A simplex based primal-dual algorithm for the perfect b-matching problem: A study in combinatorial optimization algorithm engineering
Abstract
A primary bottleneck in the development of large scale engineering decision systems is the inability to solve large specially structured combinatorial optimization problems. Many problems for which large instances have been solved rely on special purpose algorithms that exploit problem structure (2, 18, 25, 26). Implementation of such highly specialized algorithms can be quite complicated--the situation is exacerbated as problem complexity increases. Implementation can be made more manageable by using general methods and computational tools (e.g. linear programming relaxations and cutting planes) but computational efficiency must be forfeited since linear programming solvers are essentially "black boxes" and are unable to exploit problem physics. A simplex based primal-dual algorithm has been engineered for solving the perfect b-matching problem (PBMP). Although standard computational tools are used, we are able to exploit problem physics by controlling primitive operations of the simplex method such as selection of entering and exiting variables. Specifically, a cutting plane approach is used to solve the PBMP so that only 1/2 integral solution values are encountered enroute to the optimal solution, allowing for economical detection of violated facet defining constraints. In addition to providing tremendous computational efficiencies over previous cutting plane approaches, this algorithm shows that for the case of the PBMP there is a link between special purpose and cutting plane algorithms. Future work will focus on extending the insights gained from the PBMP to more generalized problems such as the connected cover problem (CCP). Ideally, we would like to gain insights from this method so that algorithms could be developed that integrate a simplex based primal-dual framework for solving linear programming relaxations with special purpose separation algorithms that are related to problem structure. A concomitant benefit could be the ability to reverse-engineer special purpose algorithms without the tremendous effort currently required to do so.
Degree
Ph.D.
Advisors
Reklaitis, Purdue University.
Subject Area
Chemical engineering|Mathematics|Industrial engineering
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