Main research interests: machine learning, computational aspects in economics and game theory, algorithms
Main research interests include parallel algorithms and languages
My research interests are in Theoretical Computer Science, with an emphasis on
Algorithms for Uncertain Environments (particularly approximation algorithms for stochastic problems,
and online algorithms) and Metric Embeddings.
Interests: Several topics in Theoretical Computer Science, including the theory of error-correcting codes, approximation algorithms & non-approximability, pseudorandomness, probabilistically checkable proofs, algebraic algorithms.
In my research I love to think about the design and analysis of (randomized) algorithms for combinatorial problems. Often I mix this broad and classical area with ideas from distributed computing, information theory, and (network) coding theory. I am always on the lookout for new topics and problems to play with.
A common theme in my research is the use of randomness. It is surprising and sometimes almost unbelievable how much simpler and more efficient algorithms can become if allowed to make random decisions.
I work on the design and performance analysis of computer systems. Much of this work is theoretical,
involving proving new theorems in stochastic processes and queueing theory and also finding ways to
solve really hard Markov chains. I look for students who are very strong in math and probabilistic thinking.
I am particularly interested in resource allocation/scheduling policies for distributed computer systems.
This includes inventing and analyzing new load sharing policies, routing policies, cycle stealing policies,
power management policies and multi-class/multi-server prioritization. I am also interested in heavy-tailed
workloads and their effect on scheduling.
Some techniques that my group and I have invented include:
(i) Recursive Dimensionality Reduction (RDR), a technique that allows one to reduce a Markov chain that grows
unboundedly in many dimensions to a Markov chain that grows unboundedly in only one dimension, by using
the idea of busy period transitions;
(ii) Recursive Renewal Reward (RRR), a technique that allows one to obtain closed-form solutions for many
one-dimensional infinite repeating Markov chains that come up in computer systems;
(iii) The All-Can-Win Theorem, a demonstration that scheduling policies which are biased towards favoring small
jobs can also be preferable to large jobs.
My main interest is in algorithm design and parallel computation. For
the last several years I have worked on fine-grain parallel algorithm
design. The main goal is to find parallel algorithms for problems which,
naively, seem to require a sequential one. The hope is two fold. First,
we find algorithms which are much faster than sequential algorithms
without using too many processors. Many parallel algorithms are known
which have much faster parallel running times but they use many more
processors. Thus, the goal is to decrease the time T without
substantially increasing the processor-time product PBT. The processor-
time product is also known as the work. Second, these new algorithms
should uncover fundamental paradigms which will be applicable to a wide
class of problems.
We enumerate some of the ongoing projects:
Sparse Matrix Calculations.
Possibly one of the most fundamental computational intensive operations
performed on a computer is finding solution to linear systems Ax = b. One
can either solve the system by using an iterative method, obtaining an
approximate solution, or directly obtaining an exact solution. In this
discussion we restrict our attention to direct methods, but a similar
approach holds for certain iterative methods as well. If the matrix A is
dense, i.e., most of the entries are nonzero, then sequential and some
parallel methods use n³ work. But if the matrix A is sparse in a nice way
then much better bounds are known on the amount of work needed. The
basic idea is to view A as the incidence matrix of a graph G. In the
case when G is a planar graph we can use the fact that every planar graph
has a separator of size O(Vn). Thus one obtains sequential and parallel
methods for solving these linear systems in approximately n³´² work. A
separator for a graph is a subset of vertices whose removal separates the
remaining graph into two roughly equal size disconnected graphs. We have
produced and are still investigating algorithms for finding efficient
separators for planar graphs.
Graph Separators in 3-Dimensions.
As discussed above, graph separators have many applications. Our main
goal is to determine which graphs have small separators and construct
efficient algorithms for finding these separators. As a corollary to the
fact that planar graphs have O(Vn) separators one gets O(Vn) separators
for such planar graphs as the Voronoi diagram and the Delaunay graph in
2-dimensions. We are presently trying to determine whether or not Voronoi
diagrams in 3-dimensions have small separators, O(n²´³) size. We have
been able to show that many important and natural graphs in 3-dimensions
have O(n²´³) separators. In particular, we have shown that a k-nearest
neighborhood graph has O(n²´³). Other graphs with small separators in
3-dimensions include several natural graphs associated with the
finite-element method in numerical analysis.
Parallel Algorithm Design.
In general we have produced many different and varied parallel
algorithms. One important paradigm which we have found and are still
using today is known as parallel tree contraction. The basic idea is to
find a tree in the underlying structure of the problem and simply
contract it in parallel. In a graph the tree might be a spanning tree and
in a parsing problem it might be a parse tree. An extremely large number
of problems can be efficiently parallelized using parallel tree
My main research areas are in complexity theory, cryptography,
combinatorics, and probability. I am also interested in inference,
learning theory, and randomized algorithms. I love counter-intuitive
Here is a list of questions I find interesting (and which I am unable to
My main research interests are in market design, game theory, optimization (integer programming, search, stochastic optimization, and convex optimization), mechanism design, electronic commerce, artificial intelligence, multiagent systems, auctions and exchanges, automated negotiation and contracting, equilibrium finding, algorithms for solving games, markets for advertising, kidney exchange, voting, coalition formation, preference elicitation, normative models of bounded rationality, resource-bounded reasoning, computational economics, privacy, machine learning, multiagent learning, and networks.
Many of our systems have been commercially fielded.
The following are just some examples of my current projects:
Solving Games, Poker.
We design algorithms for abstracting games and algorithms for solving for the equilibrium of a game, i.e., the best way to play it. For example, we have developed a lossless abstraction algorithm that enabled us to exactly solve poker games over 10,000 times larger than the largest solved previously. Our algorithms have also yielded some of the best poker playing programs for Texas Hold~em. We are also developing opponent modeling and exploitation techniques. Our work also involves complexity analysis and developing new game-theoretic solution concepts.
Expressive Auctions and Exchanges.
We are developing the theory and methodology of how one should design markets when the participants have rich preferences over outcomes. One of my startups has fielded 800 of the most combinatorial markets ever developed, with a trading volume of over $60 billion.
Advertising Markets and Electricity Markets.
We are developing techniques, and founded a new startup company, in the space of expressive advertising markets, such as for TV and online advertising. We may also apply the approaches to electricity markets.
We have developed the market designs and algorithms that run the nationwide kidney exchange at UNOS. Our current research focuses on developing better models and algorithms for the dynamic problem.
Search and Integer Programming.
For example, how should those tree search algorithms branch? Also, we are developing techniques for automated and recursive decompositions.
Revenue Maximization and Automated Mechanism Design.
We are pioneering the idea of using optimization algorithms to automatically design the rules of a game (such as an auction) so that the designer's objective is maximized even though the participants play based on self-interest. An example application is revenue-maximizing combinatorial auctions. Another example is channel abstraction (bundling, automated item definition), such as in TV and Internet display advertising.
Mechanisms for Computationally Limited Agents.
If agents are computationally limited, how should they allocate their deliberation? What should good mechanisms (e.g., auctions) look like for such agents? It turns out that sometimes mechanisms that lead to greater systemwide good can be designed for computationally limited agents than unlimited ones!
You can check out my papers via my home page.
I have worked in a variety of different areas of computer science,
including amortized analysis of algorithms, self-adjusting data
structures, competitive algorithms, natural language parsing, computer
game playing, synthesis of musical sounds, and persistent data
I (jointly with coauthor Davy Temperley) wrote a parser
for English. The system (which we call a link grammar) is unlike phrase
structure parsing or context free parsing. The scheme is elegant and
simple, and our grammar captures a very wide variety of complex phenomena
in English. We (John Lafferty and I) plan to use this as a basis for a
new statistical model of language. This work on language is described in
two technical reports: CMU-CS-91-196, CMU-CS-92-181.
Consider the idealized problem of deciding
whether to rent or buy skis. You're about to go skiing. The cost of
renting skis is $20, the cost of buying them is $400. Clearly if you
knew that you were going to go skiing more than twenty times, then you
could save money by immediately buying skis. If you knew that you would
go skiing fewer than twenty times, the it would be prudent to always rent
skis. However, suppose that you can not predict the future at all, that
is, you never know until after one ski trip ends if you will ever go
skiing again. What strategy would you use for deciding whether to rent
or buy skis? Your goal is to minimize the ratio of the cost that you
incur to the cost you would incur if you could predict the future.
(Hint: you can come within a factor of two.)
Since the simple principle behind this example turns out to be very
useful we have given it a name. A competitive algorithm is an on-line
algorithm (it must process a sequence of requests, and it must process
each request in the sequence immediately, without knowing what the future
requests will be), whose performance is within a small constant factor of
the performance of the optimal off-line algorithm for any sequence of
My collaborators and I have discovered a surprising variety of practical
problems for which there exist very efficient competitive algorithms. We
have also developed a partial theory of competitive algorithms. However
there remain many interesting open problems, from discovering competitive
algorithms for specific problems, to answering general questions about
when such algorithms exist.
Data structure problems are typically formulated in
terms of what types of operations on the data are required, and how fast
these operations should take place. A worst-case analysis of the
performance of a data structure is a bound on the performance of any
operation. An amortized analysis of a data structure bounds the
performance of the structure on a sequence of operations, rather than a
single operation. It turns out that by only requiring amortized
efficiency (rather than worst-case), a variety of new and elegant
solutions to old data structure design problems become possible. My
collaborators and I have devised a number of such solutions (splay trees,
skew heaps, fibonacci heaps, self-adjusting lists, persistent data
structures, etc.), and I continue to have a strong interest in data
structures and amortized analysis.
Interactive Network Games:
I wrote and maintain the internet chess
server. This very popular service allows people from all over the world
to play chess, observe others playing, and communicate with each other in
various ways. A rating is computed for each player as a function of his
or her performance. You can try it yourself with ``telnet ics.uoknor.edu
My research is in extremal
and probabilistic combinatorics; in particluar, recent work
has been on additive number theory, the Shannon capacities
of graphs, the combinatorics of cellular automata, random
graphs and randomized algorithms. Shannon capacity is a
prime example of the close link between extremal
combinatorics and information theory. This graph
invariant arises as a measure of the zero-error perfomance
of an associated noisy communication channel but leads to
very natural (and notoriously difficult) questions in
extremal combinatorics. The aim of the work on cellular
automata is not the study of models that have particular
applications to other scientific fields (there are many
such applicable models, but very few of them have
succumbed to rigorous mathematical treatment) but rather
to study those models that appear most ameniable to
mathematical analysis, with the goal of developing methods
that can be applied to other, more intricate models.
There is no ugly mathematics, only
mathematics that I do not understand. Some mathematics not only I do
not understand, but I do not understand why I do not understand. For
some I do not even understand why nobody understands. That latter kind
is the mathematics that I like to work on. Thus far, this led me to
working on a variety of problems that are a mix of combinatorics,
geometry, and number theory. The concrete problems change often, so to
learn what I do now, knock on my office door!
Research interests lie in combinatorics, discrete optimization and
theoretical computer science. Current focus is on probabilistic
aspects, in particular, the study of the asymptotic properties of
random graphs and the average case analysis of algorithms. Recent work
has been on a randomized algorithm for approximating the volume of
convex bodies, the Hamilton cycle problem in various models of a random
graph and the use of martingale inequalities in the study of random
graphs. I have also been working on exact algorithms for NP-hard
problems with a good average case performance. I also have a strong
interest in complexity theory.
I am interested in problems that lie at the vibrant intersection of Combinatorics, Probability, and Theoretical Computer Science. These include questions in Ramsey Theory (and more generally Extremal Combinatorics), where the Probabilistic Method has found many surprising applications in otherwise deterministic settings. My work also considers discrete systems that are inherently probabilistic, from static structures such as random graphs to dynamic processes such as randomized algorithms and combinatorial games. As these fields are closely related, there is much opportunity for developments from one side to find applications in another.
I am broadly interested in Discrete Math, Probability, and Theory of Computing: Markov chains, Isoperimetry and Functional Analysis, Combinatorics, Computational Number Theory, and Algorithms.
I am interested in Discrete Probability, Combinatorics, Convex Geometry, and Applications to Data Analysis.
My primary area of research is in Discrete Mathematics, primarily Graph Theory and Combinatorics. I also have an interest in equity and inclusion in mathematics, in particular, the impacts that race and mathemactics culture and learning have on one another.
Current research is in combinatorial optimization and graph theory,
with emphasis on the algorithmic aspects. In graph theory, a problem on
the factorization of graphs stated by Lovasz is shown to be solvable in
polynomial time. Current focus is on the study of perfect graphs,
balanced 0,1 matrices and the max-flow min-cut property. Another line
of research deals with NP-hard combinatorial optimization problems
through the study of relaxations. Using polyhedral combinatorics,
contributions are made to the theory of the set covering, location,
vehicle routing and traveling salesman problems.
Primary research interests involve the application of operations
research techniques to logical inference and related areas. These
include inference problems in propositional logic, probabilistic logic,
Bayesian networks and predicate logic, as well as problems in logic
circuit verification and machine learning. Hooker's approach is to
exploit the mathematical structure of logic problems, such as the
polyhedral structure of propositional logic. He also has interests in
transportation problems and optimal location on networks, as well as in
the philosophical presuppositions of artificial intelligence.
My research involves the development of new theory and algorithms for solving problems in large-scale convex
optimization as well as structured nonconvex optimization. It is mainly motivated by applications in decision
making under uncertainty, machine learning and high dimensional statistical inference domains.
My research interests are broadly in theoretical computer science and
operations research. I work on the design, analysis and evaluation of
algorithms. I am currently working on scheduling theory, distributed
computing, and the foundations of machine learning.
My research interests are concerned with the mathematical and computational
optimization algorithms, especially interior-point methods for convex
Currently, I am investigating properties of condition numbers for linear
programming. Like traditional condition numbers for linear equations, these
numbers aim to capture how changes in the data of the linear program affect
properties of the solutions. The study of condition numbers lies in the
intersection of optimization and numerical analysis; it combines ideas from
interior-point methods, convex analysis, perturbation theory, and complexity
Research interests include approximation algorithms, combinatorial optimization, and computational biology.
Michael Trick's Operations Research Page.
Research is in combinatorial and network optimization. Recent work has
been on computational aspects of integer and network optimization along
with data structures for efficient implementation. Current research has
concentrated on hard integer programs with underlying network
structures. Also interested in the application of algorithm design and
complexity theory in the social and biological sciences.