# About strongly polynomial time algorithms for quadratic optimization over submodular constraints

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## Abstract

We present new strongly polynomial algorithms for special cases of convex separable quadratic minimization over submodular constraints. The main results are: an O(*NM* log(*N*^{2}/*M*)) algorithm for the problem*Network* defined on a network on*M* arcs and*N* nodes; an O(*n* log*n*) algorithm for the*tree* problem on*n* variables; an O(*n* log*n*) algorithm for the*Nested* problem, and a linear 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.

## Keywords

Quadratic programming Submodular constraints Kuhn-Tucker conditions Lexicographically optimal flow Parametric maximum flow## Preview

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## Copyright information

© The Mathematical Programming Society, Inc. 1995