Graphs with tiny vector chromatic numbers and huge chromatic numbers Extended Abstract Uriel Feige Michael Langberg Gideon Schechtman Department of Computer Science and Applied Mathematics Weizmann Institute of Science, Rehovot 76100 feige,mikel,gideon @wisdom.weizmann.ac.il Abstract Karger, Motwani and Sudan (JACM 1998) introduced the notion of a vector coloring of a graph. In particular they show that every -colorable graph is also vector - colorable, and that for constant , graphs that are vector - colorable can be colored by roughly colors. Here 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 there are graphs that are vector -colorable but do not have independent sets significantly larger than (and hence cannot be colored with significantly less that colors). For we show vector -colorable graphs that do not have independent sets of size , for some con- stant . This shows that the vector chromatic number does not approximate the chromatic number within factors better than polylog . As part of our proof, we analyze “property testing” al- gorithms that distinguish between graphs that have an in- dependent set of size , 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. 1. Introduction An independent set in a graph is a set of vertices that do not induce any edges. The size of the maximum independent set in is denoted by . For an integer ,a coloring of is a function which assigns colors to the vertices of . A valid coloring of is a coloring in which each color class is an independent set. The chromatic number of is the smallest for which there exists a valid coloring of . Finding and are fundamental NP-hard prob- lems, closely related by the inequality . Given , the question of estimating the value of ( ) or finding large independent sets (small colorings) in have been studied extensively. Let be a graph of size . Both and can be approximated within a ratio of [10, 6]. It is known that unless NP=ZPP, neither nor can be approximated within a ratio of for any [12, 7]. Under stronger complexity as- sumptions, there is some such that neither prob- lem can be approximated within a ratio of [14]. The approximation ratios for these problems significantly improve in graphs that have very large independent sets, or very small chromatic numbers. The algorithms achieving the best ratios in these cases [13, 1, 3, 11] are all based on the idea of vector coloring, introduced by Karger, Motwani and Sudan [13]. A vector -coloring of a graph is an as- signment of unit vectors to its vertices, such that for every edge, the inner product of the vectors assigned to its end- points is at most (in the sense that it can only be more neg- ative) . Every -colorable graph is also vector -colorable (by identifying each color class with one vertex of a perfect -dimensional simplex centered at the ori- gin). Moreover, unlike the chromatic number, a vector - coloring (when it exists) can be found in polynomial time us- ing semidefinite programming (up to arbitrarily small error in the inner products). Given a vector -coloring of a graph, Karger, Motwani and Sudan show how to color a graph with roughly colors, where is the maximum degree in the graph. (In comparison, the technique of inductive color- ing might use colors.) In fact, they show how to find an independent set of size roughly . Combined with other ideas, this leads to coloring algorithms and algorithms