Knapsack with Small Items in Near-Quadratic Time
Karl Bringmann

TL;DR
This paper presents a near-quadratic time algorithm for Bounded Knapsack, significantly improving previous algorithms and nearly matching the conditional lower bounds, thus resolving a long-standing open problem.
Contribution
The authors develop a near-quadratic time algorithm for Bounded Knapsack, matching the conditional lower bounds and resolving the open question for both 0-1 and Bounded Knapsack.
Findings
Achieved an $ ilde O(n + w_{ ext{max}}^2)$ time algorithm for Bounded Knapsack.
Resolved the open problem of whether Knapsack can be solved in near-quadratic time.
Provided a near-optimal algorithm matching conditional lower bounds.
Abstract
The 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 Knapsack parameterized by the maximum item weight and the number of items . A conditional lower bound rules out that Knapsack can be solved in time for any [Cygan, Mucha, Wegrzycki, Wlodarczyk'17, K\"unnemann, Paturi, Schneider'17]. This raised the question whether Knapsack can be solved in time . This was open both for 0-1-Knapsack (where each item can be picked at most once) and Bounded Knapsack (where each item comes with a multiplicity). The quest of resolving this question lead to algorithms that solve Bounded Knapsack…
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Taxonomy
TopicsOptimization and Packing Problems · Complexity and Algorithms in Graphs · Algorithms and Data Compression
