DP-BASED ALGORITHM AND FPTAS FOR THE KNAPSACK SHARING AND RELATED PROBLEMS

In the knapsack sharing problem (KSP), formulated previously, we considered a game-theoretic situation in which two or more players (agents) compete for their share of capacity in a knapsack with their respective sets of items. As an extension of this problem, we formulate the extended knapsack sharing problem (XKSP). This is actually a family of KSP-like problems, and we present a dynamic programmingbased (DP-based), pseudo-polynomial time algorithm to solve XKSP to optimality in a unified way. XKSP is shown to be NP-hard, but due to the existence of this pseudo-polynomial time algorithm, it is only weakly NP-hard. Next, we develop an algorithm to solve the problem approximately in polynomial time by decomposing it into a series of subproblems. Furthermore, we introduce a scaling factor into the DP computation to obtain a fully polynomial time approximation scheme (FPTAS) for XKSP with two agents. Extension to the case of more than two agents is discussed, together with a non-DP-based PTAS.