Solving the convex cost integer dual network flow problem

In this paper, we consider a convex optimization problem where the objective function is the sum of separable convex functions, the constraints are similar to those arising in the dual of a minimum cost flow problem (that is, of the form pi(i) - pi(j) less than or equal to w(ij)), and the variables are required to take integer values within a specified range bounded by an integer U. Let m denote the number of constraints and (n+m) denote the number of variables. We call this problem the convex cost integer dual network flow problem. In this paper, we develop network flow based algorithms to solve the convex cost integer dual network flow problem efficiently. We show that using the Lagrangian relaxation technique, the convex cost integer dual network flow problem can be reduced to a convex cost primal network flow problem where each cost function is a piecewise linear convex function with integer slopes. We next show that the cost scaling algorithm for the minimum cost flow problem can be adapted to solve the convex cost integer dual network flow problem in O(nm log(n(2)/m) log(nU)) time. This algorithm improves the best currently available algorithm and is also likely to yield algorithms with attractive empirical performance.