We typically have seminars on Wednesday at noon in Malone 228. All seminar announcements will be sent to the theory mailing list.

Speaker: Amitabh Basu

Affiliation: JHU

Title: Cutting Planes and Geometry of Numbers

Abstract: We survey some recent results in cutting plane theory for integer programming. Cutting Planes give a way to reduce the search space for the optimal solution in an integer optimization problem. The results we will present are very recent connections between cutting planes and covering/tiling properties of subsets of euclidean sets. Important structural information about cutting planes can be translated to geometric questions like: Does a particular compact subset B of R^n cover all of R^n when we consider all of its translates by integer vectors. This connects to very classical problems in the geometry of numbers and deep theorems like the Venkov-Alexandrov-McMullen theorem on tilings, and the geometry of zonotopes can be leveraged. Research in this area of integer optimization is very much work-in-progress; we will close the presentation with an invitation to join our quest with some open problems.

Speaker: Grigory Yaroslavtsev

Affiliation: University of Pennsylvania

Title: Parallel Algorithms for Geometric Graph Problems

Abstract:

I will describe algorithms for geometric graph problems in the modern parallel models inspired by MapReduce. The talk will be self-contained, including a formal introduction of the modern theoretical computational models used to capture computations in massively parallel “MapReduce”-like systems. It will also include a sample of major open problems in the area.

For example, for the Minimum Spanning Tree (MST) problem over a set of points in the two-dimensional space, our algorithm computes an approximate MST. Our algorithms work in a constant number of rounds of communication, while using total space and communication proportional to the size of the data (linear space and near linear time algorithms).

I will also show how the main ideas from the MST algorithm can be captured within a general “Solve-and-Sketch” algorithmic framework that we develop. Besides MST it also applies to the approximate Earth-Mover Distance (EMD) and the transportation cost problem. Algorithms designed in the “Solve-and-Sketch” framework have implications which go beyond parallel models. In particular, our work implies new near-linear time algorithms for EMD cost and transportation cost in the plane. Other implications include algorithms in the streaming with sorting model.

Joint work with Alexandr Andoni, Krzysztof Onak and Aleksandar Nikolov.

Speaker: Michael Dinitz

Affiliation: Johns Hopkins University

Title: Approximating Graph Spanners

Abstract:

Graph spanners (subgraphs which approximately preserve distances) have been studied extensively since the 1980’s. Many of the known results are about the optimal tradeoffs between various parameters, particularly the stretch and size of the spanner. But there has been some recent progress on a different and less developed line of research: fixing the allowable stretch, and optimizing the size. This turns spanners into more of a computational problem, and allows us to use many of the standard techniques from approximation algorithms (convex relaxations in particular). In this talk we will give an overview of some of the progress in this area, its limitations, and some possible future directions.

Speaker: David Harris

Affiliation: University of Maryland – College Park

Title: Lopsidependency in the Moser-Tardos framework: Beyond the Lopsided Lov\'{a}sz Local Lemma

Abstract: The Lopsided Lovasz Local Lemma (LLLL) is a powerful probabilistic principle which has been used in a variety of combinatorial constructions. While this principle began as a general statement about probability spaces, it has recently been transformed into a variety of polynomial-time algorithms. The resampling algorithm of Moser & Tardos is the most well-known example of this. A variety of criteria have been shown for the LLLL; the strongest possible criterion was shown by Shearer, and other criteria which are easier to use computationally have been shown by Bissacot et al, Pegden, and Kolipaka & Szegedy.

We show a new criterion for the Moser-Tardos algorithm to converge. This criterion is stronger than the LLLL criterion, and in fact can yield better results even than the full Shearer criterion. This is possible because it does not apply in the same generality as the original LLLL; yet, it is strong enough to cover many applications of the LLLL in combinatorics. We show a variety of new bounds and algorithms. A noteworthy application is for $k$-SAT, with bounded occurences of variables. As shown in Gebauer, Szabo, and Tardos, a $k$-SAT instance in which every variable appears $L \leq \frac{2^{k+1}}{e (k+1)}$ times, is satisfiable. Although this bound is asymptotically tight (in $k$), we improve it to $L \leq \frac{2^{k+1} (1 – 1/k)^k}{k-1} – \frac{2}{k}$ which can be significantly stronger when $k$ is small.

Speaker: Matthew Andrews

Affiliation: Alcatel-Lucent Bell Labs

Title: Understanding Sponsored Content in Mobile Data Networks

Abstract:

Sponsored content is a mechanism in which content providers can pay the operator of a wireless network to make their content free to end users. Such offerings have recently been introduced in both the US and Asia and they raise many challenging questions regarding which sites should be candidates for sponsoring and how much the service provider should charge the content provider.

In this talk we introduce a number of models that aim to capture the interactions between the service provider, the content provider and the end users in a sponsored content offering. We show that it is possible to design the system so that it is win-win-win for all players. In many settings the problem is a generalization of the “Adwords” problem that arises in the design of sponsored search. We also show how to analyze network traffic and content provider financial data in order to calculate the input parameters for these models.

Speaker: Jeremy Fineman

Affiliation: Georgetown University

Title:

How to Fix Exponential Backoff: Achieving Constant Throughput and Robustness with Polylog Attempts

Abstract:

Randomized exponential backoff is employed in many domains to coordinate access to a shared resource or communication channel. Despite the ubiquity of the protocol, exponential backoff has poor general performance guarantees. Most notably, exponential backoff neither achieves constant throughput in a general online setting, nor is it robust to corrupted or jammed messages. This talk considers a new backoff protocol that achieves constant throughput, even in the presence of an adaptive adversary that jams (or blocks access to) the shared resource at certain times. The protocol also makes relatively few attempts to access the resources, which means that each agent does not expend too much energy. Specifically, we show that the expected energy per agent is O(log^2(n+J)), where n is the number of contenders and J is the amount of time the adversary jams.

Speaker: Alex Slivkins

Affiliation: Microsoft Research – New York

Title: Bandits with Resource Constraints

Abstract:

Multi-armed bandits is the predominant theoretical model for exploration-exploitation tradeoff in machine learning, with countless applications ranging from medical trials, to communication networks, to Web search and advertising, to dynamic pricing. In many of these application domains the learner may be constrained by one or more supply/budget limits, in addition to the customary limitation on the time horizon. We introduce a general model that encompasses such problems, combining aspects of stochastic integer programming with online learning. A distinctive feature (and challenge) in our model, compared to the conventional bandit problems, is that the optimal policy for a given problem instance may significantly outperform the policy that always chooses the best fixed action. Our main result is an algorithm with near-optimal regret relative to the optimal policy. Also, we extend this result to contextual bandits, and detail an application to dynamic pricing.

Mohammad Hajiaghayi

University of Maryland – College Park

Title: Parameterized and Promised Streaming: Matching and Vertex Cover

Abstract:

As graphs continue to grow in size, we seek ways to effectively

process such data at scale. The model of streaming graph processing, in

which a compact summary is maintained as each edge insertion/deletion

is observed, is an attractive one. However, few results are known for

optimization (often NP-hard) problems over such dynamic graph streams.

In this talk, we introduce a new approach to handling graph streams,

by instead seeking solutions for the parameterized (and promised) versions of

these problems. Here, we are given a parameter k and the objective is to

decide whether there is a solution bounded by k. By combining

kernelization techniques with randomized sketch structures, we obtain the

first streaming algorithms for the parameterized versions of Maximal

Matching and Vertex Cover. We consider various models for a graph stream on n

nodes: the insertion-only model where the edges can only be added, and

the dynamic model where edges can be both inserted and deleted.

Gordon Wilfong

Alcatel-Lucent Bell Labs

Title: Optimal Path Encoding

Abstract: Packet networks need to maintain state in the form

of forwarding tables at each switch. The cost of this state

increases as networks support ever more sophisticated per-flow

routing, traffic engineering, and service chaining. Per-flow or per-path

state at the switches can be eliminated by encoding each

packet’s desired path in its header. A key component of such a

method is an efficient encoding of paths.

We introduce a mathematical formulation of this optimal path encoding

problem. We prove that the problem is APX-hard, by

showing that approximating it to within a factor less than 8/7

is NP-hard. We then present an algorithm

approximating the optimal path-encoding problem to within a

factor 2. Finally, we provide empirical results illustrating the

effectiveness of the proposed algorithm.

Joint work with A. Hari (Bell Labs) and U. Niesen (Qualcomm)