CSDM - Twice-Ramanujan Sparsifiers

Nikhil Srivastava
Yake University
September 21, 2009

We prove that every graph has a spectral sparsifier with a number of edges linear in its
number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our
sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs.
In particular, we prove that for every d > 1 and every undirected, weighted graph G =
(V,E,w) on n vertices, there exists a weighted graph H = (V, F, ~w) with at most dn edges
such that for every x ∈ ℜV ,

 

CSDM - The Completeness of the Permanent

Amir Yehudayoff
Institute for Advanced Study
September 22, 2009

In his seminal work, Valiant defined algebraic analogs for the classes P and NP, which are known today as VP and VNP. He also showed that the permanent is VNP-complete (that is, the permanent is in VNP and any problem in VNP is reducible to it). We will describe the ideas behind the proof of this completeness of the permanent.

CSDM - Span Programs and Quantum Query Algorithms

Ben Reichardt
University of Waterloo
September 29, 2009

The general adversary bound is a lower bound on the number of input queries required for a quantum algorithm to evaluate a boolean function. We show that this lower bound is in fact tight, up to a logarithmic factor. The proof is based on span programs. It implies that span programs are an (almost) equivalent computational model to quantum query algorithms.

The Completeness of the Permanent

Amir Yehudayoff
Institute for Advanced Study
October 6, 2009

In his seminal work, Valiant defined algebraic analogs for the classes P and NP, which are known today as VP and VNP. He also showed that the permanent is VNP-complete (that is, the permanent is in VNP and any problem in VNP is reducible to it). We will describe the ideas behind the proof of this completeness of the permanent.

CSDM - On The Complexity of Circuit Satisfiability

Ramamohan Paturi
University of California at San Diego
October 12, 2009

We present a gap theorem regarding the complexity of the circuit satisfiability problem.
We prove that the success probability of deciding Circuit Satisfiability for deterministic
circuits with n variables and size m is either 2−n or 2−o(n) when restricted to probabilistic
circuit families {Cn,m} where the size of Cn,m is bounded by 2o(n)poly(m).