Submodularity of Some Classes of Combinatorial Optimization Games

Some situations concerning cost allocation are formulated as combinatorial optimization games. We consider a minimum coloring game and a minimum vertex cover game. For a minimum coloring game, Deng{Ibaraki{Nagamochi 1] showed that deciding the core nonemptiness of a given minimum coloring game is NP-complete, which implies that a good characterization of balanced minimum coloring games is unlikely to be obtained and Deng{Ibaraki{Nagamochi{Zang 2] showed that a minimum coloring game is totally balanced if and only if the underlying graph is perfect. For a minimum vertex cover game, Deng{Ibaraki{Nagamochi 1] showed that a minimum vertex cover game has the nonempty core if and only if the size of a minimum vertex cover of the underlying graph is equal to the size of a maximum matching of the graph, and Deng{Ibaraki{Nagamochi{Zang 2] showed that a minimum vertex cover game is totally balanced if and only if the underlying graph is bipartite. In this note, we characterize submodular minimum coloring games and submodular vertex cover games in terms of forbidden subgraphs. That is, a minimum coloring game is submodular if and only if the underlying graph contains no induced subgraph isomorphic to K 1 K 2 and a minimum vertex cover game is submodular if and only if the underlying graph contains no subgraph isomorphic to P 3 or K 3. A relationship with matroids is also stated.