ACO Faculty

Computer Science Department

Tepper School of Business

Department of Mathematical Sciences

Computer Science Department

  • Guy Blelloch

    Main research interests include parallel algorithms and languages

  • Avrim Blum

    My main research interests are in machine learning theory, on-line algorithms, and approximation algorithms.

    Machine Learning Theory.
    Machine learning theory aims to provide a mathematical analysis of, and algorithms for, the problems and issues that arise in getting machines to learn from experience. One issue I have been investigating is the following that comes up in concept learning. Suppose in the learning scenario there are many potential features or attributes that examples might have, even though each individual example has only a few of them. Can one produce algorithms whose performance depends only on the size of the examples and not on the size of the entire feature space? Also, what if only a small number of features are actually relevant to the function being learned? Under what conditions can one quickly hone in on the relevant features? I have been working on theoretical models that bring out these issues, and on algorithms that solve these problems for some simple concept representations. A second direction I have been working on is learning concepts that change over time, and finding algorithms that can quickly adapt to such changes.

    On-line Algorithms.
    In the area of on-line or competitive algorithms I am particularly interested in theoretical problems of scheduling and path planning in unknown environments. These problems often involve examining the following issue: how do you trade off the cost of exploring versus the costs associated with not exploring and then having, perhaps, to back up from a wrong decision? In work with Prabhakar Raghavan and Baruch Schieber at IBM, we were able to provide good algorithms for the problem of traveling from a start to a destination in an unmapped plane filled with oriented rectangular obstacles (the robot does not know the positions of the obstacles and only finds out when the obstacles are encountered). There are many important open problems in this area, such as how to make good use of partial prior knowledge, and how to use randomization to beat known deterministic approaches.

    Approximation Algorithms.
    Many important problems turn out to be NP-hard, implying it is unlikely there will be algorithms for them that are both efficient and optimal in the worst case. However, for a large number of such problems there are fast algorithms that find solutions close to optimal, or that can be shown will find optimal solutions for large classes of instances (e.g., random instances). Two problems in particular that I have been interested in are the classical graph coloring problem and an NP-hard string-matching problem motivated by issues in DNA sequencing. The latter is the following problem: given a collection of strings of characters, find the shortest superstring that contains each string as a consecutive block. We (myself and several collaborators) were able to show that a simple and commonly used greedy procedure achieves provably good worst-case bounds, thus providing some theoretical justification for past experimental evidence. One open problem I am currently thinking about is whether there might be a different algorithm for which one could show even better bounds.

  • Mor Harchol-Balter

    I am interested in the performance analysis and design of computer systems, particularly distributed systems. I work on finding analytical models which capture the important characteristics of a computer system and allow me to redesign the system to improve its performance.

    I believe that many fundamental conventional wisdoms on which we base system designs are not well understood and sometimes false, leading to inferior designs. One problem is that many of our existing beliefs stem from analysis which is based on Markovian workloads (exponentially-distributed job sizes). However todays measured workloads show much greater variability in job sizes than previously assumed and furthermore show heavy tails. In light of current empirical measurements, my research challenges existing age-old heuristics. Here are just a few examples:

    Problem areas I currently work in include: Web server design and implementation, distributed Web servers, distributed supercomputing servers , routing and scheduling in networks .

  • John Lafferty

    My research interests include: speech and natural language processing, statistical learning algorithms, applied probability and information theory, algorithms for finite groups.

  • Bruce Maggs

    My research area is parallel algorithms and architectures. For the past few years my research has focused on how to build the communications system that underlies a parallel computer. My goal has been to develop mechanisms for moving data between processors (or between processors and memory) that are both practical and provably effective. To this end, I have designed, analyzed, and implemented algorithms for routing, sorting, and tolerating faults in parallel computers. Most of this work has been theoretical.
    Parallel algorithms and architectures is an area in which there is a lot of synergy between theory and practice. While the practice of parallel computing has suggested many problems for theoreticians to tackle, the theory has had an impact on the practice as well. For example, the theory of area and volume-universal routing networks motivated the use of a fat-tree interconnection network to route data between processors in the Connection Machine Model CM-5 supercomputer. As another example, extensive experiments indicate that the fastest way to sort on a Connection Machine Model CM-2 is to use a ``Samplesort'' algorithm, which is based on a pair of theoretical sorting algorithms due to Huang-Chow and Reif-Valiant.
    Recently, a new paradigm has been suggested for parallel supercomputing. Instead of constructing a special-purpose interconnection network for each parallel computer, the idea is to use high-speed networking technology to connect a collection of processors or workstations. Examples of this technology include ATM switches and HIPPI channels. Although this technology was originally developed for routing telephone calls, in principle there is no reason why it cannot be used to route messages between processors or workstations in a parallel or distributed computer. One of the advantages of using this technology for parallel computing, rather than building special-purpose communications networks, is that independent of the market for parallel computers, the technology will be developed by the telecommunications industry.
    Many of the tasks that arise in the context of special-purpose networks also arise in the high-speed networking setting. There are important differences, however. In the special-purpose networks domain, the designer of a routing algorithm has much tighter control over the network and can make strong performance guarantees. For example, the algorithm may guarantee that the maximum number of packets queued at any node will never be large or that the maximum delay experienced by any packet will never be large. In the high-speed networking domain, the network may come with only weak performance guarantees. For example, the topology of the network may be irregular, or the network may not guarantee to deliver its packets. The designer of a routing protocol for one of these networks is more concerned with issues such as deadlock, or what to do if a message never reaches its destination. On the other hand, some of the techniques that have already developed for special-purpose networks may be even more applicable in the high-speed networking domain. For example, although we know several techniques for tolerating faults in a network such as a butterfly, in practice these faults rarely occur. In a high-speed networking environment, however, links may be much less reliable. Similarly, congestion (or ``hot spots'') may be much more pronounced in a network with irregular topology.
    My latest plan is to study the theory behind using high-speed networking technology to support parallel supercomputing. I'm hoping to find the same synergy between theory and practice in this realm that I found when studying special-purpose networks.

  • Gary Miller

    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 contraction.

  • Andrew W. Moore

  • Steven Rudich

    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 puzzles.

    Here is a list of questions I find interesting (and which I am unable to answer):

    1. If a problem has no polynomial-time algorithm, does this mean that no QUANTUM computer can solve it in polynomial-time?
    2. Is there a graph of size O(n) which contains every n-node forest as a vertex-induced subgraph?
    3. [Due to Avrim Blum] Suppose I am using one of two methods to pick poly(n) n-bit strings: a) Select them uniformly at random; b) Fix some boolean function f on log(log(n)) bits, fix as ``relevant'' log(log(n)) bit positions randomly chosen from 1 to n-1, pick each n-bit string by randomly choosing the first n-1 bits and appending f of the relevant bits. Is there an efficient method to look over the list I produce and guess which method I am using?
    4. Can the security of secret-key agreement be provably reduced to the existence of a one-way function?
    5. Is there an efficient secret sharing scheme for all monotone access structures?
    6. P=NP? P=BPP? P=PSPACE? Or more modestly, is there a function in NP which can't be computed by linear-sized, log-depth circuits?
    7. Is there a reasonable combinatorial property which would make any function which possessed it hard to compute?

  • Daniel Sleator

    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 structures.

    Natural Language:
    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.

    Competitive Algorithms:
    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 requests.
    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 Structures:
    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 5000''.

  • Tepper School of Business

  • Egon Balas

    Research has theoretical and algorithmic aspects. The former revolves mainly around polyhedral combinatorics, that is, attempts at describing various combinatorial structures by systems of linear inequalities. Recent examples are the polyhedral characterization of the convex hull of incidence vectors of perfectly matchable subgraphs of a graph; and the identification of new families of facets of the asymmetric traveling salesman polytope, the three-index assignment polytope and the job shop scheduling polyhedron. Another line of research involves the characterization of classes of graphs for which the maximum-weight clique problem(NP-complete in general) is polynomially solvable. Several such classes have let to improved algorithms for the maximum-weight clique problem on arbitrary graphs. Other algorithmic results include efficient solution or approximation methods for the 0-1 knapsack problem, the asymmetric traveling salesman and the prize collecting traveling salesman problems, set covering, traffic assignment in communication satellites, the minimum makespan problem of job shop scheduling, etc.

  • Gerard Cornuéjols

    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.

  • John Hooker

    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.

  • Javier Pena

    My research interests are concerned with the mathematical and computational aspects of optimization algorithms, especially interior-point methods for convex optimization. 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 theory.

  • R. Ravi

    Research interests include approximation algorithms, combinatorial optimization, and computational biology.

  • Michael Trick

    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.

  • Mathematics Department

  • Thomas A. Bohman

    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.

  • Alan Frieze

    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.

  • Oleg Pikhurko

    I work in various areas of discrete mathematics including extremal graph theory and Ramsey theory. It is always fascinating when the ideas and techniques from other areas (such as, for example, linear algebra or Fourier analysis) apply to combinatorial problems and vice versa. One of such powerfull tools, which interests me very much, is the probabilistic method. My recent projects deal with the hypergraph Turan problem, first-order formulas on graphs, and some questions from combinatorial number theory.

  • Richard Statman

    Principal research interests lie in the theory of computation with special emphasis on symbolic computation. In particular, current research involves lambda calculus and combinatory algebra. This area underwent extensive development in the first half of this century, and then lay dormant until Dana Scott's fundamental work in the 1970s. Part of what has emerged from Scott's work is that lambda calculus forms the foundation of functional programming at both the semantic and syntactic levels. The area has been revived by an influx of theoretical problems directly related to design and implementation issues.

  • Reha Tütüncü

    My main area of research is optimization with a recent focus on the development, analysis, and implementation of interior-point methods for the solution of linear and semidefinite programming problems. The ideas concerned lie at the confluence of mathematical programming, convex analysis, numerical linear algebra, and computational mathematics. Also of interest to me is using optimization techniques to price financial options.

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