Approximation Algorithms for NP-hard Optimization Problems

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It is generally believed that efficient algorithms do not exist for finding an optimal solution to NP-hard opti- mization problems. A natural way to deal with the inability to find exact solutions efficiently is to trade the quality of solution for the computation time. Approximation algorithms do precisely that. Approximation algorithms not only provide an approximate solution in polynomial time, they also provide a certificate of optimality for the solution. One can always refine and tune an approximation algorithm to specific class of instances arising in practice, thereby improving the performance ratio. Primary goal of this research is to further the theory and praxis of the design of approximation algorithms. We will design approximation algorithms for problems arising in the bioinformatics, facility location, schedul- ing, and machine learning domains. Our approach for designing approximation algorithms has theoretical underpinnings in combinatorial algorithms, linear and semi-definite programming. Linear programming based approaches have enjoyed a great deal of success in the recent past. The basic framework entails de- scribing the optimization problem as an integer linear program or a non-linear program over integer variables. A suitable linear programming relaxation or a semi-definite programming relaxation is constructed. The re- laxation is solved using efficient algorithms. A fractional solution thus obtained is converted to an integral solution using some scheme. Care is taken to ensure that the process of converting the fractional solution to an integral solution does not increase the cost of the solution too much. Another approach is to simultane- ously construct an integral primal solution and candidate dual solution (possibly fractional) iteratively using the primal-dual schema. Primal-dual schema has the advantage that one can work with an exponential sized formulation without having to resort to a separation oracle. Approaches based on the primal-dual schema have been very successful for certain types of optimization problems. Integrality gap of an integer program- ming formulation is the gap in the cost of the optimal fractional and the cost of the optimal integral solution. Approximation algorithms based on linear or semi-definite programming have performance ratio no better than the integrality gap of the relaxation. Therefore integer programs with small integrality gap are critical to the success of such approximation algorithms. The immediate program is i) to develop relaxations with bounded integrality gap for the optimization problems of interest or show none exists, ii) to develop combi- natorial algorithms for solving the relaxations where possible, and iii) to develop provably good strategies for converting the fractional solutions to the relaxations to integral solution. There has been considerable research activity on the hardness of approximations in the last few years and several deep results on the hard- ness of approximations have been obtained. The focus of this research is on the design of approximation algorithms and we will draw on the results on hardness of approximations to guide the program. On the theoretical front we will push the frontier in approximation algorithms design. Research conducted as part of this project will have prospect for commercialization and will be of immediate interest to the industry. Knowledge generated will be protected using patents (where applicable) and disseminated in high quality journals and conferences. The program will produce highly skilled manpower; skilled in the use of discrete optimization theory and tools.