MaxSAT Encoding for MC-Net-Based Coalition Structure Generation Problem with Externalities

Abstract

Coalition Structure Generation (CSG) is a main research issue in the domain of coalition games. A majority of existing works assume that the value of a coalition is independent of others in the coalition structure. Recently, there has been interest in a more realistic settings, where the value of a coalition is affected by the formation of other coalitions. This effect is known as externality. The focus of this paper is to make use of Maximum Satisfiability (MaxSAT) to solve the CSG problem where externalities may exist. In order to reduce the exponentially growing number of possible solutions in the CSG problem, we follow the previous works by representing the CSG problem as sets of rules in MC-nets (without externalities) and embedded MC-nets (with externalities). Specifically, enlightened by the previous MC-net-based algorithms exploiting the constraints among rule relations to solve the CSG problem, we encode such constraints into weighted partial MaxSAT (WPM) formulas. Experimental results demonstrate that an off-the-shelf MaxSAT solver achieves significant improvements compared to the previous algorithm for the same set of problem instances.

Extending MaxSAT to Solve the Coalition Structure Generation Problem with Externalities Based on Agent Relations

“…In comparison, the solver chosen for RWPM is Sat4j, which performed best according to our previous experiment [24]. Figure 1 depicts the average computation time for solving problem instances by RWPM and AWPM.…”

“…Given a CSG problem with m agents and n rules, the number of hard clauses for representing the transitive laws in AWPM and RWPM is m · (m − 1) · (m − 2) /2 and n · (n − 1) · (n − 2) /2 [24], respectively. In our experiments, #rules = #agents, which leads to the same number of clauses for representing the transitive laws.…”

“…By using the embedded MC-net, Ueda et al [21] extended the formalization of CSG in [20] and developed a direct encoding algorithm to handle the externalities among the CSG problem. Later on, Liao et al [24] encoded Ohta and Ueda's algorithms into Boolean propositional logic and showed that weighted partial MaxSAT solvers can contribute as a more efficient constraint optimization tool to solving the CSG problem.…”

“…Instead of encoding rule relations into propositional logic [24], we capture a more fine-grained relation, i.e., agent relation, and show that this agent relation-based encoding manages to solve the CSG problem with higher efficiency. More specifically, problem instances for PFGs with 120 agents (rules) could be solved within 4 seconds on average, while the rule relations-based WPM, which was directly encoded from [20], [21], requires averagely 9 seconds to solve the same set of instances.…”