Break the 1/4 approximation barrier for linear-query non-monotone SMK

Determine whether a linear-query algorithm, deterministic or randomized, can achieve an approximation ratio strictly greater than 1/4 for non-monotone submodular maximization under a knapsack constraint.

Background

The paper studies non-monotone submodular maximization under a knapsack constraint (SMK) in the value-oracle model, with particular emphasis on algorithms using a number of oracle queries linear, up to parameter-dependent factors, in the ground-set size. Before this work, randomized linear-query algorithms achieved a 1/4−ε approximation, whereas the best deterministic linear-query guarantee was 1/5−ε.

The paper establishes a deterministic (1/4−ε)-approximation using O(n log²(1/ε)/ε²) value queries, thereby matching the best previously known randomized ratio. Consequently, the remaining explicitly identified barrier is whether either deterministic or randomized linear-query algorithms can surpass the 1/4 approximation threshold.