The Bounded Knapsack problem is one of the most fundamental NP-complete
problems at the intersection of computer science, optimization, and operations
research. A recent line of research worked towards understanding the complexity
of pseudopolynomial-time algorithms for Bounded Knapsack parameterized by the
maximum item weight wmaxβ and the number of items n. A
conditional lower bound rules out that Bounded Knapsack can be solved in time
O((n+wmaxβ)2βΞ΄) for any Ξ΄>0 [Cygan, Mucha,
Wegrzycki, Wlodarczyk'17, K\"unnemann, Paturi, Schneider'17]. This raised the
question whether Bounded Knapsack can be solved in time O~((n+wmaxβ)2). The quest of resolving this question lead to
algorithms that run in time O~(n3wmax2β) [Tamir'09],
O~(n2wmax2β) and O~(nwmax3β)
[Bateni, Hajiaghayi, Seddighin, Stein'18], O(n2wmax2β) and
O~(nwmax2β) [Eisenbrand and Weismantel'18], O(n+wmax3β) [Polak, Rohwedder, Wegrzycki'21], and very recently
O~(n+wmax12/5β) [Chen, Lian, Mao, Zhang'23].
In this paper we resolve this question by designing an algorithm for Bounded
Knapsack with running time O~(n+wmax2β), which is
conditionally near-optimal.Comment: 28 page