# “Just a Spoonful of Sugar …”

Tim Roughgarden sent me the following email. I found this idea so refreshing that I thought I should share more widely. Well done PC!

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PC meetings are typically all work and no play. But the FOCS ’12 PC, in an act of rebellion, has decided to spend a day disucssing their **own** results, before they spend two days discussing everyone else’s. All are invited to attend.

**What**: Research talks by members of the FOCS ’12 PC

**When**: Thursday June 14th, roughly 10AM-5PM.

**Where**: Gates 415 (in the Stanford CS building)

**Who**: the speakers are (in alphabetical order)

1. Nikhil Bansal, “Min-Max Graph Partitioning”

2. Harry Buhrman, “Position-based cryptography”

3. Swastik Kopparty, “List-Decoding Multiplicity Codes”

4. Rafael Pass, “An Epistemic Approach to Mechanism Design”

5. Sofya Raskhodnikova “Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy”

6. Aaron Roth “Privately Computing Equilibria”

7. C Seshadhri “Understanding simple triangle enumeration algorithms on real-world graphs”

8. Berthold Vöcking “A Universally-truthful Approximation Scheme for Multi-unit Auctions”

9. David P Woodruff “Tight Bounds for Distributed Functional Monitoring”

Abstracts are at the end of this email. A more detailed schedule will be posted in a few days, with a link available from http://theory.stanford.edu/~tim/

Hope to see you there!

Tim

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Min-Max Graph Partitioning

Abstract: I will talk about graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part.

The two main versions we consider are where the k parts need to be of equal-size, and where they must separate a set of k given terminals. We design O( \sqrt{log n log k})-approximations for these problems, substantially improving upon previous results.

The main tool we use is a new O(\sqrt{\log n \log k}) approximation algorithm for the small set expansion problem, and a new randomized uncrossing procedure which may be of independent interest.

Based on joint work with Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Naor and Roy Schwartz.

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Position-based cryptography

On 20 July 1969, millions of people held their breath as they watched, live on television, Neil Armstrong set foot on the Moon. Yet Fox Television has reported that a staggering 20% of Americans have had doubts about the Apollo 11 mission. Could it have been a hoax staged by Hollywood studios here on Earth? Position based cryptography may offer a solution. This kind of cryptography uses the geographic position of a party as its sole credential. Normally digital keys or biometric features are used.

A central building block in position-based cryptography is that of position-verification. The goal is to prove to a set of verifier that one is at a certain geographical location. Protocols typically assume that messages can not travel faster than the speed of light. By responding to a verifier in a timely manner one can guarantee that one is within a certain distance of that verifier. Quite recently it was shown that position-verification protocols only based on this relativistic principle can be broken by attackers who simulate being at the claimed position while physically residing elsewhere in space.

Because of the no-cloning property of quantum information (qubits) it was believed that with the use of quantum messages one could devise protocols that were resistant to such collaborative attacks. Several schemes were proposed that later turned out to be insecure. Finally it was shown that also in the quantum case no unconditionally secure scheme is possible. We will review the field of position-based quantum cryptography and highlight some of the research currently going on in order to develop, using reasonable assumptions on the capabilities of the attackers, protocols that are secure in practice.

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List-Decoding Multiplicity Codes

Multiplicity Codes allow one to encode data with just an epsilon fraction redundancy, so that even if a constant fraction of the encoded bits are corrupted, any one bit of the original data can be recovered in sublinear time with high probability.

I will talk about a new result showing that these codes also tolerate a large fraction of errors:

(1) they can achieve “list-decoding capacity”,

(2) they can be locally list-decoded beyond half their minimum distance.

In particular, we give the first polynomial time algorithms for decoding multiplicity codes upto half their minimum distance.

The first of these results is based on solving some kinds of algebraic differential equations.

The second is based on a family of algebraically repelling “space-filling” curves.

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An Epistemic Approach to Mechanism Design

We introduce an epistemic framework for analyzing mechanisms. This framework enables mechanism designers to define desirability of outcomes not only based on players’ actual payoff types and their beliefs about the payoff types of other players (as in the classic models), but also based on higher order beliefs of the players (i.e., beliefs about beliefs about … the payoff types of the players). In this framework, we may also use epistemic solution concepts to analyze what outcomes are consistent with different levels of rationality: a player is k-level rational if he is rational and considers all other players (k-1)-level rational; following Aumann, we consider a very weak notion of rationality: player i is *rational* if he uses a strategy \sigma such that for every alternative strategy \sigma’, i considers some world possible where \sigma performs at least as well as \sigma’.

We showcase this framework in the context of single-good auctions, presenting an interim individually-rational mechanism with the following revenue guarantee: for any k\geq 0, any outcome consistent with all players being (k+1)-level rational guarantees the seller a revenue of G^k – \epsilon (for any \epsilon > 0), where G^k is the second highest belief about belief about … (k times) about the highest valuation of some player. We additionally show that no interim individually rational mechanism can guarantee a revenue of G^k – \epsilon for any constant \epsilon, if only assuming players are k-level rational (as opposed to (k+1)-level rational). Taken together, these results demonstrate the existence of a “revenue-rationality

hierarchy”: strictly higher revenue may be extracted by assuming players satisfy higher levels of rationality.

Towards analyzing our mechanism and proving our lower bounds, we introduce an iterative deletion procedure of dominated strategies that precisely characterizes strategies consistent with k-level rationality.

Prior knowledge of mechanism design or epistemic logic will not be assumed.

Joint work with Jing Chen and Silvio Micali.

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Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy.

Abstract:

A function f : D -> R has Lipschitz constant c if d_R(f(x), f(y)) <= c d_D(x, y) for all x, y in D, where d_R and d_D denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) Intuitively, a Lipschitz constant of f is a bound on how sensitive f is to small changes in its input. Lipschitz constants are important in mathematical analysis, the theory of differential equations and other areas of mathematics and computer science. However, in general, it is computationally infeasible to find a Lipschitz constant of a given function f or even to verify that f is c-Lipschitz for a given number c.

We initiate the study of testing (which is a relaxation of the decision problem described above) and local reconstruction of the Lipschitz property of functions. A property tester, given a proximity parameter epsilon, has to distinguish functions with the property (in this case, Lipschitz) from functions that are epsilon-far from having the property, that is, differ from every function with the property on at least an epsilon fraction of the domain. A local filter reconstructs a desired property (in this case, Lipschitz) in the following sense: given an arbitrary function f and a query x, it returns g(x), where the resulting function g satisfies the property, changing f only when necessary. If f has the property, g must be equal to f.

We design efficient testers and local filters for functions over domains of the form {1,…,n}^d, equipped with L1 distance, and give corresponding lower bounds. The testers that we developed have applications to programs analysis. The local filters have applications to data privacy. The application to privacy is based on the fact that a function f of entries in a database of sensitive information can be released with noise of magnitude proportional to a Lipschitz constant of f, while preserving the privacy of individuals whose data is stored in the database (Dwork, McSherry, Nissim and Smith, TCC 2006). We give a differentially private mechanism, based on local filters, for releasing a function f when a purported Lipschitz constant of f is provided by a distrusted client.

Based on a FOCS 2011 paper with M. Jha and recent work with P.

Awasthi, M. Jha and M. Molinaro

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Privately Computing Equilibria (with applications to equilibrium selection and repeated games)

We consider “large games” in which the choice of action of each individual agent can have an arbitrarily large effect on his own utility, but only a bounded effect on the utility of any other agent.

In such games, we study the problem of computing correlated equilibria while preserving the (differential) privacy of the utility function of each agent. We give a tight information theoretic lower bound on the accuracy to which approximate equilibria can be computed while preserving differential privacy, and two algorithms. Our first algorithm is computationally efficient, but has a suboptimal dependence on the number of actions in the game (It continues to match the information theoretic lower bound in games with O(1) actions). The second algorithm is not computationally efficient, but gives a tight approximation factor that depends only logarithmically on the number of actions in the game.

Finally, we mention a couple of game theoretic implications of the ability to compute private equilibria. We get an approximately truthful equilibrium selection mechanism for large games, and a way to analyze noisy repeated games with growing populations that eliminates folk theorem equilibria, and collapses all of the equilibria of the repeated game to the equilibria of the single shot game.

This is joint work with Michael Kearns, Mallesh Pai, and Jon Ullman.

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Understanding simple triangle enumeration algorithms on real-world graphs

Abstract: Triangle enumeration is an important practical (and interesting theoretical) problem for real-world graphs. This work is motivated by the following observation: simple, embarrassingly parallel heuristics for triangle enumeration perform rather well on certain classes of real-world graphs (like infrastructure networks and web networks). On the other hand, it is very easy to come up with terrible worst-case examples for these heuristics. Can we give some theoretical reason for why these algorithms work?

We analyze a basic binning algorithm for triangle counting (which is quite commonly used among practitioners) for graphs generated by the edge-configuration model. This is a fairly common model studied by statisticians and physicists who wish to construct graphs with a given degree-sequence. We prove that the running time is basically proportional to the four-thirds moment of the degree distribution.

This is a very pleasant discovery: moments of degree distributions of real-world graphs are considered a very important property (think of heavy-tail and power law), and it is this property that allows speedup in simple triangle enumeration algorithms.

Joint work with Jonathan Berry, Luke Fostvedt, Daniel Nordman, Cynthia Phillips, and Alyson Wilson

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A Universally-truthful Approximation Scheme for Multi-unit Auctions

We present a randomized, polynomial-time approximation scheme for multi-unit auctions. Our mechanism is truthful in the universal sense, i.e., a distribution over deterministically truthful mechanisms.

Previously known approximation schemes were truthful in expectation which is a weaker notion of truthfulness assuming risk neutral bidders. The existence of a universally truthful approximation scheme was questioned by previous work showing that multi-unit auctions with certain technical restrictions on their output do not admit a polynomial-time, universally truthful mechanism with approximation factor better than two.

Our new mechanism employs VCG payments in a non-standard way: The deterministic mechanisms underlying our universally truthful approximation scheme are not maximal in range and do not belong to the class of affine maximizers which, on a first view, seems to contradict previous characterizations of VCG-based mechanisms. Instead, each of these deterministic mechanisms is composed of a collection of affine maximizers, one for each bidder which yields a subjective variant of VCG.

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Tight Bounds for Distributed Functional Monitoring

We resolve several open questions in the area of distributed functional monitoring. For the purposes of this talk, we can think of there being k parties, each holding their own private vector, and the parties want to compute a statistic on the sum v of the k vectors. We show that to compute a 1+eps-approximation to the support of v or to the Euclidean norm of v, k/eps^2 communication is necesary up to polylogarithmic factors. These lower bound are matched by known upper bounds. We also mention results on approximating the Euclidean norm in the model in which the private vectors are evolving with time and a central coordinator must maintain an approximation to |v|_2 at all times. We improve the previous k^2/poly(eps) communication bound to k/poly(eps), which resolves a question of Cormode, Muthukrishnan, and Yi.

Joint work with Qin Zhang

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