Abstract |
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A subset S of a poset (partially ordered set) is convex if and only if S contains every poset element which is between any two elements in S. Poset convex subsets arise in applications that involve precedence constraints, such as in project scheduling, production planning, and assembly line balancing. We give a strongly polynomial time algorithm which, given a poset and element weights (of arbitrary sign), finds a convex subset with maximum total weight. This algorithm relies on a reduction to a maximum weight filter (or closure) problem in a poset about twice the size of the given poset; the latter problem is wellsolved as a minimum s-t cut problem. We also use this reduction to construct a compact, ideal extended formulation for the convex hull Cp of the characteristic vectors of all convex subsets in poset P. We define a class of alternating inequalities that are valid for Cp and admit a linear time separation algorithm based on Dynamic Programming (DP). Furthermore, whenever the point to separate is actually in Cp the associated DP value functions induce a feasible solution to the extended formulation. This implies that the alternating inequalities and nonnegative inequalities suffice to describe Cp. We conclude by showing that this polyhedral description is minimal, and thus also admits a linear time separation algorithm. |