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>> Think this it was funny he's a professor. Before that with Professor Richard Watson to. Me he's been. A pioneer leader. Very successful researcher many of us including There are very important and complexity. He wanted at least enough of the division of large many orthotics that most recently was that.

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Just. Thanks Larsen the ocean and thanks to all of you for coming back here. Yeah I'll be talking about a body of work which I did I mean it's actually broader than that it's more sort of a leg but I'll touch upon some works I did with Josh back and see cause an undergrad at CMU and started spitting at Stanford and what these things are about as tried to explain.

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The source of algorithms along with the for that I will let me get to sort of the book rather than met our principle is they can ask OK fine Some problems are easy and some are hard. You know maybe a little of this. So we can ask that probably loud enough so interviews classifying them as easy and hard we can also ask why are some problems easy and why are they hard so what does the underlying mathematical structure which enables out of them straight exist or somehow when you don't have that structure things become hard and given the very basket of problems in many different ways you can solve them cannot hope to answer these in good generality but there's been major subsidies for one of these plans of constraint satisfaction problem and some of it's widgets and.

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So the idea here is that. The structured underlying to get to the structure non-linear problem you're actually going to look at the symmetries in the solution space so what these polymorphisms which was in the title are is informally an operation under which the solution space is closed and I started from examples which will really cost less this notion and then we will formalize.

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OK So let's start with a very familiar example in equations take anything rationals integers finance field and take a system or lead equations over that ring. So these have the following very nice closure property that if I take 3 different solutions then X. one X. 2 X. 3 then X. one minus X. 2 plus 60 is all sorts of get more generally any I find combination is it's closer but I find combinations and since these exist in everything I just chose this particular to find combination one and mines one so then say that the following operation which takes treating elements and odd preserving element locally would stick if A.B.C. and I'll put in a minus be perceived say that it's a polymorphism for this example and idea here is that when they do this operation a good coordinate way so you can pick 3 different solutions and just a play F. coordinate locally without consideration of the other way and you will guarantee to get a new solution again this is of course an easy problem by various methods and the continuous example let's take linear programming explicit equal to B. and B. just think about the feasibility and now we know this property that it's closer to convex convex commission to Frederick 2 solutions any convex combination is a solution and and B. Of course knowing optimization that complexity is a prescription for delegate them the lack of context usually means things are hard so this is somehow what I'm going to talk about is some sense discrete analogues of.

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That phenomena the convexity from and yet the polymorphism is there a whole host of polymorphisms for any kaita you can mix 2 things F. of a B. to click rationals and then or put a day plus one month later B. That's a problem arms and this is solvable by other methods Yeah.

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Yes Yes I'm not saying that this it's not at all obvious why. What the leader of them even has to do with the spoiler modest but there seems to be those underlying principle that this polymorphisms maybe give you a hint that there should be an algorithm off one of them does not make use of the volume of them in any obvious way to get that's a that's an important discrimination.

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Case. Though in some of the examples I will give you that's really playing a. Crucial role in the UK So yeah there's a huge leap from here to here. But let me do one more let me know more of the discrete case so let's take a fairly easy problem to Sat so I have variables and to enable classes and I need to find satisfying assignment so here what happens is that if I take any 3 satisfying assignments to an arbitrary 2 sat instance and they take their coordinate majority you can check that that will also satisfy the 2 seconds is pretty easy to check and then the closure probably therefore is that if I take the solutions that a majority co-ordinate ways is also a solution and the polymorphism then there's just the 3 big majority function you just take 3 bits and I'll put the majority that is a polymorphous and besides them and formalize this couple of slate stone and this is of course all of them by a couple of minutes which you know easy to do so all these examples we saw that there were some polymorphous sums and they were all saw algorithms the link between these is not clear except that both of these exist but now let's take from Paul said Let's call the police act suppose you have a clause of with 30 then you can try to do what he just did before that we can take the satisfying assignments take their majority that's not going to be a satisfying assignment in Germany.

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OK but you might wonder OK that was just majority maybe there are some other clever polymorphous and one can show that they're basically not interesting polymorphisms for treason and I'll define interesting truckie and indeed this problem is simply. Yes it will be constancy. Of some some pondered Well yeah this is a little bit murky don't make it precise there's just a sort of warm up give you some flavor of things so what polymorphisms out of local operations to combine multiple solutions and produce a new solution that's the word poly there to say is that it takes many solutions and data symmetry which is a hierarchy of symmetry that other than a endomorphism are automorphisms.

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And what we now know all this remarkable body of work and constrain satisfaction to this is that Palami time algorithms exist for C.S.P. which I'll define if and only if they are interesting polymorphs. So there's a very precise and remarkable link between when you have algorithms and mathematical structure which is really very satisfying a complete theory for a relatively rich class of problems OK so I'll tell you about that story a little bit because that's the backdrop for our work and then we will get tools some of the newer stuff in a band which tries to build upon that they could do more general sceptics beyond see a space that was in the past so one of the C.S.P. probably all of you have seen this.

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But just to refresh and set up notation you have some an instance of a C.S.P. looks like this you have some N. variables which are supposed to be values from some domain let's say the 014000000000 problem and you have a collection of constraints and one of the constrains the constraints of simply going to take some local subsets of variables and constrain them by some predicate so if we have I allow you to predicate B. one and P 2 which is 2 kinds of predicate abstractly that is called a template so the template defines a problem and what it can do is for each you can take a subset of variables a couple of variables and constrain them by a predicate drawn from the set top.

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And the question obvious and C.S.P. is I'll focus for now one decision version is that an assignment of values from deed to the way bills and we would satisfy is on the streets OK so if you so this must look some family have to do sadly as happens or it just takes more abstractly where you have arbitrary predicates alone and the predicates define the problem and the complexity of the problem is going to define on the 2nd gun depend on the said gun a little bit more precisely so you have a template gamma which is simply a finite set of predicates or some the mean and C.S.P. of gamma is the class of problems see a species where the constraints are allowed to be drawn from the sky so a general instance will look like this will have end variables and you will have some apples constrained by some P.S. and all the B I's must be drawn from Com and the question is is that an assignment which satisfies all these constraints OK that and this captures a host of problems just to give you a brief sample so you can ask if a graph is by a party by the following thing the domain is 01 and you have a single predicate with just checks X. is not equal to one so if you have a collection of those then you are simply asking Is there a to coloring of a graph so these actors domain as you know one and you have these and of classes so and can also capture things like graph going to 50 by saying the S. must be set to 0 and T. must be set to one and you can have implication constraints and this will capture that as being not bad for Mr.

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These are all bullion problems but this works for non bully and domains too for example generalising by partners for any thing is a very canonical C.S.P. you have a single single predicate which checks things are not equal and the domain is 12 that captures whether the graph is capable of not and 2 more problems which are important in approximation are also C S P's.

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And these are it doesn't matter what they are but the label covered in any games these are basically C S P's of added the 2 so each of these people is depend on 2 rebels and one of them the the relation is a function the other one relation is a politician so these are 2 others this would have any class of problems so the main thing is depending if you change government to get a new problem and.

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And the broader goal is for which government is this problem easy which guys at heart because we know that this is easy this is hard this is easy this is hard this is heart disease so why why did this happen. Well not so I have OK but I didn't mean to say that but I'm but it is easy because this is the definition version so to decide if you need games and satisfiable is action itself and if you are really bored in my talk and think about what polymorphism makes that possible look at what is a polymorph have said this too many times so here is a ball definition so you have some arbitrary predicate about it.

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And a poly morphism is just a hierarchy of symmetry for this predicate OK so a function of some added the M. from is a polymorph some of P. If any of are following with its very natural definition whenever I take M. topples in P. R. which satisfy P. and I play F. column wise because these are M. of these I cannot play them column I still get a new value in B. This will also say again imagine this thing where you write up a matrix where the whole satisfy P R play if column wise you will get a new predicate satisfying you'll get a new topic since.

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So this combines any Solutions to be to a solution to another solution and that's what a single predicate will be called that set of functions of arbitrary out of is not fixed it can be any fine a dowdy so that fall of P. the set of all such functions and many how many predicates you want to be a polymorphism for Adama if you are polymorphism for each predicate in that's.

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Just my back to be simpler yet only just my back to the same thing because that in the C.S.P. instance you'll be constraining a particular couple by a particular P.N. so you can change them so you must map B. to B. and you should do that for every one of those people into.

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It but actually later I will change the speed to Cuban to come to promises if and when they come here but for now does the definition hopefully the definition is clear I have some examples in the next layer. But you might wonder does this follow one of them exists or done maybe the set this empty to be a reason the set is never empty no matter what Damas what it wonders function can do is you give me M. things it can simply copy the 2nd level obvious at this point so for any i either what are these declared functions which ignores all the input and simply outputs they coordinate is going to be a point and look in German pictorially to simply copy some old bill that's of course concepts so there's a set is never empty and at a high level interesting means that you have something other than the same silly did it.

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And they'll make their presence again I just wrote down the definition of polymorph them again and we saw some exam one example of an effort to set the single predicate let's say exotic like. And the negations as well all of them majority of 3 will be a polymorph some Let's take one more example suppose you have this predicate chicks odd number of things and one among 3 things how do I get a problem or for some.

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We're turns out OK Suppose I take 5 of these each of them as an R. each was an odd number of ones a combined them in a way where the new couple also has an odd number of ones they just use that are times aren't as odd so you just play a part of the function on each of the columns so bad if you're 5 here you could have taken any odd sabbatical of 3 Betty of 7 because if any are bad it is a polymorphous for the spring on one final example let's say one in 3 sat you have a pretty good would say exactly one out of this one OK now if you try to think it's a nice comment all exercise can prove that the only polymorph some subjects regard as a valid and this is a hard C.S.P. these 2 lead to easy C S P's the just lead equations OK So with that preamble the ball then.

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Even without polymorphisms the big goal in the C.S.P. theory was given a dumb one can you tell me what is the complexity of the associated C.S.P. problem again that was very influential for the remarkable paper by Federer and while even them ninety's which had a lot of built of stuff but also put forth the they part of me conjecture would said that for every such a gamma it is either Palami times are local or N.P. complete you cannot have this lad never been to the media here so there's only 2 complexities are possible and and in fact there was lots of reason to believe in this conjecture one of which was already 20 years before that Schaeffer had proved this the domain was born.

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For every problem gum out of the bullion domain it was either and be complete audit was Paloma time soluble because it can be expressed as one of the 6 easy cases in the 1st 2 are trivial very good just satisfy everybody by putting everybody to 0 or one then you have to say that and then you have the Hans saddened and then leading questions or whatever this is the predicate be X. X. X. or Y. Z. So these 6 cases are easy or when you can express your problem in the 6 cases it's easy otherwise you because this is really a long result for B. years old and there was a refinement of this conjecture in 2005 that said that it made it in some sense stronger by not only thing that experience be complete but actually told you the boundary it said that the problem isn't B. if and only if it had with some non-trivial polymorphism otherwise it and because so this actually gave you the algebraic.

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To do so at this point all of these people are like universal algebra so these people got really excited because they were doing very abstract things and universal algebra which really studies disclosure operations and so on and now they have this applied field of country and satisfaction where they could have planes and indeed that was.

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After And. And just to give a polymorphic view of the share for see them indeed for here for see them all of these 6 easy cases you have some polymorphs I already mentioned the 2nd one of the 1st 2 are trivial the problem of some of the constants and the 3rd and the 6 they mention are bad it is an odd modalities that you just stick to one side solutions you and them that would be hard since onset means that as at most one negated the true concerns of the form X 102 an extreme place explain again that the world is the do one of them has and one of them a lot and I think I got them connectives Yes.

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I'll come to that didn't perform non-trivial means non dictatorial but it's a little more subtle than that for. Government's. Yes So if you have anything interesting you will have one of these and yeah that might seem something that's actually the public that's what gives them these palm of some sort of any structure you know to protect.

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All these things will improve quickly so and finally after last year after a long line of work can settle this conjectured for generality and really the non-trivial part here was that if you had a non-trivial problem honestly had to get out of if you lacked the amount of sums that it's hard was is relatively easy because of the part of the difficulty in the computer and why is it just related to that so you might wonder after all this maybe this polymorph something was a fluke but it really doesn't you know the polymorphism really does capture the complexity of peace because you have this thing called the Gallo connection with says that if you have 2 templates and one has more poorly one of them some the other then it's an easier problem it reduces to OK And in fact it's an if and only for certain natural class of gadget reduction so this is really just winders you know B.B. all of these things that we don't use for tactile please add on this problem but that problem we can ask when said vendor such a reduction exists only exists if you have more polymorphous and extra real quality of this is that you could take a problem like one in 3 sack which only has to get more of something and it's and be hard and equality is that if you only have the good of polymorphisms then you must be and B because then you only can dig on my tool to be one in the SAT and then that reduces to and it also says to really that if you have the same polymorphisms you have the same complexed told in some sense it is the correct tool to study.

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I guess conceptually operationally may not be but turns out even that isn't an algebraic they got to make lately and was that it's in polymer time if and only if it does non-trivial while in one of the non-trivial advance of pro-science question means that for bullion it's either a dictator orders complement So that's the generally trivial polymorphism for large of thing it's some something a little bit more complex called the near unanimity doesn't matter what it is but basically it's some function F. which obey some identities which prevented from being an addicted so morally you could just think that if you have something which is genuinely a non dictatorial function then you're an interesting point look at the details of this are not important it's OK So that's the connection and.

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Any questions. Yeah in somebody's complex way and it builds on lots of tools so it's not. So it's not like a disused give me this polymorph summon the algorithm Pops old. Yeah. You know. Most of the they would say that if you have that then you lose girls in elimination so they have these 2 basic algorithms those in elimination and linear programming or local algorithms and they mix them into.

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More of those 2 things but you have to combine them in sort of devilishly clever ways so little bit address more questions or the polymorph them have a lot of structure so there's something called a clone in the language one thing is they contain all dictators as I said and they also cross and badly composition so by which I just mean that you know I've just written some examples and you can imagine you can just draw in arbitrary tree put variables down our bridges label each in turn order by a polymorphism the composting will be appointments and that is simply because a pile in one of them opened something also in the same predicate so you can look us oblique I'm bored.

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And because of these 2 things they have they don't have a very discreet structure and because of this it's also a quality of this is also a close friend of mine or else it just means that it doesn't matter you can just identify somebody Abel's out of Italy and that will stick to function will also be polymorphous them and because of this once you how one interesting polymorphous them it'll actually get the man it lead to a whole family of interesting problems for example the moment you have a majority of 3 you can get any odd majority just by the use in this mine of property and that's what really happens in these algorithms Once you have a single non-trivial amount of them that can be amplified to give you a very rich set of polymorph sums which then prescribe algorithms can lead to other things and if you don't have any of them at all becoming so you have this very strong Descartes I mean also in the structural space you that have a it's not whether you know one of them or not it's very you have very rich bottom of some sort not because of these projects.

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OK So all this was a bit of a preamble so you might wonder if OK The conjecture is now broadly appealing to the maybe that's should go home or never goes home I guess but Pierre that is a good reason not to go home one thing is that I only talked about satisfiability of C.S.P. and that's the 1st 2 columns of us 2 laws but they're also an optimisation you can ask tell me if it's maybe it's not satisfy but what's the maximum number of constrains you can satisfy there's a dichotomy for that as well there is a dichotomy for counting the number of solutions and these actually are actually a pain before these and these are actually easier to obtain at a high level that's because most of the problems you had are hot so I mean it's very rare that problems here these are exactly one of them stooped so these tend to be typically hard sort of strong to be able to pull the dichotomy and that was done and then you can also talk about approximation the sions and and where the president does what gives conditional connection between polymorphisms and this and I'll might mention later this fall of one of them is is finding enough to not only tell you whether you are in P.R. not but you can actually even predict the best exponent in your mind it can tell you whether to in principle whether you have a $1.00 to the end of them or a $1750000000.00 so you can even study on so there's still these are more or less done so these things that is more work to do.

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But I didn't notice in some sense to go even beyond the stable and I asked him for an even broader problem morphic principle is a true that interesting of them is in play efficient algorithms and even broader context provided you are careful enough to interpret interesting in their vision OK So that's the hope that will be really nice the more the better we get more said settings where we can explain the source of algorithms and the onset of hard problems.

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And what we have done is develop study of these things called Promise E.S.P.'s as an instance of the principles and this also brought to the fore some nice connections to find in complexity and comics optimisation So I hope to tell you a little bit in the 2nd half of the talk in the story so in fact in under a croc in is one of the key players in the C S P dichotomy algebra approach he gave a total of $33.00 any time all said that it's actually more closer to the beginning and I think one of the reasons the main reason he said that was basically he got really excited about this line of work one promise and constraint satisfaction which is what I'm going to tell you about for the next you know 1520 minutes maybe so what is promised consensus faction so that the thing with exactly is this is that typically these problems at heart most of that if you take a random template it's not going to have any problem are for them and that leads to a hard C.S.P. so we would like to cope with these saw promise E.S.P. is one B. to cope with this is just some locals related topics mission algorithms it's just approximation in a different sense so what do you do in a promise E.S.P. is that in sort of satisfying the predicates in government you try to satisfy every last one of those predicates here once again it's best given by an example of the most canonical examples I could give you a graph and tell you it can be colored seekers so the predicate is not equal or domain C. and then I can allow you to use more couples so I still have to satisfy all the constraints but I do myself leave it by using a more relaxed pretty good that's one example another example is a variant of satisfiability which in some sense is what led us to this whole line of work so I'll talk about that briefly So this is a version of back with the promise so I'll give you a guess that problem which is of course hard and be hard but now I tell you that this instance is nice enough that there is a satisfying assignment which gets at least a little strong in each class.

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So it's sort of in some sense richly satisfied but then I want to ask you to satisfy it in a normal way get at least one out of the keep because one in 3 said to be a little of the sack but you can imagine some of them and one of the complexity of this by some simple gadget reduction by just repeating 3 sad things or 3 times you can prove that if I tell you killed 3 of the care of satisfied doesn't really help someone telling you one in 3 satisfiable is not going to help you sitting behind another one Interestingly if I tell you that in each clause a majority of things are satisfied but then this problem becomes easy and effective and it's basically a generalization of the 2 sat a lot of them are very nice exercise in saying out of class so but now OK What about the in-between reading for example like what about board of fights and so is easy or hard and a few years back we proved that in fact what do you know which is what you might guess that.

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That's the boundary of easy so if you get a majority of them satisfy bill then it's easy but if I just tell you that less than 2 or satisfiable in each class that doesn't help me satisfy the instance at all to that end. And and does this work we also formally put forth promise E.S.P. which are.

[00:27:16]
Defined in ways. Or we do it in action from cover something so unfortunately this for instance even though just in the hardest thing it requires for example the P.C. people and there is and actually this polymorphically also explains because of this Gulliver connection this does have some non-trivial polymorph some of the telephone sharpening so you can for example just reduce it from pre-set to Shaun's.

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So but now that it requires a hammer as big as the P.C. thing I don't. Yes I'm all for this you had a very simple election for the city. Or the algorithm as there are poor times you can do one of the random walk a lot of them where you just start with an arbitrary time and then just go to classes randomly by flipping around a term that works for exactly similar reasons as to sac and there's also early L.B.L. of them which is which if you want to if you want to dumbest ago.

[00:28:18]
Yes song I'm happy to waterboard backed off. So what are the promise the S.P.V. are going to start to take from these examples and just do it do the most natural relaxation so now instead of just having a collection of pretty good you have a collection of pairs of predicates where Q Why is a week of pretty good then again so whenever a stroke you a would be automatically true and then you have to C.S.P. instances which are look exactly the same except the P A's are replaced by Q.S. So now I give you an the problem say such watching will be that I give you an instance of the strict but I ask you to find an assignment with satisfied as axed OK so you can imagine this is the coloring this is forthcoming and so abstractly you can have a collection of P. A and Q. A and you can call this promise E.S.P. P.C.S. B. of become a Q. and other special given P. and Q. are the same gamma as of course he is so there's just general a C S P's in natural.

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And now of course of the ambitious goal before Pabst be in Q Tell me when they're easy and they can't and just to make sure we have the same bit examples so deal of guess that is basically the following things are the one true when there are piste once we talk to one is true and unless you have at least one case of Clearly Q one lacks a skew be one and you also have the other predicate because you have to know negations of them is this problem is trivial maybe just a day everybody to one so that the 2nd predicate basically allows you to capture the negation because it tells you to ask you allows you to say that $1000000.00 in a Geisha.

[00:29:53]
Just a compact way of reading the occasion so here is another a nice example I can give you one in 3 stars which is hard. Not only called the savages hard but then again the lead them of the promises by the supremacy of people P. is true if exactly 11 is one of the 3 as one Q. is true of either one or 2 of them are meant to pique you relax so now that I can give you a one in 3 sentences and ask you to find an article 3 and by the way this is closely related to hyper graph coloring the script in C. and so on that are problems there can capture let's just pick the simplest one and then coloring of the color of a graph will be is one of make the domain then but then demands of the variables lay in 123 but then Q. will do that but there's really any pair of pretty good for being placed you defines as B.'s and the point is a lot of natural problems can be gassed and.

[00:30:53]
Yes I'm still all the predicates I mean bloating the P. and Q. are fixed once and for all and not part of the input since I mean it makes sense to ask the other question but it's out of scope of the species OK so now I'm going to talk about how to study P.C.'s piece and then we're going to do it with this polymorphic lens So what are the polymorphism for a promise E.S.P. so you can have a pair of predicates B. and Q. Q. weaker than B. and as before that just make one change to the previously so you have a polymorphic F. is a polymorphism of become a Q If whenever I pick M. Rause which satisfy P. and I apply F. I get a new row which satisfies not be but Q. so that's and read so it combines a nice implosions of people solution of Q. and then I can then order that by Paul of become a Q. and for a collection of templates I just intersect them and not that once again this is never empty because I can begin to functions that even satisfy B. and therefore.

[00:31:54]
Also Q.. OK so that's polymorphisms really not in it just combines so. Well beat that that that we don't know but we have some pieces of that puzzle the grand thing we don't so that's the story I'll tell you I mean. So OK saw him and towards that they need to know that this is in fact the correct order to study this and then it turns out we can extend our connection to say that if you have more poly if a pair has more polymorphisms then indeed it does reduce to the other thing OK so this part is still true so in principle polymorphisms are the right tool to study P.C.'s So if you have 2 sets of the same pile of polymorphisms in the promise for world they will have the same complexed.

[00:32:47]
But a major difference and which makes this theory much harder and you know it's not clear to. People The hope will go much farther than we have come so far is that you lose this property of closure composition. Recall that for C.S.P. is polymorphisms of closer composition Now this is no longer true because when you open the output of a promise polymorphism is no longer in P. this week of Q So it just doesn't lend itself to composition saw in some Therefore what makes it more tricky therefore is that you know it's not like an all or nothing kind of thing it's not like if you are one promised file in one of them you can amplify it you really need any whole family of polymorphisms before you start getting out of them's and.

[00:33:33]
That makes the quality of vanity easy and minutes hard much more nuanced because it's not just about a single function but about interesting families. And indeed I'll show you examples where there are some of them but they stop existing at some point and the problem is still hot.

[00:33:50]
Any questions for them yes. Yes. Yes Yes otherwise the feelings are not that in particular so going back to the example so really interesting is a piece E.S.P.'s when for example this problem one in 3 start is hard not all equally sad as hard but this promise E.S.P. is easy and it has 4 and it has problems and by the way the same thing is true also for K. or.

[00:34:24]
Both of those individually at heart so any basically easy cases of P.T.S.D. where neither P. and Q. are easy which we know have the yeah definitely that other examples and I'll make that more clearance OK So just going back to this what are the polymorphisms I guess this is exactly as well time for some of the question so to your prefer to decide I said is easy so one of the polymorphisms of turns out once again any autumn authority is a polymorphous OK So majority of an odd number of by dissatisfying assignment I just mean at least 3 of them are cruel this must be a satisfying assignment like just the approved by picture so if you have to order for sack and they take 5 rows each has at least 2 ones then when they take a majority there must be a column with one and the proof is just literally this one line but let's just do it in this example in the Middle East that are going to be a total of 10 once played because at least to win this war there must be a column with at least 2.5 once and then for at least 3 months that's way the majority and this proof is the same because in any order majority much like to set it has the same it shares this feature with 2 set that any odd majority is a pointless and indeed that was OK So that's that's the case saw I just wrote that the case here that this is in fact he's.

[00:35:49]
And this part and then we have this our main result in the work was actually earlier what was the hardest thing that we said that if you take hold of 2 P. plus one side to be slightly become less than Killer 2 then you are hard and why is that in this case it's not that there are no polymorphisms there are polymorphisms but there are only there who does so for example than 2 or 5 sack the only polymorph sums that depend on so they're not as as silly as Dick did this but in some sense they are silly because you can make that a B. 1000 but they are still only dependent so more concretely majority of 3 is a polymorphism for 2 or of 35 set but this problem is still hard because majority of 5 will not be a polymorphs need there is no poly one of them have at it before or more so in some order there are some sort of local cases of polymorphisms but they're not very interesting they cease to exist OK And this is what makes the line between easy and hard hard to get done because it's not like if you have some polymorph and you need it to become easier so it's more nuanced and OK so.

[00:37:00]
So I have some stuff on coloring I think I'm going to skip this. But I just want to advertise this one thing so basically just to summarize the slate in just one sentence based following this feast studied approximate coloring which is a very natural C.S.P. and we're very able to use this polymorphic framework to get some new hardness results for calling for example 6 coloring or 4 color of a graph is hard but one of these pesky questions in the field for those new it is really that we didn't know that given to think of a graph of course it's hard to call it a 3 colors it's also hard to tell the 4 colors and this gave another proof of that directly 3 proves but we didn't know that 5 coloring was hard and that has finally been resolved this year by these guys in a beautiful paper where they actually give it an action from some hyper after learning problem to those problems is known to be hard and what to me exciting here is that they really are.

[00:37:53]
To me it seems that the squid not have been discovered if we hadn't developed this polymorphically this is really developed understood in this polymorphic language this time all prove that this problem is fewer Palomar sums than this problem and this is hard so that must be. 458 is exactly the number of added the 2 polymorphisms of time so it's a number of colorings.

[00:38:20]
And it's. Yeah so it's number of 5 colorings of the Square Enix. That's the number 4. There are some aspects of the city so I'm quite excited by this that they have still able to even though there is no composition here they have able to develop some elements of a composition preety And I think they'll be more advanced.

[00:38:45]
Level covered. Yeah that was that was not that was a was about 15 years old 20032000 Forbes So classic. So again polymorphism and P.C. S.P. tractability some older the principle seems to be this that if you have limited polymorphisms then your heart so for example the only poly Marfan's out who does you are hard and the something we can formalize and if you have a rich enough polymer of them's you get a vision of what example of you are all majority good.

[00:39:17]
But the point is as one clear dividing line between what is limited and what does the rich enough and so we need more examples and in fact I got I mean might well not exist in this case this is because it's possible at least for the nonbeliever but what you have managed to do is that for some special cases they can sure they called me in particular if you take a bully in problem and you listed the P.S. and Q. ways to be symmetric predicates namely their value only depends on the number of ones in the input then there is a dichotomy in order that the 2 examples of what I mean example satisfy this trachea.

[00:39:56]
And so here are 3 of them is that every such C.S.P. is either and B. R. and B. completes that which is a dichotomy and the criterion is a falling so you have polynomial time solvable if the polymorphous thems have either all our data B.'s all our majorities of all 80 function that helped define the 2nd part of their complements or doesn't that's a technicality otherwise it's OK So this is the main on the show and what isn't it the function A.B. is basically like a majority except it's a signed thing so it takes it sees the number ones in the order positions number ones and even positions and sees for the darting seeds even OK and.

[00:40:39]
OK this is a strange thing and one of the algorithms and these work in general the algorithm for padded these goes in elimination for majorities linear programming and for 80 is linear equations because of these sums and these algorithms are known things but they just sort of kick in in these cases.

[00:40:58]
Linear Programming and. So this will be linearly Yeah so this is linear equations or integers and does cause an emission or the field if to it yet but it's. So that only 2 really out of them linear programming which captures local algorithms and we need equations which captures more global and the hardness basically is which is where the main one in some sense was that if you missed these rich families then you're only.

[00:41:31]
Polymorphs and then you become so i won't tell you will it seems depending on the time I have. OK so I just do what I want to do is now I'll give you a little bit of flavor of the algorithms how the palm of them sticking by of themselves brands are present.

[00:41:56]
Yeah yeah. They don't help. They don't help here. Also because these are like exact versions I mean you do want to satisfy all the constraints of these or feasibility problems so they don't tend to. I mean there are cases where as D.P. helps the basic it's D.P. that approximation.

[00:42:19]
OK so how does the majority kick in. So let me actually skip this. Actually And maybe I will show you this because you might wonder why polymorph some give you a lot of them's So let's do 2 sat in a more complex way let's say the natural for 2 sat right where you just do the obvious constrains for next year bar then you put this constrain why a plus one minus one ideas at least one and you find such a solution and the obvious rounding so we know this works late so if it's more than half you make it one if it's less than half you may conceal to revisit Yeah that's right and Gabby are what happens when Y. is half that is a big problem because any said to be used feasible by setting every way is half let's ignore that issue from now and come back to it but suppose you had a nice solution that was half as forbidden then what will we do or than the majority works and of course you can directly see it in this case but you can also have a more elaborate way to see it which is that if you have a class for example X. one bottom extra bar then we want to say that when you move the threshold rounding you will satisfy this class and elaborate way to see this is that you can write this as a climax combination of these satisfying assignments and you line up a large number of copies of those satisfying assignments and then you can just and because these are all satisfying assignments of this class if you play majority you must get a satisfying ascent because majority of the parliament and then what does a majority going to do well here the number of wonderfully one faction so the majority is just doing threshold OK So that was quick don't worry but basically the point is my job is intimately connected potential.

[00:44:01]
OK And the fact that it's a poly more of them means that when you do this threshold around in every class this variable participates in will be set to the bottom out of some comes in. In the analysis of the other OK. So here's another. The case which is the.

[00:44:20]
One in 3 and not all equal here it has to this is a case which has a P. as a polymorphism you can check that but but let me just tell you the algorithm because it's very cute. It's one line out of the believe. Suppose you want to satisfy all this P.C.'s B. I can but I don't the following conditions right for that matter have exactly one of X. A just say the summers one and if solving this or 01 is the same as solving the original one in problem but now I can do that of course but send be hard so instead I will find indeed a solution to their body which surely exists because that is it on solution so you can solve linear systems or indeed as efficiently do that and then you just do assign based from if the Z. is not always positive you give it one of its non-positive you can see and why is this a kind of thing to do because Nord that to satisfy the scums trained not all 3 can be positive because the sum then will be at least 3 so it is one of them so you cannot have all of them to be one the zoning will not set all variables to one in any constraint it will also not set all variables 0 because of all of them are non positive you cannot add a case that's then to have a look at the men under the hood the fact that it works more abstractly ill because it has these problems OK so that's really but there's already interesting because even though the problem is bullion you really have to use an algorithm which works on integers an infinite domain to get the sun.

[00:45:49]
Shows that richness of these P.C.'s be framer quality in very stylized cases like this OK. And there is a way to blend these 2 give algorithms whenever you have any symmetric families of point OK so now. Winter they finish. In the 5 minutes OK I'll try to say a few things connecting documentation and so on so how do you handle this half bad news this is of course a pesky thing because in fact that LP actually the M.P. just simply doesn't work for percent because you can just set way to be hard for everything and that will satisfy all the so.

[00:46:30]
So the idea is that of course there is a way to ask why does by using ad hoc methods but we can ask for a more principled approach because that is also useful Paul tackle more different threshold of sums which might have the shores of different places not just one so one ideas that I do have ideas to solve LP and more principle ways to solve it on a different thing than national.

[00:46:54]
Q. but for something else well you could just solve it all or integers that would be good but of course that's this is a linear program not you need equations as in be hard but then we can solve all was summering which I why it's one half OK but let's actually dickering with our wives all the actual numbers other than integers 0 join with through.

[00:47:14]
And this actually works and again I like to mention this also because this there is a polymorphism here there's a convex combination of 2 solutions which only uses core visions in the ceiling and that should prescribe that there is an algorithm and indeed we were able to confirm even though the pathway from the Poly One of them to the algorithm is completely unclear one of them has nothing to do with per se with the Palomar For some this is why we thought we should be able to pull this theorem which is that if you have a linear program which is feasible or integers Let's say you can find a solution.

[00:47:47]
Which is not going to integers but it's in the story and once you do this you will never get one half and you're. OK So these are just some asides we get in optimization So let's get more ambitious suppose in terms of our winding half I just want to like the middle one 3rd of the interim.

[00:48:08]
That would be nice also like I want. It would be cool but this is unfortunately and be hard because this will actually solve those if you laid a similar deal before the SAT The only obstacle is that you can set everybody to be one 3rd of something so so this will lead to not do this.

[00:48:27]
But then say OK can I build this can you do what can you do for this problem in particular can you give an algorithm which beats the trivial runtime of to the end because you can always into the in time find as you know one solution we're dealing with things just feasible or 01 can you do this and it turns out this is something OK gentle result here is that they give you a linear program which is feasible or $1.00 So it has an integer solution then we can find a solution which is very close to the 2 boundaries that actors of his body and of and this but just the result we can come of the 2 endpoints in time which improves over to the end proportionally to the boundary So for example for Gamma is one 3rd to get a 4th or to the end how to get them.

[00:49:11]
And the sum all smoothly into place between linear programming and Z. in the on. Yes it will give you an algorithm for 3 that's exactly. The. So in particular maybe. If you find a solution for this in particular the silly mediately get added to minus 2 or kids at the end of them which generally is a shining celebrated algorithm and it's also more general it gives you a lot of time for any some pressure or punishment so that it covers that but it's more gentle than that and very briefly just because I think this might intrigue a lot of people is a lot of our friends can actually give you very fine grained information tool even for C. a speech at N.P. hard but there's something called partial polymorphs thems which govern the smallest C. for which there is a C. to the nth time and so there's some theory and I've just put one work which develops this and in particular the way that they have proved is that there is a Gallup connection if you have more partial polymorphisms then proceed to the end I look at them for this game plays one for the other and roughly this is done by giving reductions we do not introduce any oxalate even so they preserve the exponent of that and so and that's what this partial polymorphisms capture and using this there are some poor results this paper proves For example shows that out of all and be hard bullion C S P's the easiest one in terms of having the smallest C. is one in the set even though we don't know what that sees but the the one with and this has a pretty good algorithm already one point not 6 to the end or something but this does have a sports.

[00:50:47]
And you might wonder what is the part of polymorphous I'm I haven't defined it but you can just guess it's just a poly one of them which is a partial function so it may not be defined on some inputs at all for example for a case that the following is a part of partial one of some take to be some large any number.

[00:51:04]
Of the dozen different employed corner cases if you are having weight of the number one of the next is less than one in good fraction. That's more than one minus one of our key a fractional performance in between on opening. Day and you can prove that this is a part of problem of them and in some sense the reason that becomes harder as the game becomes bigger seems to be because this function is more passion for larger because it's undefined it's only defined under 2 or diffraction and that actually based of the 2 minus toward the reason we get a 2 minus 2 or algorithm is because that partial polymorph them is only defined on a tortilla fraction and that's proportional improvement to the birth because this was just a sort of give you a little bit of these are on hold these things go I'll skip there's a lot of them sketch but basically it's a substantial generalization of shellings algorithm and it's based on random walks to gradually come much to solution near the boundary point OK so I'm ready to conclude so basically promise E.S.P. is.

[00:52:09]
We have developed some result as a chore some glimpse of this phenomenon that if you have a bitch in a family of problem of sums you get out of atoms and if you don't have that if you're very limited and then you get hardness but the body within these is not clear and technical obstacle to study this is that there is no composition allowed here so you really have to work with the family of polymorphisms rather than single polymorphs.

[00:52:35]
And so really to develop this theory for that I think we also need more examples interesting examples of both the algorithmic part of the hardness part which can then guide us towards maybe some conjecture a dividing line between these 2 and there are some general questions but also very specific questions which we don't know which we would love to know the answer to for example I mentioned this 5 coloring was a 63 coloring thing 6 coloring of the color of a graphic still don't know as in be hard and then later problem has been that you may not have seen that you can also ask Suppose I give you a graph and I tell you it has a homomorphism into an arc cycle that has 7 cycle now Kenny 3 colored so I'm some all sort of making a promise E.S.P. problem by giving you more structure than 3 going but then asking you want to OK this should also be hard but we don't understand B. and so there are very concrete questions we don't know definitely which theory is also.

[00:53:28]
OK So to summarize there is this polymorphisms which I had out of symmetries in the solution space and there seemed an interesting such operations exist it seems to govern algorithms and this is a quantum is really fully established for satisfiability of C.S.P. as well as many variants like daunting for experimental tractability and so on but for Promise E.S.P.'s the T.T. is still very new and we need to develop it much more and what means exciting is also there are 2 very advanced methodology is to study C S P's one is the P.C.P. technology and the other is this school polymorphism based algebra technology and really promise E.S.P. is you need to some hole in a blender need to get the best of both of these 2 and get them to work together which is quite interesting and it seems to have which are possibilities in terms of connections to exponential time algorithms optimizations So let me.

[00:54:38]
You mean the Optima Max E.S.P. type thing yeah that was that I went as though they had the table so there you have to be again and you have to work with families of polymorphisms and you need to somehow say that these polymer sums genuinely depend on all the inputs so you say that if you have a polymorphism but every variable the very Norway table has sizable influence you know something like majority and so on then.

[00:55:05]
Then you get all of them and then what are the quality of the palm of them the way one said that if a sample Rause what each was had its face being with some probably see when I play F 2 that column wise trip satisfy the pretty good would probably be S. And that means that I will get a seek a minus approximation so or so and so it's a much more analytic notion.

[00:55:32]
Yes. Yes yes. OK. Yeah I don't know what you are so one of the things which underlies this new development of the 3 was 5 coming is that they have and I just break view of this label cover problem which is one of these canonical things. But I still don't know if it's a whole whole much deeper that connection is and whether there is some sort of approval of it if I have a view which which is which might help you with questions.

[00:56:24]
Or. No no I think football. If you're right if you're right the levels of feeling of Charlie Adams that would be enough for the Tribune to go but for the basic thing should not be done because for any to start instance even if it's not satisfy a book.

[00:56:47]
They may be noisy don't solution but all have says on this satisfied. OK Thank you guys.