Cutting Planes Cannot Approximate Some Integer Programs

For every positive integer l, we consider a zero-one linear program describing the following optimization problem: maximize the number of nodes in a clique of an n-vertex graph whose chromatic number does not exceed l. Although l is a trivial solution for this problem, we show that any cutting-plane proof certifying that no such graph can have a clique on more than rl vertices must generate an exponential in min{l, n/rl}(1/4) number of inequalities. We allow Gomory-Chvatal cuts and even the more powerful split cuts. Previously, exponential lower bounds were only known for the case r = 1. (c) 2012 Elsevier B.V. All rights reserved.