Polynomial algorithms for p-dispersion problems in a 2d Pareto Front
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Having many best compromise solutions for bi-objective optimization problems, this paper studies p-dispersion problems to select p⩾ 2 representative points in the Pareto Front(PF). Four standard variants of p-dispersion are considered. A novel variant, denoted Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2d PF. Firstly, it is proven that 2-dispersion and 3-dispersion problems are solvable in O(n) time in a 2d PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2d PF. The Max-Min p-dispersion problem is proven solvable in O(pnlog n) time and O(n) memory space. The Max-Sum-Min p-dispersion problem is proven solvable in O(pn^3) time and O(pn^2) space. The Max-Sum-Neighbor p-dispersion problem is proven solvable in O(pn^2) time and O(pn) space. Complexity results and parallelization issues are discussed in regards to practical implementation.
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