Faster separation of 1-wheel inequalities by graph products

Using graph products we present an O(|V ||E| + |V | log |V |) separation algorithm for the nonsimple 1-wheel inequalities by Cheng and Cunningham (1997) of the stable set polytope, which is faster than their O(|V |) algorithm. There are two ingredients for our algorithm. The main improvement stems from a reduction of separation problem to multiple shortest path problems in an auxillary graph having only 6|V | vertices and 9|E| arcs, thereby preserving low sparsity. Then Johnson’s algorithm can be applied exploiting that preserved sparsity of the original graph which is maintained in the auxillary graph. In contrast, Cheng and Cunningham’s auxillary graph is by construction dense, |E| = O(|V |), so application of Johnson’s algorithm provides no large improvement.