DIMACS Discrete Math-Theory of Computing Seminar

Title: Graphs with tiny vector chromatic numbers and huge chromatic numbers

Speaker: Michael Langberg, Weizmann Institute, Rehovot, Israel

Date: Tuesday, November 26, 4:30 PM

Location: CoRE Center, room 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract.

Karger, Motwani and Sudan (JACM 1998) introduced the notion of a vector coloring of a graph. In particular they show that every \$k\$-colorable graph is also vector k-colorable, and that for constant \$k\$, graphs that are vector \$k\$-colorable can be colored by roughly \$\Delta^{1 - 2/k}\$ colors. Here \$\Delta\$ is the maximum degree in the graph. Their results play a major role in the best approximation algorithms for coloring and for maximal independent set.

We show that for every positive integer \$k\$ there are graphs that are vector \$k\$-colorable but do not have independent sets significantly larger than \$n/\Delta^{1 - 2/k}\$ (and hence cannot be colored with significantly less that \$\Delta^{1 - 2/k}\$ colors). For \$k = O(\log n/\log\log n)\$ we show vector \$k\$-colorable graphs that do not have independent sets of size \$(\log n)^c\$, for some constant \$c\$. This shows that the vector chromatic number does not approximate the chromatic number within factors better than \$n/{\mbox{polylog}} n\$.

As part of our proof, we analyze ``property testing'' algorithms that distinguish between graphs that have an independent set of size \$n/k\$, and graphs that are ``far'' from having such an independent set. Our bounds on the sample size improve previous bounds of Goldreich, Goldwasser and Ron (JACM 1998) for this problem.

Joint work with U. Feige and G. Schechtman.