Remaining with Integral relaxed integer linear programming

Question

When i relax my binary integer program (0,1) to LP (0< =x <=1), the solution seizes to be optimal since some of the x values are fractional. Is there a way i can relax and still get an optimal solution. Thanks

Answer 1

No, in general that doesn't work. A tipical case when you can relax and still get an optimal solution is for totally unimodular matrices, see this Wikipedia article for an introduction to the topic. Moreover, the integrality gap will depend on the particular formulation you use as a model. In general, when you have relaxed the integer program and get a fractional solution, you could use cutting planes and a branch-and-cut schema in order to get feasible integer solutions that have a relatively small integrality gap.