School of Mathematics

New Tools for Graph Coloring

Rong Ge
Princeton University
April 19, 2011

How to color $3$ colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time algorithm uses $n^{0.2130}$ colors.

We explore the possibility that more levels of Lasserre Hierarchy can give improvements over previous algorithms. While the case of general graphs is still open, we can give analyse the Lasserre relaxation for two interesting families of graphs.

Quantum Fingerprints that Keep Secrets

Dmitry Gavinsky
NEC Research Laboratories
April 18, 2011

In a joint work with Tsuyoshi Ito we have constructed a fingerprinting scheme (i.e., hashing) that leaks significantly less than log(1/epsilon) bits about the preimage, where epsilon is the error ("collision") probability. It is easy to see that classically this is not achievable; our construction is quantum, and it gives a new example of (unconditional) qualitative advantage of quantum computers.

Graph Sparsification by Edge-Connectivity and Random Spanning Trees

Nick Harvey
University of Waterloo
April 11, 2011

A "sparsifier" of a graph is a weighted subgraph for which every cut has approximately the same value as the original graph, up to a factor of (1 +/- eps). Sparsifiers were first studied by Benczur and Karger (1996). They have wide-ranging applications, including fast network flow algorithms, fast linear system solvers, etc. Batson, Spielman and Srivastava (2009) showed that sparsifiers with O(n/eps^2) edges exist, and they can be computed in time poly(n,eps).