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

• Published in: Journal of the Operations Research Society of Japan Vol. 62; no. 1; pp. 1 - 14
• Main Authors: Yamada, Takeo, Kataoka, Seiji
• Format: Journal Article
• Language: English
• Published: Tokyo The Operations Research Society of Japan 31.01.2019; Nihon Opereshonzu Risachi Gakkai, Operations Research Society of Japan
• Subjects: Algorithms; approximation; Combinatorial optimization; DP-based algorithm; Dynamic programming; FPTAS/PTAS; Game theory; Information sharing; knapsack sharing problem; Polynomials; Scaling factors
• ISSN: 0453-4514; 1878-6871; 2188-8299; 2188-8299; 1878-6871
• DOI: 10.15807/jorsj.62.1

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 programming-based (DP-based), pseudo-polynomial time algorithm to solve XKSP to optimality in a unified way. XKSP is shown to be {\cal NP}-hard, but due to the existence of this pseudo-polynomial time algorithm, it is only weakly {\cal 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.