Christopher Umans

Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005]. Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).

Simple extractors for all min-entropies and a new pseudorandom generator

A “randomness extractor” is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction that produces good extractors for all min-entropies. Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al. [2001b]. Using our improvements, we obtain, for example, an extractor with output length m = k/(log n)O(1/α) and seed length (1 + α)log n for an arbitrary 0 < α ≤ 1, where n is the input length, and k is the min-entropy of the input distribution.A “pseudorandom generator” is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the Nisan-Wigderson generator [Nisan and Wigderson 1994], and turns worst-case hardness directly into pseudorandomness. The parameters of our generator match those in Impagliazzo and Wigderson [1997] and Sudan et al. [2001] and in particular are strong enough to obtain a new proof that P = BPP if E requires exponential size circuits.Our construction also gives the following improvements over previous work:---We construct an optimal “hitting set generator” that stretches O(log n) random bits into sΩ(1) pseudorandom bits when given a function on log n bits that requires circuits of size s. This yields a quantitatively optimal hardness versus randomness tradeoff for both RP and BPP and solves an open problem raised in Impagliazzo et al. [1999].---We give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits. This technique also give a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.

Loss-less condensers, unbalanced expanders, and extractors

An extractor is a procedure which extracts randomness from a detective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important derandomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits required is suboptimal when the input source has low-min-entropy. Significant progress toward overcoming this bottleneck has been made, but so far has required complicated recursive techniques that lose the simplicity of Trevisans construction. We give a clean method for overcoming this bottleneck by constructing {\em loss-less condensers}. which compress the n-bit input source without losing any min-entropy, using O(\log n) additional random bits. Our condensers are built using a simple modification of Trevisans construction, and yield the best extractor constructions to date. Loss-less condensers also produce unbalanced bipartite expander graphs with small (polylogarithmic) degree D and very strong expansion of (1-\epilon)D. We give other applications of our construction, including dispersers with entropy loss O(\log n), depth two super-concentrators whose size is within a polylog of optimal, and an improved hardness of approximation result.

Pseudo-random generators for all hardnesses

(MATH) We construct the first pseudo-random generators with logarithmic seed length that convert s bits of hardness into sΩ(1) bits of 2-sided pseudo-randomness for any s}. This improves [8] and gives a direct proof of the optimal hardness vs. randomness tradeoff in [15]. A key element in our construction is an augmentation of the standard low-degree extension encoding that exploits the field structure of the underlying space in a new way.

The Minimum Equivalent DNF Problem and Shortest Implicants

We prove that the Minimum Equivalent DNF problem is ?P2-complete, resolving a conjecture due to Stockmeyer. We also consider the complexity and approximability of a related optimization problem in the second level of the polynomial hierarchy, that of finding shortest implicants of a Boolean function. We show that when the input is given as a DNF, this problem is complete for a complexity class above coNP utilizing O(log2n)-limited nondeterminism. When the input is given as a formula or circuit, the problem is ?P2-complete, and ?P2-hard to approximate within factors of n1/2?? and n1??, respectively.

Group-theoretic algorithms for matrix multiplication

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

A group-theoretic approach to fast matrix multiplication

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra /spl Copf/[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present machinery and examples to support (1), including a proof that certain families of groups of order n/sup 2+o(1)/ support n /spl times/ n matrix multiplication, a necessary condition for the approach to yield exponent 2. Although we cannot yet completely achieve both (1) and (2), we hope that it may be possible, and we suggest potential routes to that result using the constructions in this paper.

Simple extractors for all min-entropies and a new pseudo-random generator

We present a simple, self-contained extractor construction that produces good extractors for all min-entropies (min-entropy measures the amount of randomness contained in a weak random source). Our construction is algebraic and builds on a new polynomial-based approach introduced by A. Ta-Shma et al. (2001). Using our improvements, we obtain, for example, an extractor with output length m=k/sup 1-/spl delta// and seed length O(log n). This matches the parameters of L. Trevisans (1999) breakthrough result and additionally achieves those parameters for small min-entropies k. Our construction gives a much simpler and more direct solution to this problem. Applying similar ideas to the problem of building pseudo-random generators, we obtain a new pseudo-random generator construction that is not based on the NW generator (N. Nisan and A. Widgerson, 1994), and turns worst-case hardness directly into pseudo-randomness. The parameters of this generator are strong enough to obtain a new proof that P=BPP if E requires exponential size circuits. Essentially, the same construction yields a hitting set generator with optimal seed length that outputs s/sup /spl Omega/(1)/ bits when given a function that requires circuits of size s (for any s). This implies a hardness versus randomness trade off for RP and BPP that is optimal (up to polynomial factors), solving an open problem raised by R. Impagliazzo et al. (1999). Our generators can also be used to derandomize AM.

Complexity of two-level logic minimization

The complexity of two-level logic minimization is a topic of interest to both computer-aided design (CAD) specialists and computer science theoreticians. In the logic synthesis community, two-level logic minimization forms the foundation for more complex optimization procedures that have significant real-world impact. At the same time, the computational complexity of two-level logic minimization has posed challenges since the beginning of the field in the 1960s; indeed, some central questions have been resolved only within the last few years, and others remain open. This recent activity has classified some logic optimization problems of high practical relevance, such as finding the minimal sum-of-products (SOP) form and maximal term expansion and reduction. This paper surveys progress in the field with self-contained expositions of fundamental early results, an account of the recent advances, and some new classifications. It includes an introduction to the relevant concepts and terminology from computational complexity, as well a discussion of the major remaining open problems in the complexity of logic minimization

Hamiltonian cycles in solid grid graphs

A grid graph is a finite node induced subgraph of the infinite two dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP complete. We give a polynomial time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed by A. Itai et al. (1982). In fact, our algorithm can identify Hamiltonian cycles in quad quad graphs, a class of graphs that properly includes solid grid graphs.