6 ( b ) . Combinatorial Optimization via LPs

The flavor of a combinatorial optimization problem is to maximize an objective function over a set of valid points. In the case of TSP, the valid set of points S ⊆ {0, 1}( n 2) is the set of valid tours on the graph and the objective function is the minimum weight tour. In the cases that we will be interested in, the objective function will be linear. To make the problem amenable to linear programming, we look at the convex hull of S, which we will denote by conv(S). The caveat here is that the polytope thus formed will have large number of facets and thus will be exponentially big to even represent. The question that we will be addressing in this lecture is, for what sets S will the polytope corresponding to conv(S) have a small number of facets? We note that even if conv(S) has exponentially many facets, it can still have an efficient LP based algorithm as long as there is a seperation oracle.