Paroids: A generic environment for local search
Abstract
Perhaps the most important development of the past decade in discrete optimization has been the emergence of a coherent complexity theory providing formal definitions of the classes of problems tractable to different kinds of algorithms. The theory has isolated a vast problem family, NP-Complete, for which is widely accepted that no formally efficient algorithm can be produced for any of its members. This conjecture has renewed mathematical interest in the heuristic/approximate approaches long used in an ad hoc way to tackle hard combinatorial problems. To date, most research on approximate combinatorial algorithms has been either very problem specific or if generic, rooted in linear programming. This research is directed to an alternative approach. The goal is to open the door to truly generic research in combinatorial heuristics by isolating and proving the viability of a canonical combinatorial environment in which heuristics can be structured, compared and applied to numerous specific models. We will define a new matroid based combinatorial structure called paroid. A number of classes of paroids are introduced, and their relation to classical models is shown. Structural properties of paroids and their relation to matroids are presented. Two optimization problems that arise from paroids are introduced, and "natural" reductions of well-known discrete models into the paroid optimization environment are also shown. The models studied are: k-Matroid Intersection, Matching, Traveling Salesman Problem, Vertex Packing, Graph Partitioning, and Knapsack. We also present a local search procedure, called paroid search which generalizes a number of problem specific algorithms. Some of these generalized procedures include Lin Kerninghan for the Traveling Salesman Problem, the greedy for independence systems, and an optimal algorithm for 2-Matroid Intersection. A number of algorithmic properties are shown, including PLS-Completeness and reachability.
Degree
Ph.D.
Advisors
Rardin, Purdue University.
Subject Area
Industrial engineering
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