Heuristics, LPs, and Generalizations of Trees on Trees

Abstract

We study a class of models, known as overlay optimization problems, with a "base" subproblem and an "overlay" subproblem, linked by the requirement that the overlay solution be contained in the base solution. In some telecommunication settings, a feasible base solution is a spanning tree and the overlay solution is an embedded Steiner tree (or an embedded path). For the general overlay optimization problem, we describe a heuristic solution procedure that selects the better of two feasible solutions obtained by independently solving the base and overlay subproblems, and establish worst-case performance guarantees on both this heuristic and a linear programming relaxation of the model. These guarantees depend upon worst-case bounds for the heuristics and linear programming relaxations of the unlinked base and overlay problems. Under certain assumptions about the cost structure and the optimality of the subproblem solutions, the performance guarantees for both the heuristic and linear programming relaxation of the combined overlay optimization model are 33%. We also develop heuristic and linear programming performance guarantees for specialized models, including a dual path connectivity model with a worst-case performance guarantee of 25%, and an uncapacitated multicommodity network design model with a worst-case performance guarantee (approximately) proportional to the square root of the number of commodities.