Existence of output‑polynomial algorithms for the bi‑objective knapsack problem

Determine whether there exists an output‑polynomial time algorithm for computing the full Pareto set of the bi‑objective 0–1 knapsack problem, where solutions are evaluated by simultaneously minimizing total weight and maximizing total profit, and the running time must be polynomial in the input size and in the size of the Pareto set.

Background

In multi‑objective optimization, the Pareto set can be exponentially large, so one seeks output‑polynomial algorithms whose running time is polynomial in both the input size and the output size. For the bi‑objective knapsack problem, the classical Nemhauser‑Ullmann dynamic programming algorithm computes intermediate Pareto sets for prefixes of the item list.

This work constructs instances showing the Nemhauser‑Ullmann algorithm does not have output‑polynomial running time, leaving open whether any algorithm with output‑polynomial running time exists for the bi‑objective knapsack problem.