Fully Dynamic Algorithms for Knapsack Problems with Polylogarithmic Update Time

TL;DR

This paper introduces dynamic algorithms for the multiple knapsack problem that efficiently maintain near-optimal solutions with polylogarithmic update times, addressing a gap in the literature for non-graph packing problems.

Contribution

It presents the first dynamic algorithms for knapsack problems with worst-case polylogarithmic update times, supporting implicit solutions and efficient queries.

Findings

Achieves polylogarithmic worst-case update time for dynamic knapsack solutions.

Supports efficient queries like item packing and solution value retrieval.

Bridges the gap between dynamic algorithms and packing problems.

Abstract

Knapsack problems are among the most fundamental problems in optimization. In the Multiple Knapsack problem, we are given multiple knapsacks with different capacities and items with values and sizes. The task is to find a subset of items of maximum total value that can be packed into the knapsacks without exceeding the capacities. We investigate this problem and special cases thereof in the context of dynamic algorithms and design data structures that efficiently maintain near-optimal knapsack solutions for dynamically changing input. More precisely, we handle the arrival and departure of individual items or knapsacks during the execution of the algorithm with worst-case update time polylogarithmic in the number of items. As the optimal and any approximate solution may change drastically, we only maintain implicit solutions and support certain queries in polylogarithmic time, such as the packing of an item and the solution value. While dynamic algorithms are well-studied in the context of graph problems, there is hardly any work on packing problems and generally much less on non-graph problems. Given the theoretical interest in knapsack problems and their practical relevance, it is somewhat surprising that Knapsack has not been addressed before in the context of dynamic algorithms and our work bridges this gap.