Succinct Relaxations for Some Discrete Problems

A discrete problem can be relaxed by taking the continuous relaxation of an integer programming formulation An equivalent relaxation is obtained by projecting this relax ation onto the original continuous variables The projection is simple for piecewise linear functions xed charge problems and some disjunctive constraints This allows one to solve much smaller relaxations without sacri cing the quality of bounds In particular the projected relaxations for some classical network design and warehouse location problems are minimum cost network ow problems a fact that can dramatically accelerate their solution A relaxation for a problem with discrete elements is often obtained by adding integer variables to the model The integrality constraints not only capture the discrete element but can be dropped in order to obtain a continuous relaxation of the model An equivalent relaxation can be obtained however by projecting the traditional continuous relaxation onto the original continuous variables Projection can generate a large number of inequality constraints but in some important special cases it does not In such cases one can obtain a relaxation that is of the same quality as the conventional one but much more succinct due to the absence of integer variables Occasionally the projected relaxation has special structure that the convention relaxation lacks and can be more easily solved for that reason as well Once integer variables are removed from the relaxation they can be eliminated entirely Logical expressions can be used to express the discrete elements of a problem perhaps more naturally as noted in Branching on logical possibilities or propositional variables can re place branching on integer variables This may even allow branching to terminate sooner than in an integer programming context as argued in In any case the use of projected relax ations almost certainly accelerates the solution of the problem because it provides the same bounds as traditional integer programming and does so more rapidly because the relaxations are smaller This paper examines three projected relaxations First Beaumont s use of elementary inequalities to relax logical disjunctions is brie y reviewed and strengthened due to the usefulness of disjunctions for expressing the discrete aspect of a problem Next a simple convex hull relaxation of piecewise linear functions is presented Finally it is noted that xed charge problems also have a compact projected relaxation In particular the projected relaxation of xed charge network ow problems including warehouse location problems has the structure of a minimum cost network ow problem whereas the conventional relaxation does not This permits a far more rapid solution of the relaxation than is otherwise possible Because solving the relaxation consumes nearly all the solution time in branch and bound algorithms projected relaxations can dramatically accelerate the solution of these problems The relaxations presented here are simple Nonetheless they and their advantages are normally overlooked Elementary Inequalities Beaumont showed that a single elementary inequality relaxes a disjunction of linear inequal ities