Yuval Filmus

Threshold models for competitive influence in social networks

The problem of influence maximization deals with choosing the optimal set of nodes in a social network so as to maximize the resulting spread of a technology (opinion, product-ownership, etc.), 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 linearthreshold 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 [2] about a general cascade model for competitive diffusion. Finally, we suggest a natural model that is amenable to the greedy approach.

A Tight Combinatorial Algorithm for Submodular Maximization Subject to a Matroid Constraint

We present an optimal, combinatorial 1-1/e approximation algorithm for monotone sub modular optimization over a matroid constraint. Compared to the continuous greedy algorithm (Calinescu, Chekuri, Pal and Vondrak, 2008), 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 non-oblivious potential function, which is also monotone sub modular. In our previous work on maximum coverage (Filmus and Ward, 2011), the potential function gives more weight to elements covered multiple times. We generalize this approach from coverage functions to arbitrary monotone sub modular functions. When the objective function is a coverage function, both definitions of the potential function coincide. The parameters used to define the potential function are closely related to Pade approximants of exp(x) evaluated at x = 1. We use this connection to determine the approximation ratio of the algorithm.

Automatic web-scale information extraction

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.

Triangle-intersecting Families of Graphs

A family of graphs F is triangle-intersecting if for every G,H 2 F, G\H contains a triangle. A conjecture of Simonovits and Sos 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 of size 1 2 ( n 2) . 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 dierent, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.

Monotone submodular maximization over a matroid via non-oblivious local search

We present an optimal, combinatorial

A quasi-stability result for dictatorships in Sn

1-1/e

From Small Space to Small Width in Resolution

approximation algorithm for monotone submodular optimization over a matroid constraint. Compared to the continuous greedy algorithm [G. Calinescu et al., IPCO, Springer, Berlin, 2007, pp. 182--196] our algorithm is extremely simple and requires no rounding. It consists of the greedy algorithm followed by a local search. Both phases are run not on the actual objective function, but on a related auxiliary potential function, which is also monotone and submodular. In our previous work on maximum coverage [Y. Filmus and J. Ward, FOCS, IEEE, Piscataway, NJ, 2012, pp. 659--668], 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 curvatu...

Invariance Principle on the Slice

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 [6], and first proved in [10]. 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.

The Power of Local Search: Maximum Coverage over a Matroid

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a “black-box” technique for proving space lower bounds via a “static” complexity measure that works against any resolution refutation—previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

A stability result for balanced dictatorships in Sn

The non-linear invariance principle of Mossel, O’Donnell, and Oleszkiewicz establishes that if f(x1,… ,xn) is a multilinear low-degree polynomial with low influences, then the distribution of if f(b1,…,bn) is close (in various senses) to the distribution of f(G1,…,Gn), where Bi ∈R {-1,1} are independent Bernoulli random variables and Gi ∼ N(0,1) are independent standard Gaussians. The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans–Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO’s invariance principle works for any two vectors of hypercontractive random variables (X1,… ,Xn),(Y1,… ,Yn) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variables X1,… ,Xn are independent, as are Y1,…,Yn. The independence condition is crucial to the proof of the theorem, yet in some cases we would like to use distributions X1,… ,Xn in which the individual coordinates are not independent. A common example is the uniform distribution on the slice ([n]k) which consists of all vectors (x1,…,xn)∈{0,1}n with Hamming weight k. The slice shows up in theoretical computer science (hardness amplification, direct sum testing), extremal combinatorics (Erdős–Ko–Rado theorems), and coding theory (in the guise of the Johnson association scheme). Our main result is an invariance principle in which (X1,…,Xn) is the uniform distribution on a slice ([n]pn and (Y1,… ,Yn) consists either of n independent Ber(p) random variables, or of n independent N(p,p(1-p)) random variables. As applications, we prove a version of Majority is Stablest for functions on the slice, a version of Bourgain’s tail theorem, a version of the Kindler–Safra structural theorem, and a stability version of the t-intersecting Erdős–Ko–Rado theorem, combining techniques of Wilson and Friedgut. Our proof relies on a combination of ideas from analysis and probability, algebra, and combinatorics. In particular, we make essential use of recent work of the first author which describes an explicit Fourier basis for the slice.