Approximation algorithms for the generalized incremental knapsack problem

We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given set n items, each associated with non-negative weight, T time periods non-decreasing capacities $$W_1 \le \dots W_T$$ . When item i is inserted at t, gain profit $$p_{it}$$ ; however, remains in for all subsequent periods. The goal to decide if when insert item, subject time-dependent capacity constraints, objective maximizing our total profit. Interestingly, setting subsumes as special cases number recently-studied problems, known be strongly NP-hard. Our first contribution comes form polynomial-time $$(\frac{1}{2}-\epsilon )$$ -approximation problem. This result based on reformulation single-machine sequencing which addressed by blending dynamic programming techniques Shmoys–Tardos algorithm assignment Combined further enumeration-based self-reinforcing ideas new structural properties nearly-optimal solutions, turn into quasi-polynomial approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, finding rules out possibility that APX-hard.