Linear programming

In this chapter we discuss the role of linear programming (LP) in the design and analysis of combinatorial approximation algorithms. Our emphasis is on NP-hard problems in combinatorial optimization. One aspect of their computational hardness is that such problems lack a good characterization of optimal solutions. Thus, approximating the optimum often involves finding a tight-as-possible bound on the optimal value that can be computed efficiently. LP is a powerful tool in deriving such bounds. The starting point is usually a formulation of the combinatorial optimization problem as an integer linear program. As a concrete example, consider the problem of VERTEX COVER. Given an undirected graph G = (V, E) with nonnegative weights on the vertices w : V → N, we wish to find a minimum-weight set of vertices V′ ⊂ V such that for every e ∈ E, e ∩ V′ ≠ ∅. This is a well-known NP-hard problem (see Ref. [1]), and here is a natural way to express it as an integer linear program. For every i ∈ V assign an indicator variable xi ∈ {0, 1}, indicating whether or not i ∈ V′. The constraints e ∩ V′ ≠ ∅ can be expressed as xi + x j ≥ 1, where e = {i, j }. The resulting program is (formula present) An ideal bound on the optimum can be derived by optimizing the same objective function over the convex hull of the integer solutions. As the vertices of the convex hull are integer solutions, this would yield an optimal solution. Unfortunately, the fact that VERTEX COVER is NP-hard implies that we are not aware of a concise representation of this linear program. In particular, the convex hull has an exponential number of facets, corresponding to an exponential number of linear constraints. A polynomial-time algorithm that, given a vector x ∈ RV, finds a violated constraint or verifies that x is in the convex hull (a so-called separation oracle) is unlikely to exist. However, we can compute a lower bound on the optimum by relaxing the integrality constraints (6.2). Thus we get the following linear programming relaxation for VERTEX COVER: (formula present) Notice that in an optimal solution there is no reason to set any variable xi to a value greater than 1, so we do not have to add explicitly the inequalities xi ≤ 1, ∀i ∈ V.