Shalev Ben-David

Separations in query complexity using cheat sheets

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grovers algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique, which builds upon the techniques of Ambainis et al. [STOC 2016]. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al. [STOC 2016], to be shown in a unified manner.

Data stability in clustering: A closer look

We consider the model introduced by Bilu and Linial (2010) [13], who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations, e.g. on the order of O(n) for max-cut clustering. Awasthi et al. (2012) [6] consider center-based objectives, and Balcan and Liang (2012) [9] analyze the k-median and min-sum objectives, giving efficient algorithms for instances resilient to certain constant multiplicative perturbations. Here, we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms. We show there is little room to improve these results by giving NP-hardness lower bounds for both the k-median and min-sum objectives. On the other hand, we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial, leaving only a narrow range of resilience parameters for which clustering is interesting. We also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability. Our results provide a close examination of the consequences of assuming stability in data.

Separations in Communication Complexity Using Cheat Sheets and Information Complexity

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ≈ 1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to Goos, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.

Randomized Query Complexity of Sabotaged and Composed Functions.

We study the composition question for bounded-error randomized query complexity: Is R(f circ g) = Omega(R(f)R(g))? We show that inserting a simple function h, whose query complexity is onlyTheta(log R(g)), in between f and g allows us to prove R(f circ h circ g) = Omega(R(f)R(h)R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f circ g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f circ g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity. Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem from zero-error randomized query to communication complexity implies a similar result for bounded-error algorithms for all total functions.

On Rota's Conjecture and nested separations in matroids☆

Abstract We prove that for each finite field F and integer k ∈ Z there exists n ∈ Z such that no excluded minor for the class of F -representable matroids has n nested k-separations.