About strongly polynomial time algorithms for quadratic optimization over submodular constraints

The authors present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(NMlog(N²/M)) algorithm for the problem Network defined on a network on M arcs and N nodes; an O(nlogn) algorithm for the tree problem on n variables; an O(nlogn) algorithm for the Nested problem, and a linar time algorithm for the Generalized Upper Bound problem. These algorithms are the best known so far for these problems. The status of the general problem and open questions are presented as well.