IntegerLinearProgram

What is an integer linear program?

A linear program, together with additional constraints that restrict some of the variables to take on only integer values. The problem of solving integer linear programs is NP-hard.

Example: min-cost vertex cover

Input: a graph G=(V,E) with vertex costs c.
Goal: find a vertex cover S of minimum cost, where cost(S) = sum_{v in S} c(v).

Introduce a variable x(v) for each vertex v.

Minimize ∑_v c(v) x(v) subject to:

x(v) ≥ 0 for each vertex v ∈ V.

x(u)+x(v) ≥ 1 for each edge (u,v) ∈ E.

x(v) takes on only integer values for each v. (integrality constraints)

prove: each feasible solution x corresponds to a vertex cover S such that cost(x) = cost(S). (cost(x) denotes the objective function value).

prove: the optimal solution corresponds to a min-cost vertex cover.