Minimum dispersion problems Original Research Article

We consider the problem of identifying a feasible solution S under a general combinatorial optimization setting such that the sum of the absolute deviation of element weights associated with S from the median of the element weights of S is as small as possible (MADM). It is shown that MADM can be solved in polynomial time whenever an associated minsum problem can be solved in polynomial time. If this minsum problem is difficult, but permits an ε-approximation scheme, then MADM can also be solved by an ε-approximation scheme. We also consider the case when median is replaced by average (MADA). Unlike MADM, MADA is NP-hard even if the associated minsum problem is solvable in polynomial time. Further, we consider MADLP, the linear programming analog of MADM. MADLP is formulated as a linear program with an exponential number of constraints and a polynomial time algorithm is proposed to solve it.