Columbia Theory Seminar, Spring 2013

For Spring 2013, the usual time for the meetings will be Friday at 11:30 - 13:00 in the CS conference room, CSB 453. Abstracts for talks are given below the talk schedule.


  • Friday, February 8, 11:30, CSB 453: Delaram Kahrobaei: Non-commutative Algebraic Cryptography (Abstract)
  • Friday, February 15, 11:30, CSB 453: Jens Vygen: Shorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs (Abstract)
  • Friday, February 22, 11:30, CSB 453: Igor Carboni Oliveira: Constructing hard functions from learning algorithms (Abstract)
  • Friday, March 1, 11:30, CSB 453: Clément Canonne: Testing probability distributions using conditional samples (Abstract)
  • Friday, March 8, 11:30, CSB 453: Xiaorui Sun: Multi-Stage Propagation and Quasipolynomial-Time Isomorphism Testing of Steiner 2-Systems (Abstract)
  • Friday, March 15, 11:30, CSB 453: Krzysztof Onak: Superlinear lower bounds for multipass graph processing (Abstract)
  • Friday, April 12, 11:30, CSB 453: Howard Karloff: Maximum Entropy Summary Trees (Abstract)
  • Friday, April 19, 11:30, CSB 453: Augustin Chaintreau: Can Selfish Groups be Self-Enforcing? (Abstract)
  • Friday, April 26, 11:30, CSB 453: Daniel Kane: Diffuse Decompositions of Polynomials (Abstract)
  • Friday, May 3, 11:30, CSB 453: Esther Ezra: On Relative Approximations in Geometric Hypergraphs (Abstract)
  • Friday, May 17, 11:30, CSB 453: Yoram Moses: Time/Space of Digital Communication: Using Time to Coordinate Actions in Distributed Systems (Abstract)
  • Tuesday, June 18, 13:30, CSB 453 (note unusual time and date): Brendan Juba: Efficient reasoning in PAC Semantics (Abstract)
  • Friday, July 19, 11:00, CSB 453 (note unusual time): Angelina Vidali: Mechanism Design for Scheduling with Uncertain Execution Time (Abstract)

  • Talk Abstracts


    Friday, February 8:
    Non-commutative Algebraic Cryptography
    Delaram Kahrobaei
    CUNY Graduate Center

    Abstract: Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are based on number theory and depend, in particular, on the structure of abelian groups. The strength of computing machinery has made these techniques potentially susceptible to attack. As a result, an active line of research has arisen to develop cryptosystems and key exchange protocols using noncommutative cryptographic platforms. This line of investigation has been given the broad title of noncommutative algebraic cryptography. Research in this area was initiated by two public key protocols that used the braid groups, one by Ko, Lee et. al. and one by Anshel, Anshel and Goldfeld. The study of these protocols and the group theory surrounding them has had a large effect on research in infinite group theory. In this talk I survey a couple of these noncommutative group based methods and discuss several ideas in abstract group theory that have arisen from them. I will also allude to my recent work in this area.


    Friday, February 15:
    Shorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs
    Jens Vygen
    University of Bonn

    Abstract: We prove new results and obtain polynomial-time algorithms with improved approximation guarantees for the graphic traveling salesman problem and some related problems.

    For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-T-join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic s-t-path-TSP as a special case. Our improved approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4/3.

    The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

    This is joint work with Andras Sebo.


    Friday, February 22:
    Constructing hard functions from learning algorithms
    Igor Carboni Oliveira
    Columbia University

    Abstract: Fortnow and Klivans proved the following relationship between efficient learning algorithms and circuit lower bounds: if a class of boolean circuits C contained in P/poly is exactly learnable with membership and equivalence queries in polynomial-time, then [;EXP^{NP};] is not contained in C (the class [;EXP^{NP};] was subsequently improved to EXP by Hitchcock and Harkins). In this paper, we improve on these results and show

    - If C is exactly learnable with membership and equivalence queries in polynomial-time, then DTIME([;n^{\omega(1)};]) is not contained in C. We obtain even stronger consequences if C is learnable in the mistake-bounded model, in which case we prove an average-case hardness result against C.

    - If C is learnable in polynomial time in the PAC model then PSPACE is not contained in C, unless PSPACE is contained in BPP. Removing this extra assumption from the statement of the theorem would provide an unconditional separation of PSPACE and BPP.

    - If C is efficiently learnable in the Correlational Statistical Query (CSQ) model, we show that there exists an explicit function f that is average-case hard for circuits in C. This result provides stronger average-case hardness guarantees than those obtained by SQ-dimension arguments (Blum et al. 1993). We also obtain a non-constructive extension of this result to the stronger Statistical Query (SQ) model.

    Similar results hold in the case where the learning algorithm runs in subexponential time.

    Our proofs regarding exact and mistake-bounded learning are simple and self-contained, yield explicit hard functions, and show how to use mistake-bounded learners to "diagonalize"' over families of polynomial-size circuits. Our consequences for PAC learning lead to new proofs of Karp-Lipton-style collapse results, and the lower bounds from SQ learning make use of recent work relating combinatorial discrepancy to the existence of hard-on-average functions.

    Joint work with Adam Klivans (UT Austin) and Pravesh Kothari (UT Austin).


    Friday, March 1:
    Testing probability distributions using conditional samples
    Clément Canonne
    Columbia University

    Abstract: One of the most fundamental problem paradigms in statistics is that of inferring some information about an unknown probability distribution D, given access to independent samples drawn from it. More than a decade ago, Batu et al. initiated the study of problems of this type from within the framework of /property testing/ - a setting where, given an unknown "massive object" an algorithm can access only by making a small (sublinear) number of "local inspections" (queries) of the object, the goal is to determine whether the object has a particular property. The algorithm must accept if the object has the desired property, and reject if the object is far from every object with the property. In distribution property testing, the "massive object" is an unknown probability distribution D over an N-element set, and the tester accesses the distribution by drawing independent samples from it.

    We study a new framework for such a task, by considering distribution testing algorithms that have access to a /conditional sampling oracle/. This is an oracle that takes as input a subset S of the domain [N]={1,...,N} of the unknown probability distribution D and returns a draw from the conditional probability distribution D restricted to S. This new model allows considerable flexibility in the design of distribution testing algorithms; in particular, testing algorithms in this model can be adaptive.

    We study a wide range of natural distribution testing problems in this new framework and some of its variants, giving both upper and lower bounds on query complexity. These problems include testing whether D is the uniform distribution U; testing whether D = D* for an explicitly provided D*; testing whether two unknown distributions D1 and D2 are equivalent; and estimating the variation distance between D and the uniform distribution.

    At a high level our main finding is that the new "conditional sampling" framework we consider is a powerful one: while all the problems mentioned above have Omega(N^(1/2)) sample complexity in the standard model (and in some cases the complexity must be almost linear in N), we give poly(log N, 1/eps)-query algorithms (and in some cases poly(1/eps)-query algorithms independent of N) for all these problems in our conditional sampling setting.

    Joint work with Dana Ron (Tel-Aviv University) and Rocco Servedio (Columbia University).


    Friday, March 8:
    Multi-Stage Propagation and Quasipolynomial-Time Isomorphism Testing of Steiner 2-Systems
    Xiaorui Sun
    Columbia University

    Abstract: A standard process for testing graph isomorphism is to first assign distinct labels to a small subset of vertices of an input graph, and then propagate and create new labels across the graph, aiming to assign distinct and isomorphism-invariant labels to all vertices in the graph. This process is usually referred to as the individualization based refinement for canonical labeling of graphs.

    In this talk, we consider a framework for analyzing an extended version of this individualization-based refinement process over Steiner 2-systems. A Steiner 2-system consists of a set of points and a set of lines such that each line passes the same number of points and each pair of points uniquely determines a line. Applying this framework, we show that isomorphism of Steiner 2-systems with n lines can be tested in quasipolynomial time exp(polylog(n)).

    An essentially identical result from a rather different perspective was obtained simultaneously by Laszlo Babai and John Wilmes. These results improved the previous best exp(O(n^{1/4}))-time algorithm of Spielman, and the quasipolynomial-time algorithm of Babai and Luks for the special case when the line size is polylogarithmic.

    Join work with Xi Chen and Shang-Hua Teng.


    Friday, March 15:
    Superlinear lower bounds for multipass graph processing
    Krzysztof Onak
    IBM Research

    Abstract: I will present [;n^{1+\Omega(1/p)};] lower bounds for the space complexity of p-pass streaming algorithms for the following problems on n-vertex graphs:

    Before our result, it was known that these problems require [;\Omega(n^2);] space in one pass, but no [;n^{1+\Omega(1)};] lower bound was known for any p >= 2.

    Our result follows from a communication complexity lower bound for a communication game in which the players hold two graphs on the same set of vertices. The task of the players is to find out whether the sets of vertices reachable from a specific vertex in exactly p+1 steps intersect. The game requires a significant amount of communication only if the players are forced to speak in a specific difficult order.


    Friday, April 12:
    Maximum Entropy Summary Trees
    Howard Karloff
    AT&T Labs--Research

    Abstract: Given a very large, node-weighted, rooted tree on, say, n nodes, if one has only enough space to display a k-node summary of the tree, what is the most informative way to draw the tree? We define a type of weighted tree that we call a "summary tree" of the original tree, that results from aggregating nodes of the original tree subject to certain constraints. We suggest that the best choice of which summary tree to use (among those with a fixed number of nodes) is the one that maximizes the information-theoretic entropy of a natural probability distribution associated with the summary tree, and we provide a (pseudopolynomial-time) dynamic-programming algorithm to compute this maximum entropy summary tree, when the weights are integral. The result is an automated way to summarize large trees and retain as much information about them as possible, while using (and displaying) only a fraction of the original node set. We also provide an additive approximation algorithm and a greedy heuristic that are faster than the optimal algorithm, and generalize to trees with real-valued weights.

    This is joint work with Ken Shirley.


    Friday, April 19:
    Can Selfish Groups be Self-Enforcing?
    Augustin Chaintreau
    Columbia University

    Abstract: Algorithmic graph theory has thoroughly analyzed how, given a network describing constraints between various nodes, groups can be formed among these so that the resulting configuration optimizes a global metric. In contrast, for various social and economic networks, groups are formed de facto by the choices of selfish players. A fundamental problem in this setting is the existence and convergence to a self-enforcing configuration: assignment of players into groups such that no player has an incentive to move into another group than hers. Motivated by information sharing on social networks – and the difficult tradeoff between its benefits and the associated privacy risk – we study the possible emergence of such stable configurations in a general selfish group formation game.

    Our paper considers this general game for the first time, and it completes its analysis. We show that convergence critically depends on the level of collusions among the players – which allow multiple players to move simultaneously as long as all of them benefit. Solving a previously open problem we exactly show when, depending on collusions, convergence occurs within polynomial time, non-polynomial time, and when it never occurs. We also prove that previously known bounds on convergence time are all loose: by a novel combinatorial analysis of the evolution of this game we are able to provide the first asymptotically exact formula on its convergence. Moreover, we extend these results by providing a complete analysis when groups may overlap, and for general utility functions representing multi-modal interactions. Finally, we prove that collusions have a significant and positive effect on the efficiency of the equilibrium that is attained.

    Joint work with Guillaume Ducoffe and Dorian Mazauric.


    Friday, April 26:
    Diffuse Decompositions of Polynomials
    Daniel Kane
    Stanford University

    Abstract: We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i. We then present some applications of this result.


    Friday, May 3:
    On Relative Approximations in Geometric Hypergraphs
    Esther Ezra
    New York University

    Abstract: Motivated by range counting problems, relative approximations have become a central tool in Computational Geometry. In fact, they were introduced by Har-Peled and Sharir, who derived them from the seminal work of Li, Long, and Srinivasan in the theory of Machine Learning. In this talk I will discuss properties of relative approximations, and present improved upper bounds for their size under certain favorable assumptions. Our approach is probabilistic, where we apply the constructive version of the Asymmetric Lovasz Local Lemma.


    Friday, May 17:
    Time/Space of Digital Communication: Using Time to Coordinate Actions in Distributed Systems
    Yoram Moses
    Technion

    Abstract: Communicated messages play a central role in coordinating events in distributed computer systems. In purely asynchronous systems, Lamport's "happened-before" relation, based on message chains, has been shown to govern causality and coordination. This talk will consider the role that guarantees regarding message transmission times play. The existence of clocks and such timing information significantly affect the ways in which sites can coordinate their actions. This talk will characterize communication structures that are necessary and sufficient for coordinating basic patterns of actions in the presence of clocks. In addition, it will draw parallels with time/space diagrams and discuss causality. This talk is based on joint work with Ido Ben Zvi.


    Tuesday, June 18:
    Efficient reasoning in PAC Semantics
    Brendan Juba
    Harvard University

    Abstract: Machine learning is often employed as one step in a larger application, serving to perform information extraction or data mining for example. The rules obtained by such learning are then used as inputs to a further analysis. As a consequence of the inputs to this analysis having been learned, however, its conclusions can only be theoretically guaranteed to satisfy the same standard as the inputs---that of "PAC Semantics" (Valiant, 2000). In this talk, we explore the benefits of taking the problems of learning and logical reasoning together, capturing a relatively general family of such applications. Briefly, the benefitis that we can simultaneously (1) handle incomplete information, (2) utilize rules that are a reasonable but imperfect fit for the data, and (3) reach conclusions more efficiently than is achieved by known algorithms for reasoning from rules alone. Precisely, we consider a problem of testing the validity of a "query" formula (hypothesis) against incomplete data. We present algorithms for soundly verifying such queries that succeed whenever there exist learnable rules that suffice to complete a simple proof of the query, matching what can be achieved by such a two-stage learning and reasoning process.


    Friday, July 19:
    Mechanism Design for Scheduling with Uncertain Execution Time
    Angelina Vidali
    Duke University

    Abstract: We study the problem where a task must be executed, there are multiple machines/agents that can potentially perform the task, and our objective is to minimize the expected sum of the agents' processing times. Each agent does not know exactly how long it will take him to finish the task; he only knows the distribution from which this time is drawn. These times are independent across agents and the distributions fulfill the monotone hazard rate condition. Agents are selfish and will lie about their distributions if this increases their expected utility.

    We construct a mechanism that takes as input the agents' reported distributions and that outputs a schedule that minimizes the expected sum of processing times, as well as payments that make it an ex-post equilibrium for the agents to both truthfully report their distributions and exert full effort to complete the task.

    Joint work with: Vincent Conitzer


    Contact xichen-at-cs-dot-columbia-dot-edu if you want to volunteer to give a talk (especially for students!). The talk can be about your or others' work. It can be anything from a polished presentation of a completed result, to an informal black-board presentation of an interesting topic where you are stuck on some open problem. It should be accessible to a general theory audience. I will be happy to help students choose papers to talk about.
    There is a mailing list for the reading group. General information about the mailing list (including how to subscribe yourself to it) is available here. If you want to unsubscribe or change your options, send email to theoryread-request@lists.cs.columbia.edu with the word `help' in the subject or body (don't include the quotes), and you will get back a message with instructions.

    Comments on this page are welcome; please send them to xichen-at-cs.columbia.edu

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