Algorithmic Aspects of the Core of Combinatorial Optimization Games

Abstract

We discuss an integer programming formulation for a class of cooperative games. We focus on algorithmic aspects of the core, one of the most important solution concepts in cooperative game theory. Central to our study is a simple (but very useful) observation that the core for this class is nonempty if and only if an associated linear program has an integer optimal solution. Based on this, we study the computational complexity and algorithms to answer important questions about the cores of various games on graphs, such as maximum flow, connectivity, maximum matching, minimum vertex cover, minimum edge cover, maximum independent set, and minimum coloring.

New techniques for cost sharing in combinatorial optimization games

“…We show how the results concerning the OCSP in [7] for covering games, and in [14] for facility location games, arise as fairly simple corollaries of the above results on P x , P x I and C x . Indeed, [7] proves that, for an IM game in which S = 2 V \{∅}, B is the identity matrix, A is binary and d = 0, the value of the optimal cost share coincides with that of the LP relaxation of Eq.…”

“…Indeed, [7] proves that, for an IM game in which S = 2 V \{∅}, B is the identity matrix, A is binary and d = 0, the value of the optimal cost share coincides with that of the LP relaxation of Eq. 4 for S = V .…”

“…Packing and covering games are examined in [7]. Their covering games are nothing but the IM games in which S = 2 V \{∅}, B is the identity matrix, A is binary and d = 0.…”

“…Their covering games are nothing but the IM games in which S = 2 V \{∅}, B is the identity matrix, A is binary and d = 0. As shown in [7], many well-known games on graphs fall into this special case.…”

“…These include, for example, the assignment game [36], the spanning tree game [16], the traveling salesman game [38], the vehicle routing game [15], the packing and covering games [7] and the facility location game [14].…”