Solving integer programs over monotone inequalities in three variables: A framework for half integrality and good approximations

Abstract

We define a class of monotone integer programs with constraints that involve up to three variables each. A generic constraint in such integer program is of the form ax− by⩽ z+ c, where a and b are nonnegative and the variable z appears only in that constraint. We devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables U. The solution is obtained from a minimum cut on a graph with O( nU) nodes and O( mU) arcs where n is the number of variables of the types x and y and m is the number of constraints. Our algorithm is also valid for nonlinear objective functions. Nonmonotone integer programs are optimization problems with constraints of the type ax+ by⩽ z+ c without restriction on the signs of a and b. Such problems are in general NP-hard. We devise here an algorithm, relying on a transformation to the monotone case, that delivers half integral superoptimal solutions in polynomial time. Such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation. When the half integral solution can be rounded to an integer feasible solution, this is a 2-approximate solution. In that the technique is a unified 2-approximation technique for a large class of problems. The results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe. The algorithm described here has a wide array of problem applications. An additional important consequence of our results is that nonmonotone problems in the framework are MAX SNP-hard and at least as hard to approximate as vertex cover. Problems that are amenable to the analysis provided here are easily recognized. The analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality.