I have just finished my PhD studies at the University of Toronto, under the supervision of Toni Pitassi.
I am attending the special semester on real analysis in computer science in the Simons institute, Berkeley.
Following that, I will spend three semester at the IAS in Princeton.
I can be contacted via yuvalf﹫cs.toronto.edu.
We consider the NP-complete optimization problem Bandwith. Anupam Gupta gave an O(log2.5n) approximation algorithm for trees, and showed that his algorithm has an approximaion ratio of O(log n) on caterpillars, trees composed of a central path and paths emanating from it. We show that the same approximation ratio is obtained on trees composed of a central path and caterpillars emanating from it.
Our result relies on the following lemma. A sequence a1, …, an has thickness Θ if the sum of any d consecutive elements is at most dΘ, for 1≤d≤n. If a sequence has thickness Θ, then the sequence obtained by ordering the elements in non-decreasing order also has thickness Θ
@mastersthesis{filmus:thesis:2002,
author = {Yuval Filmus},
title = {Bandwidth approximation of a restricted family of trees},
school = {Weizmann institute of science},
year = {2002}
}
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Extremal combinatorics studies how large a collection of objects can be if it satisfies a given set of restrictions. Inspired by a classical theorem due to Erdős, Ko and Rado, Sominovits and Sós posed the following problem: determine how large a collection of graphs on the vertex set {1, …,n} can be, if the intersection of any two of them contains a triangle. They conjectured that the largest possible collection, containing 1/8 of all graphs, consists of all graphs containing a fixed triangle (a triangle-star). The first major contribution of the thesis is a confirmation of this conjecture. This result first appeared in our paper Triangle-intersecting families of graphs with David Ellis and Ehud Friedgut.
We prove the Simonovits–Sós conjecture in the following strong form: the only triangle-intersecting families of the maximal measure 1/8 are triangle-stars (uniqueness), and every triangle-intersecting family of measure 1/8−ε is O(ε)-close to a triangle-star (stability). Our proof uses spectral methods (Hoffman's bound).
In order to prove the stability part of our theorem, we utilize a structure theorem for Boolean functions on {0,1}m whose Fourier expansion is concentrated on the first t+1 levels, due to Kindler and Safra. The second major contribution of this thesis consists of two analogs of this theorem for Boolean functions on Sm whose Fourier expansion is concentrated on the first two levels. These results appear in our papers A quasi-stability result for dictatorships in Sn and A stability results for balanced dictatorships in Sn, both with David Ellis and Ehud Friedgut.
In the same way that the Kindler–Safra theorem is useful for studying triangle-intersecting families, our structure theorems are useful for studying intersecting families of permutations, which are families in which any two permutations agree on the image of at least one point. Using one of our theorems, we give a simple proof of the following result of Ellis, Friedgut and Pilpel: an intersecting family of permutations on Sm of size (1−ε)(m−1)! is O(ε)-close to a double coset, a family which consists of all permutations sending some point i to some point j.
The thesis includes a detailed exposition of Friedgut's paper On the measure of intersecting families, uniqueness and stability, a proof of the Ahlswede–Khachatrian theorem in the μp setting, and a gentle introduction to the representation theory of Sn from the point of view of class functions.
Download BibTeX
@mastersthesis{filmus:thesis:2013,
author = {Yuval Filmus},
title = {Spectral methods in extremal combinatorics},
school = {University of Toronto},
year = {2013}
}
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Journal of the European Mathematical Society, Volume 14, Issue 3, 2012, pp. 841–885
A family of graphs is said to be triangle-intersecting if the intersection of any two graphs in the family contains a triangle. A conjecture of Simonovits and Sós from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family containing 1/8 of all graphs.. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.
The ArXiv version corrects a mistake in the proof of the stability part of Corollary 2.3. The original proof assumed that the approximating family G is odd-cycle-agreeing, and deduced that G must be a triangle-junta from uniqueness. However, showing that G is odd-cycle-agreeing requires a separate argument, found in the new version.
An alternative presentation of the results in this paper can be found in my PhD thesis.
ArXiv Presentation 1 Presentation 2 BibTeX
@article{eff0,
author = {David Ellis and Yuval Filmus and Ehud Friedgut},
title = {Triangle-intersecting families of graphs},
journal = {Journal of the European Mathematical Society},
volume = {14},
issue = {3},
year = {2012},
pages = {841--885}
}
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Submitted
We prove that Boolean functions on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn are close to being unions of cosets of point-stabilizers. We use this to give a natural proof of a stability result on intersecting families of permutations, originally conjectured by Cameron and Ku, and first proved by David Ellis. We also use it to prove a ‘quasi-stability’ result for an edge-isoperimetric inequality in the transposition graph on Sn, namely that subsets of Sn with small edge-boundary in the transposition graph are close to being unions of cosets of point-stabilizers.
An alternative presentation of the main theorem (without the second application) can be found in my PhD thesis.
ArXiv BibTeX
@article{eff1,
author = {David Ellis and Yuval Filmus and Ehud Friedgut},
title = {A quasi-stability result for dictatorships in {$S_n$}},
journal = {Submitted}
}
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Accepted to Random Structures and Algorithms
We prove that a balanced Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on Sn generated by the transpositions.
Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, the preceding paper deals with Boolean functions of expectation O(1/n) whose Fourier transform is highly concentrated on the first two irreducible representations of Sn. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.
An alternative presentation of the main theorem can be found in my PhD thesis.
ArXiv BibTeX
@article{eff2,
author = {David Ellis and Yuval Filmus and Ehud Friedgut},
title = {A stability result for balanced dictatorships in {$S_n$}},
journal = {Random Structures and Algorithms}
}
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Preprint
We prove that Boolean functions on Sn whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of n whose largest part has size at least n−t are close to being unions of cosets of stabilizers of t-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on Sn which is asymptotically sharp for sets of measure 1/poly(n). We then combine both results to obtain a best-possible edge-isoperimetric inequality for sets of size (n−t)! where n is large compared to t, confirming a conjecture of Ben-Efraim in these cases.
Preprint BibTeX
@article{eff3,
author = {David Ellis and Yuval Filmus and Ehud Friedgut},
title = {A quasi-stability result for low-degree {B}oolean functions on {$S_n$}},
journal = {Preprint}
}
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Preprint
In 1990 Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem (CCV). He and Mayr showed that NL⊆CC⊆P, and proved that in addition to CCV several other problems are complete for CC, including the stable marriage problem, and finding the lexicographically first maximal matching in a bipartite graph. Although the class has not received much attention since then, we are interested in CC because we conjecture that it is incomparable with the parallel class NC which also satisfies NL⊆NC⊆P; this implies that CC-complete problems don't have an efficient polylog time parallel algorithm. We provide evidence for our conjecture by giving oracle settings in which relativized CC and relativized NC are incomparable.
We give several alternative definitions of CC, including (among others) the class of problems computed by uniform polynomial-size families of comparator circuits supplied with copies of the input and its negation, the class of problems AC0-reducible to CCV, and the class of problems computed by uniform AC0 circuits with CCV gates. We also give a machine model for CC which corresponds to its characterizations as log-space uniform polynomial-size families of comparator circuits. The various characterizations show that CC is a robust class. Our techniques also show that the corresponding function class FCC is closed under composition. The main technical tool we employ is universal comparator circuits.
Other results include a simpler proof of NL⊆CC, a more careful analysis showing that the lexicographically first maximal matching problem and its variants are CC-complete under AC0 many-one reductions, and an explanation of the relation between the Gale–Shapley algorithm and Subramanian's algorithm for stable marriage.
This paper continues previous work of Cook, Lê and Ye which focused on Cook–Nguyen style uniform proof complexity, answering several open questions raised in that paper.
The preprint contains more results than the ArXiv version, and the presentation is different.
@article{cfl,
author = {Stephen A. Cook and Yuval Filmus and Dai L\^e},
title = {The complexity of the comparator circuit value problem},
journal = {Preprint}
}
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FOCS 2013
An approximation computation of a Boolean function f by a circuit or switching network M is a computation in which M computes f correctly on the majority of the inputs (rather than on all of them). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many subareas of complexity theory such as cryptography and derandomization. Lower bounds for approximate computation are also known as correlation bounds or average case hardness.
We obtain the first average case monotone depth lower bounds for a function in monotone P. We tolerate errors up to 1/2−1/n1/3−δ. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we establish that for every i there are functions that can be computed with no error in monotone NCi+1 but that cannot be computed without large error by monotone circuits in NCi. We provide a similar separation between monotone NC and monotone P.
Our proof extends and simplifies the Fourier-analytic technique due to Potechin and further developed by Chan and Potechin.
As a corollary of our main lower bound, we prove that the communication complexity approach for monotone depth lower bounds does not naturally generalize to the average case setting.
@inproceedings{fprc,
author = {Yuval Filmus and Toniann Pitassi and Robert Robere and
Stephen A. Cook},
title = {Average case lower bounds for monotone switching networks},
booktitle = {The 54th Annual Symposium on Foundations of Computer
Science ({FOCS} 2013)},
pages = {598--607},
year = {2013}
}
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ICALP 2011, submitted for journal publication
We give a general transformation which turns polynomial-size Frege proofs to subexponential-size AC0-Frege proofs. This indicates that proving exponential lower bounds for AC0-Frege is hard, since it is a longstanding open problem to probe super-polynomial lower bounds for Frege. Our construction is optimal for tree-like proofs.
As a consequence of our main result, we are able to shed some light on the question of weak automatizability for bounded-depth Frege systems. First, we present a simpler proof of the results of Bonet et al. showing that under cryptographic assumptions, bounded-depth Frege proofs are not weakly automatizable. Second, we show that because our proof is more general, under the right cryptographic assumptions it could resolve the weak automatizatbility question for lower depth Frege systems.
@inproceedings{fps-icalp,
author = {Yuval Filmus and Toniann Pitassi and Rahul Santhanam},
title = {Exponential lower bounds for {AC$^0$}-{F}rege imply
polynomial {F}rege lower bounds},
booktitle = {The 38th International Colloquium on Automata,
Languages and Programming ({ICALP} 2011)},
year = {2011}
}
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CCC 2012
During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers.
There has been a relatively long sequence of papers on space in resolution, which is now reasonably well understood from this point of view. For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in Alekhnovich et al., where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space.
We prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.:
ECCC BibTeX
@inproceedings{flntz,
author = {Yuval Filmus and Massimo Lauria and Jakob Nordstr\"om and
Neil Thapen and Noga Zewi},
title = {Space complexity in polynomial calculus},
booktitle = {The 27th Annual Conference on Computational Complexity ({CCC} 2012)},
year = {2012}
}
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ICALP 2013
During the last decade, an active line of research in proof complexity has been into the space complexity of proofs, and how space is related to other measures. By now, these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard “benchmark formulas” is still open, as well as the relation of space to size and degree in PC/PCR.
We prove that if a formula requires large resolution width then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least Ω(n1/2).
Our proofs use techniques recentled introduced by Bonacina and Galesi. Our final contribution is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework, and suggesting that we are still far from characterizing PC/PCR space.
Download BibTeX
@inproceedings{flmnv,
author = {Yuval Filmus and Massimo Lauria and Mladen Mik\v{s}a and
Jakob Nordstr\"om and Marc Vinyals},
title = {Towards an understanding of {P}olynomial {C}alculus: new
separations and lower bounds},
booktitle = {The 40th International Colloquium on Automata,
Languages and Programming ({ICALP} 2013)},
year = {2013}
}
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Preprint
In this note we argue that semantic cutting planes refutations are stronger than syntactic ones. In particular, we give a formula for which any refutation in syntactic cutting planes requires exponential length, while there is a polynomial length refutation in semantic cutting planes. This means that syntactic cutting planes does not p-simulate (nor simulate) semantic cutting planes.
We also give a pair of incompatible cutting planes lines which require exponential length to be refuted in syntactic cutting planes.
Download BibTeX
@article{fl,
author = {Yuval Filmus and Massimo Lauria},
title = {A separation between semantic and syntactic cutting
planes},
journal = {Preprint}
}
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STACS 2012
We present an optimal, combinatorial 1−1/e approximation algorithm for Maximum Coverage over a matroid constraint, using non-oblivious local search. Calinescu, Chekuri, Pál and Vondrák have given an optimal 1−1/e approximation algorithm for the more general problem of monotone submodular maximization over a matroid constraint. The advantage of our algorithm is that it is entirely combinatorial, and in many circumstances also faster, as well as conceptually simpler.
Following previous work on satisfiability problems by Alimonti and by Khanna, Motwani, Sudan and Vazirani, our local search algorithm is non-oblivious. That is, our algorithm uses an auxiliary linear objective function to evaluate solutions. This function gives more weight to elements covered multiple times. We show that the locality ratio of the resulting local search procedure is at least 1−1/e. Our local search procedure only considers improvements of size 1. In contrast, we show that oblivious local search, guided only by the problem's objective function, achieves an approximation ratio of only (n−1)/(2n−1−k) when improvements of size k are considered.
In general, our local search algorithm could take an exponential amount of time to converge to an exact local optimum. We address this situation by using a combination of approximate local search and the same partial enumeration techniques used by Calinescu et al., resulting in a clear (1−1/e)-approximation algorithm running in polynomial time.
We obtained our auxiliary linear objective function using linear programming. This is detailed in Ward's thesis.
Download Slides 1 Slides 2 BibTeX
@inproceedings{fw1,
author = {Yuval Filmus and Justin Ward},
title = {Maximum coverage over a matroid},
booktitle = {29th Symposium on Theoretical Aspects of Computer
Science ({STACS} 2012)},
year = {2012}
}
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FOCS 2012
We present an optimal, combinatorial 1−1/e approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm due to Calinescu, Chekuri, Pál and Vondrák, our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by local search. Both phases are run not on the actual objective function, but on a related auxiliary potential function, which is also monotone submodular.
In our previous work on maximum coverage (the preceding paper), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone submodular functions. When the objective function is a coverage function, both definitions of the potential function coincide.
Our approach generalizes to the case where the monotone submodular function has restricted curvature. For any curvature c, we adapt our algorithm to produce a (1−e−c)/c approximation. This matches results of Vondrák, who has shown that the continuous greedy algorithm produces a (1−e−c)/c approximation when the objective function has curvature c, and proved that achieving any better approximation ratio is impossible in the value oracle model.
The paper exists in many different versions:
The preprint supersedes the previous versions.
Preprint FOCS ArXiv Extended version BibTeX
@inproceedings{fw2,
author = {Yuval Filmus and Justin Ward},
title = {A tight combinatorial algorithm for submodular maximization
subject to a matroid constraint},
booktitle = {53rd Annual {IEEE} Symposium on Foundations of Computer Science ({FOCS} 2012)},
year = {2012}
}
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WINE 2010
The problem of influence maximization deals with choosing the optimal set of nodes in a social networks so as to maximize the resulting spread of a technology (opinion, product ownership and so on), given a model of diffusion of influence in a network. A natural extension is a competitive setting, in which the goal is to maximize the spread of our technology in the presence of one or more competitors.
We suggest several natural extensions to the well-studied linear threshold model, showing that the original greedy approach cannot be used.
Furthermore, we show that for a broad family of competitive influence models, it is NP-hard to achieve an approximation that is better than a square root of the optimal solution; the same proof can also be applied to give a negative result for a conjecture in Carnes et al. about a general cascade model for competitive diffusion.
Finally, we suggest a natural model that is amenable to the greedy approach.
Download BibTeX
@inproceedings{bfo1,
author = {Allan Borodin and Yuval Filmus and Joel Oren},
title = {Threshold models for competitive influence in social networks},
booktitle = {The 6th Workshop on Internet and Network Economics
({WINE} 2010)},
year = {2010}
}
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IJCAI 2013
Top-k voting is an especially natural form of partial vote elicitation in which only length k prefixes of rankings are elicited. We analyze the ability of top-k vote elicitation to correctly determine true winners with high probability, given probabilistic models of voter preferences and candidate availability. We provide bounds on the minimal value of k required to determine the correct winner under the plurality and Borda voting rules, considering both worst-case preference profiles and profiles drawn from the impartial culture and Mallows probabilistic models. We also derive conditions under which the special case of zero elicitation (i.e., k=0) produces the correct winner. We provide empirical results that confirm the value of top-k voting.
Download BibTeX
@inproceedings{bfo2,
author = {Craig Boutilier and Yuval Filmus and Joel Oren},
title = {Efficient vote elicitation under candidate uncertainty},
booktitle = {23rd International Joint Conference on Artificial Intelligence
({IJCAI} 2013)},
year = {2013}
}
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SIGMOD 2012
In this demonstration, we showcase the technologies that we are building at Yahoo! for web-scale information extraction. Given any new website, containing semi-structured information about a pre-specified set of schemas, we show how to populate objects in the corresponding schema by automatically extracting information from the website.
Work done while I was a summer intern in Yahoo! Tel Aviv.
Link BibTeX
@inproceedings{bdfjkk,
author = {Philip Bohannon and Nilesh Dalvi and Yuval Filmus and Nori
Jacoby and Sathiya Keerthi and Alok Kirpal},
title = {Automatic web-scale information extraction},
booktitle = {Proceedings of the 2012 {ACM} {SIGMOD} International Conference on Management of Data},
year = {2012},
pages = {609--612}
}
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Information Processing Letters, Volume 111, Issue 18, 2011, pp. 841–885
Ellul, Krawetz, Shallit and Wang prove an exponential lower bound on the size of any context-free grammar generating the language of all permutations over some alphabet. We generalize their method and obtain exponential lower bounds for many other languages, among them the set of all squares of given length, and the set of all words containing each symbol at most twice.
The version below corrects two typos in the proof of Proposition 6: w1∼w2 should be x(w1)∼x(w1), and in the following sentence N−1(A) should be x(N−1(A)).
Download BibTeX
@article{filmus1,
author = {Yuval Filmus},
title = {Lower bounds for context-free grammars},
journal = {Information Processing Letters},
volume = {111},
issue = {18},
year = {2011},
pages = {895--898}
}
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Information Processing Letters, Volume 113, Issue 13, 2013, pp. 457–464
We devise a method for proving inequalities on submodular functions, with a term rewriting flavour. Our method comprises of the following steps:
The crucial third step is non-constructive, since it uses compactness of the dual cone of submodular functions. Its proof uses the classical uncrossing technique with a quadratic potential function.
We prove several inequalities using our method, and use them to tightly analyze the performance of two natural (but non-optimal) algorithms for submodular maximization, the random set algorithm and local search.
Download BibTeX
@article{filmus2,
author = {Yuval Filmus},
title = {Inequalities on submodular functions via term rewriting},
journal = {Information Processing Letters},
volume = {113},
issue = {13},
year = {2013},
pages = {457--464}
}
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Logical Methods in Computer Science, Volume 9, Issue 3, 2013, Paper 7.
A code of the natural numbers is a uniquely-decodable binary code of the natural numbers with non-decreasing codeword lengths, which satisfies Kraft's inequality tightly. We define a natural partial order on the set of codes, and show how to construct effectively a code better than a given sequence of codes, in a certain precise sense. As an application, we prove that the existence of a scale of codes (a well-ordered set of codes which contains a code better than any given code) is independent of ZFC.
@article{filmus3,
author = {Yuval Filmus},
title = {Universal codes of the natural numbers},
journal = {Logical Methods in Computer Science},
volume = {9},
issue = {3},
year = {2013},
pages = {paper 7}
}
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A survey of Friedgut's research program in extremal combinatorics. Friedgut uses spectral methods — Hoffman's eigenvalue bound — to obtain tight bounds on measures of intersecting families. His method has the advantage of implying stability: families of near-maximal measure are similar to families of maximal measure.
Results surveyed:
I prepared this survey for my depth oral.
Bonet, Pitassi and Raz and, independently, Krajíček came up with the idea to use feasible interpolation as a vehicle for lower bounds on proof systems. Bonet et al. presented their construction in a somewhat ad hoc manner, and Krajíček used a theorem of Razborov generalizing Karchmer–Wigderson games. We provide a different interpretation of their arguments in terms games which are more suitable than the ones considered by Razborov.
Our account includes three flavors of arguments: ones following Bonet et al., ones following Krajíček, and ones following a suggestion of Neil Thapen. The different arguments prove lower bounds for slightly different formulas.
Alekhnovich and Razborov (Lower bounds for polynomial calculus: non-binomial case) gave a sophisticated degree lower bound for proofs in the polynomial calculus proof system. However, their proof is somewhat opaque. We present a different, more intuitive proof. We also adapt our proof to obtain related results of Galesi and Lauria.
Work done while visiting KTH in December 2012.
Snepscheut came up with the following puzzle. There is a table with four coins at the center of the four sides. The goal is to get all of the coins in the same orientation, that is, all heads or all tails. You never get to see the coins (you are blindfolded), but you can turn one or two of them. Prior to each move, the table is rotated by an arbitrary, unknown multiple of 90 degrees. How many moves do you need to reach the goal?
Dijkstra generalized the solution to 2n sides. We consider an even more general problem, in which the table has n sides, and the coins have m “states” which are modified by addition modulo m (more generally, one could think of a finite Abelian group for the state of the coins). We show that:
Andreas Blass explained how the type-theoretic identity T=1+T2 for binary trees leads to a “finitistic” identity T7=T (the solution to the former equation is a primitive sixth root of unity). More generally, he proved that two polynomials in T are “equal” (under his finitistic interpretation) if and only if they agree on a primitive sixth root of unity and in terms of cardinalities. We give an exposition of this result (using work by Fiore and Leinster), using a more intuitive definition of “strong” equality, as equality given by an algorithm which also works for infinite binary trees.
This talk was given in the Toronto Student Seminar on 16/9/2009.
We survey several modern integer factorization methods, including Pollard's ρ, Pollard's p−1, Williams' p+1, the elliptic curve method, Shanks' continued fractions algorithm, the quadratic sieve (including MPQS and Dixon's provable variant) and the number-field sieve (which is only sketched).
Originally given as a talk in the Toronto Student Seminar on 25/11/2009, this talk has proven popular and I gave it several more times.
We provide two proofs of the central limit theorem (up to Lévy's continuity theorem), one using cumulants and the other using moments. As a bonus, we also prove the asymptotic normality of the number of distinct prime factors of a ‘random’ integer. Our account follows the exposition in the book The semicircle law, free random variables and entropy.
This talk was given in the Toronto Student Seminar on 20/1/2010.
An exposition of Feige's celebrated result on the hardness of approximating set cover.
This talk was given as part of the PCP reading group on 23/1/2010.
An exposition on algorithms for matrix multiplication, in two parts:
These talks were given in the Toronto Student Seminar on 2/2/2012 and 9/2/2012, though the second one has been significantly updated since.
Cai, Pavan and Sivakumar showed that it is hard to compute the permanent even with an inversely polynomial success probability, assuming the worst-case hardness of computing the permanent. Their proof combines the LFKN protocol with Sudan's list-decoding algorithm for Reed-Solomon codes. We given an exposition of their result, as well as several results leading to it.
This talk was given in the Toronto Student Seminar on 17/9/2012.
We survey several recent algorithmic results on submodular maximization:
We also briefly survey some lower bounds:
This talk was given in the Toronto Student Seminar on 31/1/2013.
We give an exposition of Smolensky's fundamental paper on the polynomial method, a lower bound method in circuit complexity. Books usually contain a simpler argument which works only for parity, whereas Smolensky's argument also works for majority (directly). Our exposition omits the step of approximation a constant depth circuit by a low-degree polynomial.
An exposition of parts of Jan Krajíček's book Forcing with random variables and proof complexity, concentrating on lower bounds for constant depth proof systems.
Parts of this exposition has been given as talks in a reading group on the book on 13/6/2013 and 20/6/2013.
We provide a list of all inequivalent non-trival antichains on the Boolean lattice of dimension 6, excluding the empty antichain and the one containing the empty set. Alternatively, this is a list containing all inequivalent non-constant monotone Boolean functions on six inputs, given by their minterms.
Antichains depending of dimension n are given in terms of the points 1, …, n. The number of antichains of given dimension (including the two trivial cases) forms the sequence A003182.
We given a Katona-like proof that a triangle-intersecting family of graphs contains at most 1/8 of the graphs. Unfortuantely, our proof works only on up to eight vertices. We discuss several other methods which also cannot give a general proof.
Parts of this note are summarized in my thesis.
Latała and Oleszkiewicz proved the special L1 case of the Khintchine-Kahane inequality. We reformulate their proof using Fourier analysis.
Vincenzo Ciancia asked on cstheory.stackexchange about the class of regular languages satisfying the following property: whenever a word w belongs to the language, all of its positive powers wk also belong to the language. He termed these languages ‘circular languages’. Answering his question, we have shown the following:
The normal form appears in a paper by Calbrix and Nivat, and the PSPACE-completeness result follows quite easily from a paper by Kozen, as I detail in my answer. My proofs appear below.
Yaroslav Bulatov asked on mathoverflow what is the asymptotic number of self-avoiding integer walks of length n in which adjacent positions are either 1 or 2 apart. We obtain a formula for the exact number of such walks, and deduce that it is Õ(μn), where μ≈2.20556943040059.
Avinoam Braverman considered the sequence n mod x, where 1≤x≤√n, took its local minima, and plotted the results. When n is large, a curious pattern composed out of what seem to be triangles appears. We explain this phenomenon heuristically, given formulas for the envelope of the “triangles” (which turn out to be quadratic functions). We go on to describe the envelope of the plot when an arbitrary number of the local minima and local maxima operations are composed.
Suppose n points are thrown on the circumference of a unit-circumference circle, partitioning the circumference into n segments. What is the expected length of the kth smallest segment? There is a well-known formula for this expectation.
We consider the expected length of kth smallest two adjacent segments. We develop a method for computing them exactly, and compute the expectations for several small n,k. The results do not seem to fit into a nice pattern.
We provide a bijective proof for the well-known fact that the number of 123-avoiding permutations and the number of 132-avoiding permutations are both counted by Catalan numbers. Simion and Schmidt came up with a direct bijection between the two sets of permutations, and their proof is recommended over mine.
It is well-known that the until operator in linear temporal logic (LTL) cannot be expressed using next, always and eventually. We provide a simple proof of this fact.
We prove that several equivalent definitions of the Sprague–Grundy function coincide (an exercise given in a course on combinatrial games given by Aviezri Fraenkel).
Katona gave a simple proof of the Erdős–Ko–Rado theorem. We adapt his proof to the μp setting. We are also able to prove uniqueness, but not stability.
If (x,y)=1 then (x+y,xy)=1. The first part shows how to prove this using standard arguments and using the characterization of GCD as the minimal positive value obtained as an integer combination of the operands. The second part generalizes the argument to a lemma involving n variables.
We give a combinatorial proof of the identity 1/(1−x)⋅1/(1−y) = 1/(1−x−y+xy).
Stan Wagon gave fourteen proofs of the following result about tiling a rectangle: if a rectangle can be tiled using rectangles with at least one integral side, then the tiled rectangle also has at least one integral side. We paraphrase his 13th proof.
Dehn proved that if a rectangle can be tiled by rectangles whose sides are commensurable, then the tiled rectangle is also commensurable. His proof, as described in Proofs from the book, applies a homomorphism which results in possibly negative side lengths. We modify his proof so that all side lengths are positive.
The crucial ingredient is the following lemma: for each finite set of positive reals there is a basis (over the rationals) of positive reals such that every element in the set is a non-negative integral combination of base elements. We provide two proofs of this lemma, one due to us and one due to Avinoam Braverman.
We solve the following puzzle: given an arbitrary supply of AND gates and OR gates, invert n inputs using as few NOT gates as possible.
Question 8 in Chapter 2 of The Probabilistic Method asks us to show that for every n×n orthogonal matrix and 1≤k≤n, there is a column such that the squared L2 norm of its first k entries is at least k/n, or at most k/n. It also asks for an example in which this is tight. We exhibit such an example which works simultaneously for all k.
A big sheet of paper contains 2n rows consisting of all possible vectors of length n whose entries are +1 or −1. Someone changes some of the entries to zero. Show that there must be a non-empty subset of the rows summing to zero.
Probably much harder than you think it is!
We show that the range of a symmetric matrix over GF(2) always contains its diagonal. We present both our algorithmic proof and a simple proof by Noga Alon. A simplification of our proof has been given by Soltys (Lemma 9).
Démonstration d'un Théorème d'Arithmétique, Nouveaux Mémoires de l'Académie royale des Science et Belles-Lettres de Berlin, année 1770.
Démonstration d'un Théorème nouveau concernant les Nombres premiers, Nouveaux Mémoires de l'Académie royale des Science et Belles-Lettres de Berlin, année 1771.
Beweis der Transzendenz der Zahl e, Mathematische Annalen, Bd. 43, 1893, S. 220–221.
Translation from Herstein's Topics in Algebra, p. 176–178.
Transcendenz von e und π, Mathematische Annalen, Bd. 43, 1893, S. 222–224.
Über die Transzendenz der Zahlen e und π, Mathematische Annalen, Bd. 43, 1893, S. 216–219.
Über die Komposition der quadratischen Formen von beliebig vielen Variablen, Nachrichten von der k. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse, 1898, S. 309–316.
Another proof of Cauchy's theorem, American Mathematical Monthly, Vol. 66 (February 1959), p. 119.
Excerpt from Modèles non-standard en arithmétique et théorie des ensembles, Jean-Pierre Ressayre & Alec J. Wilkie.
Shalom Shabazi is the most important Jewish poet from Yemen. Following a facsimile of a diwan published by Seri and Tobi, we have copied a few of his Hebrew poems.
The Kuzari is an important Jewish theological work attributed to the medieval poet Yehuda Halevi. While originally written in Arabic, it is usually found in translation. Following an edition by Rabbi Qafih, we have copied the entire Arabic text.
My translation of Avidan's poem Menat Krav, in which Avidan, wary of the world, sleeps for 2000 years and awakens to a sci-fi future.
A patience collection by Ella von Haunstein from the early 20th century, regrettably without the diagrams.
Our own contribution to the mythological origins of the diatonic scales.
The background image is the final result of a 2D cellular automaton. For rules and animations, follow the link.
Rhythms of resistance is a world-wide network of political samba bands. I have prepared Python code for generating rhythms, along with two examples. There is also a cheat sheet if you're playing.