Holographic Algorithms

Abstract: Using the notion of polynomial time reduction computer scientists have discovered an astonishingly rich web of interrelationships among the myriad computational problems that arise in diverse applications. These relationships can be used both to give evidence of intractability, such as that of NP-completeness, as well as to give efficient algorithms. In this talk we discuss a notion of reduction, which we call a holographic reduction that is more general than the traditional one in the following sense. Instead of locally mapping solutions one-to-one it maps them many-to-many but preserves some measure such as the sum of the solutions. One application is to finding new polynomial time algorithms where none was known before. Another is to give evidence of intractability. There are pairs of related problems that can be contrasted in this manner. For example, for a skeletal version of Cook's 3CNF problem (restricted to be planar and where every variable occurs twice positively) the problem of counting the solutions modulo 2 is NP-hard, but counting them modulo 7 is polynomial time computable. Also, one can revisit the currently accepted conjectures of computer science, such as that P does not equal NP, and ask whether this new kind of reduction offers any insights either way regarding the truth of these conjectures.

Speaker's Bio: Leslie Valiant was educated at King's College, Cambridge; Imperial College, London; and at Warwick University where he received his Ph.D. in computer science in 1974. He is currently T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics in the Division of Engineering and Applied Sciences at Harvard, where he has taught since 1982. Before coming to Harvard he had taught at Carnegie-Mellon University, Leeds University, and the University of Edinburgh. His work has ranged over several areas of theoretical computer science, particularly complexity theory, computational learning, and parallel computation. He also has interests in computational neuroscience, evolution and artificial intelligence. He received the Nevanlinna Prize at the International Congress of Mathematicians in 1986 and the Knuth Award in 1997. He is a Fellow of the Royal Society (London) and a member of the National Academy of Sciences (USA).