6 research outputs found
Average case polyhedral complexity of the maximum stable set problem
We study the minimum number of constraints needed to formulate random
instances of the maximum stable set problem via linear programs (LPs), in two
distinct models. In the uniform model, the constraints of the LP are not
allowed to depend on the input graph, which should be encoded solely in the
objective function. There we prove a lower bound with
probability at least for every LP that is exact for a randomly
selected set of instances; each graph on at most n vertices being selected
independently with probability . In the
non-uniform model, the constraints of the LP may depend on the input graph, but
we allow weights on the vertices. The input graph is sampled according to the
G(n, p) model. There we obtain upper and lower bounds holding with high
probability for various ranges of p. We obtain a super-polynomial lower bound
all the way from to . Our upper bound is close to this as there is only an essentially quadratic
gap in the exponent, which currently also exists in the worst-case model.
Finally, we state a conjecture that would close this gap, both in the
average-case and worst-case models
MULTIOBJECTIVE OPTIMIZATION MODELS AND SOLUTION METHODS FOR PLANNING LAND DEVELOPMENT USING MINIMUM SPANNING TREES, LAGRANGIAN RELAXATION AND DECOMPOSITION TECHNIQUES
The land development problem is presented as the optimization of a weighted average of the objectives of three or more stakeholders, subject to develop within bounds residential, industrial and commercial areas that meet governmental goals. The work is broken into three main sections. First, a mixed integer formulation of the problem is presented along with an algorithm based on decomposition techniques that numerically has proven to outperform other solution methods. Second, a quadratic mixed integer programming formulation is presented including a compactness measure as applied to land development. Finally, to prevent the proliferation of sprawl a new measure of compactness that involves the use of the minimum spanning tree is embedded into a mixed integer programming formulation. Despite the exponential number of variables and constraints required to define the minimum spanning tree, this problem was solved using a hybrid algorithm developed in this research