# Partially-Ordered Knapsack and Applications to Scheduling

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## Abstract

In the *partially-ordered knapsack* problem (*POK*) we are given a set N of items and a partial order ≺*p* on *N*. Each item has a size and an associated weight. The objective is to pack a set *N’*⊆ *N* of maximum weight in a knapsack of bounded size. *N’* should be precedence-closed, i.e., be a valid prefix of ≺*p* . *POK* is a natural generalization, for which very little is known, of the classical Knapsack problem. In this paper we advance the state-of-the-art for the problem through both positive and negative results. We give an FPTAS for the important case of a *2-dimensional* partial order, a class of partial orders which is a substantial generalization of the series-parallel class, and we identify the first non-trivial special case for which a polynomial-time algorithm exists. We also characterize cases where the natural linear relaxation for *POK* is useful for approximation and we demonstrate its limitations. Our results have implications for approximation algorithms for scheduling precedence-constrained jobs on a single machine to minimize the sum of weighted completion times, a problem closely related to *POK*.

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