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 C[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^(2 + o(1)) support n-by-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.Comment: 12 pages, 1 figure, only updates from previous version are page numbers and copyright informatio

Pseudorandom generators and the BQP vs. PH problem

It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the Nisan-Wigderson pseudorandom generator [NW94] to fool AC_0, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof from [NW94]) can be avoided in this setting. This is a question that has been asked previously in the pseudorandomness literature [BSW03]. We then make three main contributions: (1) We show that our conjecture implies the existence of an oracle relative to which BQP is not in the PH. This entails giving an explicit construction of unitary matrices, realizable by small quantum circuits, whose row-supports are "nearly-disjoint." (2) We give a simple framework (generalizing the setting of Aaronson [A10]) in which any efficiently quantumly computable unitary gives rise to a distribution that can be distinguished from the uniform distribution by an efficient quantum algorithm. When applied to the unitaries we construct, this framework yields a problem that can be solved quantumly, and which forms the basis for the desired oracle. (3) We prove that Aaronson's "GLN conjecture" [A10] implies our conjecture; our conjecture is thus formally easier to prove. The GLN conjecture was recently proved false for depth greater than 2 [A10a], but it remains open for depth 2. If true, the depth-2 version of either conjecture would imply an oracle relative to which BQP is not in AM, which is itself an outstanding open problem. Taken together, our results have the following interesting interpretation: they give an instantiation of the Nisan-Wigderson generator that can be broken by quantum computers, but not by the relevant modes of classical computation, if our conjecture is true.Comment: Updated in light of counterexample to the GLN conjectur

The Complexity of Rationalizing Network Formation

We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to be stable. From this we would like to infer some unobserved data about the system: edge prices, or how much each vertex values short paths to each other vertex. We study two rationalization problems arising from the network formation model of Jackson and Wolinsky [14]. When the goal is to infer edge prices, we observe that the rationalization problem is easy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find prices that maximize the stability of the system. In contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex by each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a close connection between rationalization problems and the Inequality-SAT (I-SAT) problem. Finally and most significantly, we prove that an approximation version of this NP-complete rationalization problem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial algorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input graphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible assuming P ≠ NP To do this we prove a tight (1/2 + δ) -approximation hardness for a variant of I-SAT in which all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LlN_(R_+) (linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification of the recent result of Guruswami and Raghavendra [10] which achieved tight hardness for this problem without the non-negativity constraint. Our technical contributions regarding the hardness of I-SAT and MAX-LIN_(R_+) may be of independent interest, given the generality of these problem

Pseudorandomness for Approximate Counting and Sampling

We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to “boost” a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize Arthur-Merlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the “boosting” theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S_2^P ⊆ PP⊆(NP) and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice

Fast generalized DFTs for all finite groups

For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication

Fast generalized DFTs for all finite groups

For any finite group G, we give an arithmetic algorithm to compute generalized Discrete Fourier Transforms (DFTs) with respect to G, using O(|G|^(ω/2+ ϵ)) operations, for any ϵ > 0. Here, ω is the exponent of matrix multiplication

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.Comment: 10 page

The complexity of Boolean formula minimization

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ^P_2-complete and indeed appears as an open problem in Garey and Johnson (1979) [5]. The depth-2 variant was only shown to be Σ^P_2-complete in 1998 (Umans (1998) [13], Umans (2001) [15]) and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is Σ^P_2-complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ^P_2-complete under Turing reductions