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Decades of research on the 0-1 knapsack problem led to very efficientalgorithms that are able to quickly solve large problem instances tooptimality. This prompted researchers to also investigate whether relativelysmall problem instances exist that are hard for existing solvers andinvestigate which features characterize their hardness. Previously the authorsproposed a new class of hard 0-1 knapsack problem instances and demonstratedthat the properties of so-called inclusionwise maximal solutions (IMSs) can beimportant hardness indicators for this class. In the current paper, weformulate several new computationally challenging problems related to the IMSsof arbitrary 0-1 knapsack problem instances. Based on generalizations ofprevious work and new structural results about IMSs, we formulate polynomialand pseudopolynomial time algorithms for solving these problems. From this wederive a set of 14 computationally expensive features, which we calculate fortwo large datasets on a supercomputer in approximately 540 CPU-hours. We showthat the proposed features contain important information related to theempirical hardness of a problem instance that was missing in earlier featuresfrom the literature by training machine learning models that can accuratelypredict the empirical hardness of a wide variety of 0-1 knapsack probleminstances. Using the instance space analysis methodology, we also show thathard 0-1 knapsack problem instances are clustered together around a relativelydense region of the instance space and several features behave differently inthe easy and hard parts of the instance space.

Features for the 0-1 knapsack problem based on inclusionwise maximal solutions