Generalized network design polyhedra

Abstract

In recent years, there has been an increased literature on so-called generalized network design problems (GNDPs), such as the generalized minimum spanning tree problem and the generalized traveling salesman problem. In a GNDP, the node set of a graph is partitioned into "clusters," and the feasible solutions must contain one node from each cluster. Up to now, the polyhedra associated with different GNDPs have been studied independently. The purpose of this article is to show that it is possible, to a certain extent, to derive polyhedral results for all GNDPs simultaneously. Along the way, we point out some interesting connections to other polyhedra, such as the quadratic semiassignment polytope and the boolean quadric polytope.

Automatic action annotation in weakly labeled videos

“…The optimal solution is a subgraph that maximizes the above objective function. It is easy to observe that the above combinatorial optimization problem falls under the umbrella of generalized maximum clique problem (GMCP) [17] [18]. GMCP is the class of graph theory problems that generalizes the standard subgraph problems (from node to group of nodes).…”

“…For this purpose, we build a fully connected graph where all proposals in one video are connected to every proposal in all other videos and the edges between proposals capture global, fine grain and shape simiarities between proposals. Finally, we formulate the proposals matching across multiple videos as a Generalized Maximum Clique Problem (GMCP) [17,18]. The output of our method is the most action representative spatio-temporal locations in the video.…”

Editorial: Preface to the Special Issue

“…Feremans et al [5] introduce a unified framework of some known network design problems: the generalized subgraph (GS) problem. The study of the GS polytope allows one to unify and generalize results related to the generalized traveling salesman problem and the generalized spanning tree problem.…”