Polynomial Time Instances for the IKHO Problem

Figure 6

Joining 𝒳^𝑅[𝑘^,ℎ^,𝑠] with 𝒳[𝑘,ℎ,𝑠]. The solution 𝑦 belongs to the space 𝒳[𝑘,ℎ,𝑠], while 𝑦^∈𝒳^𝑅[𝑘^,ℎ^,𝑠]. Note that the knapsacks [ℎ+𝑐+𝑢+1,𝑚], where 𝑦 satisfies the capacity constraint, are complementary to the knapsacks [1,ℎ^−𝑢−1], where 𝑦^ satisfies that constraint.