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ost by clément canonne here is a first guest post from icalp 2014 kindly written by clément canonne enjoy it this post is also available in pdf any typo or typesetting issue with the html version is my responsibility on dnf approximators for monotone boolean functions eric blais in this talk eric blais presented joint work with johan håstad rocco servedio and li yang tan concerned with boolean functions more precisely the simplest representations of functions dnf disjunctive normal form formulas for a little bit of background recall that a boolean function f 0 1 n 0 1 defined on the hypercube 2 is a dnf if it can be written as that is as an or of and s one can also see such functions as being eactly those taking value 1 on an union of subcubes if one prefers i will not argue with one a nice property of dnf formulas is that they are arguably amongst the simplest of all representations of boolean functions while formulas of depth 3 are a nightmare dnfs have been extensively studied and by now everything is known about them well almost everything indeed amidst other facts we have that theorem 1 folklore every boolean function can be computed by a dnf of size 2 n 1 theorem 2 lupanov 61 this is tight parity n needs that much theorem 3 korshunov 81 kuznetsov 83 a random boolean function can be computed by dnfs of size θ 2 n log n and requires that much so are we done yet the mere presence of eric in the auditorium was a clear hint that all was not settled and as it turns out if the picture is well understood for exact computation of boolean functions by dnfs what about approximate representation of a function that is what about the size required to approximate a boolean function by a dnf if one allows error ε as a fraction of the inputs this leads to the notion of dnf approximator complexity and here again some results much more recent results theorem 4 blais tan 13 every boolean function can be approximated by a dnf of size o 2 n log n furthermore our all friend parity n only needs dnf size o 2 1 2ε n that s way better than 2 n 1 so again are we done here and again not quite this brings us to the main point of the paper namely what about monotone functions can they be computed more efficiently approximated more efficiently recall that a boolean function f is monotone if x y implies f x f y where is the coordinate wise partial order on bit strings as a starter no theorem 5 folklore every monotone boolean function can be computed by a dnf of size o 2 n n 1 2 using subcubes rooted on each min term and again this is tight for parity furthermore and quite intuitively using negations does not buy you anything to compute a monotone function and why should it indeed theorem 6 quine 54 to compute monotone boolean functions monotone dnfs are the best amongst dnfs not surprising i suppose well it s a whole new game when one one again asks only for approximations and that s the gist of the paper presented here first of all drastic savings in the size of the formulas theorem 7 blais hastad servedio tan 14 every monotone boolean function can be approximated by a dnf of size o 2 n 2 ω n 1 2 eric gave a high level view of the proof again it works by considering the subcubes rooted on each min term but in two steps regularity lemma the world would be much simpler if all subcubes were rooted on the same level of the hypercube so first reduce it to this case writing f f 1 f k each f i has this property then approximate independently each f i using a probabilistic argument via random sampling to prove there exists a good approximator for all f i s and then stitching them together and they also show it is tight this time with the majority function maj n the proof goes by a counting argument and concentration of measure on the hypercube every or almost every input is on the middle belt of the hypercube but each subcube thus has to be rooted there and each cannot cover too much so many are needed so approximation does buy us a lot but clearly using negations shouldn t should it why would allowing non monotone dnf s to approximate monotone functions ever help hint it does yep theorem 8 blais hastad servedio tan 14 for every n there exists ε n and f 0 1 6n 0 1 such that f can be ε n approximated by dnfs of size o n any monotone dnf ε n approximating f must have size ω n 2 take that intuition the upshot exact computation and approximate computation have intrinsically very different properties eric then concluded with an open question namely how to improve better understand the gap between approximating functions with monotone dnf vs approximating them with general dnf s the currently known gap in the size being quite huge almost exponential additionally how to get a separation as in the mind boggling theorem above but changing the quantifiers that is for a constant ε independent of n also can someone fix my intuition i think it s broken 1 http www cs columbia edu rocco papers icalp14 html 2 not this one posted by luca aceto at 9 44 pm no comments email this blogthis share to x share to facebook share to pinterest day 1 of icalp 2014 guest post by andrew winslow andrew winslow was one of the colleagues who kindly answered my call for guest bloggers from icalp 2014 here is guest post on the first conference day enjoy it conference welcome local organizing committee chair thore husfeldt kicked off icalp by mentioning a few stats about it university of copenhagen itu where the conference is located started in 1999 2000 students 10 theory cs faculty mostly working in track b areas in icalp parlance 1 building he also noted that itu is separate from two other universities in copenhagen hosting tcs conferences this month copenhagen university hosting sea last week and technical university of denmark hosting swat last week invited talk claire matthieu claire presented work entitled homophily and the emergence of a glass ceiling effect in social networks joint work with chen avin zvi lotker barbara keller david peleg and yvonne ann pignolet she started the talk by asking why there s only one female danish scientist inge lehmann listed on the wikipedia page for famous danish scientists out of about 60 entries why so rare looking at the dblp database data as a social network with co authorship edges gives similarly bleak results the induced subgraphs of men and women in the top 1000 nodes shows a large well connected graph and a disconnected sparse graph respectively claire and co authors developed a formal model of growing co authorship social networks including assumptions about what a glass ceiling effect means for these networks and proved that three properties are minimal sufficient set of properties to imply the effect these properties are preferential attachment a rich get richer effect where incoming students choose to work an advisor with probability proportional to the number of co authors the advisor has minority an incoming student is female with probability r 0 5 homophily students stick with an advisor of the same gender with probability 1 otherwise find a new advisor with probability p 1 incoming students are iteratively added to the graph to form growing random graphs repeatedly chosing new advisors according to homophily and preferential attachment the glass ceiling effect is defined by looking at all nodes with degree k where k k n chosen so that the number of such nodes goes to infinity with n and examining the ratio of women to men in this set if the expected ratio of female over male nodes goes to 0 as the number of nodes goes to infinity then the system is said to have a glass ceiling effect for the parameters r p chosen they are able to prove that the above three properties imply the class ceiling effect and removing any one of them does not the model was also verified experimentally using the dblp database as a final twist claire considered modifying the model to account to capture the men are intrinsically better hypothesis giving new male students two advisors rather than one this was proven not to cause the glass ceiling effect i was really struck by the meta ness of the talk and how natural the problem seems in the formal study of social networks isn t the most natural network of study the one that directly impacts our ability to formally study networks györgy dósa and jiří sgall optimal analysis of best fit bin packing dösa and sgall complete the analysis of two classic online algorithms for the standard bin packing problem given a collection of items specified by numbers in 0 1 assign these items to the fewest number of bins where each bin has an item total of at most 1 the two algorithms first fit and best fit process the items one by one placing each item permanently into an existing bin if possible otherwise placing the item into a newly created bin first fit and best fit differ only in the criteria by which they place item into existing bins using the oldest bin first fit or bin that would have the least remaining room after the item is placed into it best fit it was previously shown that for every instance with optimal solution using opt bins first fit and best fit use ceil 1 7 opt bins and there exist instances where both use 1 7 opt bins dösa and sgall improved the upper bound to 1 7 opt bins dropping the ceiling and completing the analysis of the competitive ratio of these algorithms jiří commented that he felt that such classic algorithms deserved a complete analysis and i agree the talk had an air of finality having put the 40 year old gap to rest hossein esfandiari mohammadtaghi hajiaghayi reza khani vahid liaghat hamid mahini and harald räcke online stochastic reordering buffer scheduling vahid liaghat gave this talk on a scheduling problem where items have types from a fixed universe and enter into a buffer of sized size k one can then process all items of a type flushing them from a buffer and causing new items to take their place the cost of processing a sequence of items is either the number of processing steps used block operation model or number of times the type processed changes standard model it was known that a o loglog k competitive algorithm is possible for the adversarial worst case this work studies the sitation where the input is not adversarial but is randomly sampled from some set esfandiari et al achieve an average case o 1 competitive algorithm for both the block operation and standard models for the case where an adversary designs an input that is then randomly permuted the analysis uses several lemmas regarding the optimal solutions of various buffer sizes as well as threshold algorithms algorithms that process each type as soon as a specific number of occurrences of a type is exceeded erik d demaine yamming huang chung shou liao and kunihiko sadakane canadians should travel randomly this paper attracted a few comments related to the curious title the title comes from the canandian traveler problem of study introduced by papadimitriou and yannakakis in 1991 the problem is a shortest path problem with partial information some of the edges of the path are blocked by intense canadian snowfall and cannot be traversed but this is only known after a traveler has visited an endpoint of the blocked edge the parameterized version of the problem has k blocked edges and the goal is find an algorithm with good competitive ratio compared to the shortest path that avoids all blocked edges for deterministic algorithms tight 2k 1 upper and lower bounds on the ratio are known while for randomized algorithms only a lower bound of k 1 and implied upper bound of 2k 1 were known demaine huang liao and sadakane improve the best known randomized algorithm in two ways giving a polynomial time algorithm that achieves a ratio of 2 o 1 k 1 and a pseudo polynomial time algorithm parameterized by k the number of blocked edges that achieves a 1 sqrt 2 2 k 1 ratio the ideas for both algorithms are to repeatedly compute a collection of nearly shortest paths traverse them each until blocked and repeating the process with the new information if none of the paths lead to reaching the destination a careful selection of the number of paths used then yields the bound the main difference between the two algorithms is exactly what short paths are used the pseudo polynomial time algorithm uses a list of shortest paths whereas the polynomial time algorithm only finds paths within a constant factor of the shortest dmitry gavinsky and shachar lovett en route to the log rank conjecture new reductions and equivalent formulations dmitry started by reviewing two party communication problems where the goal is to compute a combined boolean function f of inputs x and y held by the two parties while passing the fewest number of bits between the parties one can consider the matrix defined by the two inputs with an entry for every x y pair called the boolean matrix the rank of this matrix and boolean function f is the minimum number of rectangles found in some partition of the matrix into rectangular blocks of entries with the same value it is known that if a c bit communication protocol is possible then rank of the matrix is at most 2 c i e the communication complexity in bits is bounded from below by log rank f the log rank conjecture is that polylog rank f is an upper bound on the communication complexity of computing f part of this work by gavinsky and lovett is to provide a number of equivalent versions of the problem that may provide a way to make progress on the log rank conjecture for instance they show that proving the log rank conjecture for the randomized communication complexity on low rank functions implies the log rank conjecture for the deterministic version as does proving the conjecture for the information cost of f the minimum amount of information given by each party the other party in the worst case eatcs general assembly the day ended with the eatcs general assembly led by luca aceto and appearances by the many people involved in icalp and eatcs icalp 2015 will be in kyoto japan and some information about travel accomodations and kyoto were presented by kazuo iwama and icalp 2016 will be in rome italy some statistics about icalp 2014 were given by elias koutsoupias the number of submissions is up from previous years acceptance rate is about steady at 30 and the vast bulk of the submissions were to to track a as one final thought from the day the organization of the conference so far has been top notch many of the talks are online including claire matthieu s the sessions start and end on time sound and projection issues are non existent and the food drink is tasty and rotates every break posted by luca aceto at 8 51 am no comments email this blogthis share to x share to facebook share to pinterest monday july 07 2014 in copenhagen for icalp 2014 it is good to be back in lovely copenhagen and breathe the atmosphere of the danish capital city one of the first things i saw when arriving in kongens nytorv was a smiling guy show...
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