Mathematics Department - DIMACS Theory of Computing Seminar - Spring 2017

# DIMACS Theory of Computing Seminar - Spring 2017

### Organizer(s)

Swastik Kopparty, Shubhangi Saraf

### Website

http://www.math.rutgers.edu/~sk1233/theory-seminar/S17/

## Wednesday, March 22nd

### "Solving Linear Programs without Breaking Abstractions"

Time: 11:00 AM
Location: CoRE 301
Abstract: We draw connections between descriptive complexity theory and combinatorial optimization to show that the ellipsoid method for solving linear programs can be implemented in a way that respects the abstractions and symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make choices, between variables or constraints in the program unless they are distinguished by properties definable from the linear program.

In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable in FPC. We use this to show that the size of a maximum matching in a graph is definable in FPC. This refutes a conjecture first posed by Blass, Gurevich and Shelah (1999). On the way to defining a suitable separation oracle for the maximum matching program, we provide FPC formulas defining canonical maximum flows and minimum cuts in undirected weighted graphs.

This is joint work with Anuj Dawar.

## Wednesday, March 8th

### "On Phase Transitions for Spiked Random Matrix and Tensor Models"

Time: 11:00 AM
Location: CoRE 301
Abstract: A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, in which a prominent eigenvector (or low rank structure) is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughout the sciences, where the goal is often to recover or detect the planted low rank structured. In this talk we discuss fundamental limitations of statistical methods to perform these tasks and methods that outperform PCA at it. Emphasis will be given to low rank structures arising in Synchronization problems.

Time permitting, analogous results for spiked tensor models will also be discussed.

Joint work with: Amelia Perry, Alex Wein, and Ankur Moitra.

## Wednesday, March 1st

### "Time-Space Hardness of Learning Sparse Parities"

Time: 11:00 AM
Location: CoRE 301
Abstract: How can one learn a parity function, i.e., a function of the form $f(x) = a_1 x_1 + a_2 x_2 + ... + a_n x_n (mod 2)$ where $a_1, ..., a_n are in {0,1}$, from random examples?

One approach is to gather $O(n)$ random examples and perform Gaussian-elimination. This requires a memory of size $O(n^2)$ and $poly(n)$ time. Another approach is to go over all possible $2^n$ parity functions and to verify them by checking $O(n)$ random examples for each guess. This requires a memory of size $O(n)$, but $O(2^n * n)$ time.

In a recent work, Raz [FOCS, 2016] shows that if an algorithm has memory of size much smaller than $n^2$, then it has to spend exponential time in order to learn a parity function. In other words, fast learning requires good memory.

In this work, we show that even if the parity function is known to be extremely sparse, where only log(n) of the a_i's are nonzero, then the learning task is still time-space hard. That is, we show that any algorithm with linear size memory and polynomial time fails to learn log(n)-sparse parities.

Consequently, the classical tasks of learning linear-size DNF formulae, linear-size decision trees, and logarithmic-size juntas are all time-space hard.

Based on joint work with Gillat Kol and Ran Raz.

## Wednesday, February 15th

### "Equivocating Yao: Constant-Rounds Adaptively Secure Multiparty Computation in the Plain Model"

Time: 11:00 AM
Location: CoRE 301
Abstract: Yao’s circuit garbling scheme is one of the basic building blocks of cryptographic protocol design. Originally designed to enable two-message, two-party secure computation, the scheme has been extended in many ways and has innumerable applications. Still, a basic question has remained open throughout the years: Can the scheme be extended to guarantee security in the face of an adversary that corrupts both parties, adaptively, as the computation proceeds?

We provide a positive answer to this question. We define a new type of encryption, called functionally equivocal encryption (FEE), and show that when Yao’s scheme is implemented with an FEE as the underlying encryption mechanism, it becomes secure against such adaptive adversaries. We then show how to implement FEE from any one way function.

Combining our scheme with non-committing encryption, we obtain the first two-message, two-party computation protocol, and the first constant-rounds multiparty computation protocol, in the plain model, that are secure against semi-honest adversaries who can adaptively corrupt all parties. We also provide extensions to the multiparty setting (with UC-security) and applications to leakage resilience.

## Wednesday, February 8th

### "Communication requirements and Informative signaling in matching markets"

Time: 11:00 AM
Location: CoRE 301
Abstract: We study the amount of communication needed to fi nd a stable matching in a two-sided matching market with private preferences when agents have some knowledge of the preference distribution. In a two-sided market with workers and fi rms, when the preferences of workers are arbitrary and private and the preferences of firms follow an additively separable latent utility model with commonly known and heterogeneous parameters, we describe an algorithm that fi nds a stable matching with high probability and requires at most O*(sqrt{n}) bits of communication per agent. (We also show that this is the best possible under this setting.) Our algorithm is a modification of worker-proposing deferred acceptance that skips wasteful applications. Firms help workers better target applications by signaling workers that they privately like and broadcasting to the market a qualifi cation requirement to discourage those with no realistic chances from applying. Our results yield insights on how matching markets can be better organized to reduce congestion. Broadly, agents should reach out to their favorites among "gettable" partners, while waiting for their dream matches to reach out to them.

Joint work with Itai Ashlagi, Mark Braverman and Peng Shi.

## Wednesday, February 1st

### "Alice and Bob Show Distribution Testing Lower Bounds (They don’t talk to each other anymore.)"

Time: 11:00 AM
Location: CoRE 301
Abstract: We present a new methodology for proving distribution testing lower bounds, establishing a connection between distribution testing and the simultaneous message passing (SMP) communication model. Extending the framework of Blais, Brody, and Matulef [BBM12], we show a simple way to reduce (private-coin) SMP problems to distribution testing problems. This method allows us to prove several new distribution testing lower bounds, as well as to provide simple proofs of known lower bounds.

Our main result is concerned with testing identity to a specific distribution p, given as a parameter. In a recent and influential work, Valiant and Valiant [VV14] showed that the sample complexity of the aforementioned problem is closely related to the 2/3-quasinorm of p. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's K-functional. We show that this quantity is closely related to the size of the effective support of p (loosely speaking, the number of supported elements that constitute the vast majority of the mass of p). This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.

Joint work with Eric Blais (University of Waterloo) and Tom Gur (Weizmann Institute)

## Wednesday, January 25th

### "Simultaneous Communication Complexity of Two-Player Combinatorial Auctions "

Time: 11:00 AM
Location: CoRE 301
Abstract: We consider simultaneous protocols for the following communication problem: Alice and Bob each have some list of subsets of [m], A_1,...,A_a, and B_1,...,B_b, and wish to either:

(Allocation Problem): partition [m] into A sqcup B in order to maximize max_i {A cap A_i} + max_j {B cap B_j}. (Decision Problem): decide whether there exists an i, j, such that A_i cup B_ j geq C.

For interactive protocols, a (tight) 3/4-approximation is known in poly(m) communication for both problems. We show the following:

- A randomized, simultaneous protocol with O(m) communication that achieves a (tight) 3/4-approximation for the allocation problem.

- No randomized, simultaneous protocol with subexponential (in m) communication guarantees better than a 3/4-1/108 approximation for the decision problem.

I'll further discuss the implications of these results for truthful combinatorial auctions due to a recent result of Dobzinski [FOCS 16].

Joint work with Mark Braverman and Jieming Mao.

## Wednesday, January 18th - CANCELLED

### "Equivocating Yao: Constant-Rounds Adaptively Secure Multiparty Computation in the Plain Model "

Time: 11:00 AM
Location: CoRE 301
Abstract: Yao’s circuit garbling scheme is one of the basic building blocks of crytographic protocol design. Originally designed to enable two-message, two-party secure computation, the scheme has been extended in many ways and has innumerable applications. Still, a basic question has remained open throughout the years: Can the scheme be extended to guarantee security in the face of an adversary that corrupts both parties, adaptively, as the computation proceeds?

We provide a positive answer to this question. We define a new type of encryption, called functionally equivocal encryption (FEE), and show that when Yao’s scheme is implemented with an FEE as the underlying encryption mechanism, it becomes secure against such adaptive adversaries. We then show how to implement FEE from any one way function.

Combining our scheme with non-committing encryption, we obtain the first two-message, two-party computation protocol, and the first constant-rounds multiparty computation protocol, in the plain model, that are secure against semi-honest adversaries who can adaptively corrupt all parties. We also provide extensions to the multiparty setting (with UC-security) and applications to leakage resilience.

This page was last updated on February 09, 2016 at 10:04 am and is maintained by webmaster@math.rutgers.edu.