cool problems reduce to max-flow.
- mcintyre1994 13y agoOur lecturer told us that the problem was initially considered simultaneously by Russians trying to optimise the rail transport of stuff from USSR back to Russia and Americans trying to work out the cheapest way to destroy that network. Not sure if it's true, but it helps me think about the problem a lot. Edit: I clicked the link, he used them exact slides. That set of slides comes up a lot, they're great.
- tubbzor 13y agoYou are right! Edmonds-karp was developed (really just a modified Ford-Faulkerson approach with a BFS as opposed to a random augmenting path...but we'll give it to him), yielding an O(ve^2) algorithm. A russian (soviet) mathematician named Dinic nearly simultaneously, and independently, developed an O(ev^2) algorithm which we can see beats Edmonds-karp as edge density grows.
- abc_lisper 13y agoIt looks like this means we can now redirect traffic on highways to avoid congestion. I know current gps' do this now. But in future, they could suggest different paths for different users, even though they are travelling to the same approximate destination.
- mcintyre1994 13y agoThat sort of thing seems like a real possibility for self-driving cars and drones too. Swarm navigation sounds like a really interesting area actually.
- erex78 13y agoAn Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations: http://arxiv.org/pdf/1304.2338v2.pdf http://arxiv.org/pdf/1304.2338v2.pdf
- Rizz 13y agoThe paper also mentions someone else who also recently found a near linear time approximation for maximum flow: http://arxiv.org/pdf/1304.2077v1 http://arxiv.org/pdf/1304.2077v1
- ckaygusu 13y agoI just skimmed through the paper thinking maybe I can grab one thing or two. What I've found there is completely black magic.
- GarvielLoken 13y agoYour method of learnings is flawed. The specific lesson to be learned here is that they changed their view on the solution for a max-flow problem from a serial probing algorithm to a parallel electrical probing solution and lowered the time-complexity. The general lesson to be learned here is that they changed their view on the solution for a massive parallel multiple choice problem from a serial probing algorithm to a parallel probing solution and lowered the time-complexity. The actual implementation is uninteresting and will always vary according to domain and optimization, but the intuition lesson is always a valid tool that we now have. And in hindsight this was a really obvious solution, carouse one should try all the routes and identify the bottlenecks. Food for thought, single cpu core vs GPGPU.
- DannyBee 13y agoI'm always wary of these kinds of research papers because they are often not using comparative benchmarks sanely. However, that said, there are all kinds of interesting max flow problems, so hopefully it pans out in practice. Heck, even some profile based compiler optimizations can be formulated as max-flow problems.
- ahomescu1 13y agoThese papers improve time complexity, not actual running time. I'm not sure it ever makes sense to do comparative benchmarking, since you can't test on arbitrarily large inputs (which is what's important when looking at algorithm complexity). For smaller inputs (for example, N in the 1000-1 million range), constant factors impact running time significantly. To compensate, you'd have to run a benchmark of size 10^10,000,000. EDIT: For an example, let's say the current best algorithm for some problem takes O(N^2), with a small constant (let's say 100 * N^2 operations). Then someone comes up with a O(N^1.9), but with a larger constant (let's say it takes 1,000,000 * N^1.9). The newer algorithm will practically be slower for any reasonably-sized input you can throw at it, even though it's a major theoretical breakthrough.
- Rizz 13y agoAs you are talking about reasonably sized input, your constant examples should also be reasonable. 10000 times difference in the constant is not reasonable. You wouldn't even get that from going from all data in the level 1 cache to random memory access with 10 times more instructions. And "these papers" often do care about running time and will say whether it is possible to implement the algorithm efficiently, if you're lucky they'll even have done an implementation and you can see at what dataset sizes they are faster in practice. And even if it were slower in practice, research like this is very useful, because it helps gives other researchers inspiration, ever since someone thought of looking at these problems like electrical flow in 2010, there have been ever faster approximations for it, while others will find ways to implement the algorithm efficiently, just in case a naive implementation of the offered algorithm isn't efficient enough. Dismissing such research without having found any actual extreme inefficiencies in the proposed algorithm is not an attitude suited to anyone who wants to see progress. And finally, the actual speed increase in the past decades hasn't been from O(N^2) to O(N^1.9) but from O(VE^2) to O(VE), which obviously does have a real impact, and even with a non-realistic 10000 times bigger constant it would still be far faster in any graph with more 10000 edges. And graphs with billions of edges aren't unusual anymore.
- igravious 13y agoWas the theoretical lower bound of this `max-flow' algo known? Just curious. Also, are there a class of algorithms where the current practical state of the art differs greatly from the theoretical or how would I go about finding out the answer to this question? Much obliged. I kind of thought a lot of these routing problems were worked out or known not to be work-out-able. Just shows what I know :/ * sigh