In the Shapley-Scarf economy each agent is endowed with one unit of an indivisible good (house) and wants to exchange it for another, possibly the most preferred one among the houses in the market. In this economy, core is always nonempty and a core allocation can be found by the famous Top Trading Cycles algorithm. Recently, a modification of this economy, containing Q >= 2 types of goods (say, houses and cars for Q=2) has been introduced. We show that if the number of agents is 2, a complete description of the core can be found efficiently. However, when the number of agents is not restricted, the problem to decide the nonemptyness of the core becomes NP-hard already in the case of two types of goods. We also show that even the problem to decide whether an allocation exists in which each agent strictly improves compared to his endowment, is NP-complete.
To overcome the shortage of cadaveric kidneys available for transplantation, several countries organize systematic kidney exchange programs. The kidney exchange problem can be modelled as a cooperative game between incompatible patient-donor pairs whose solutions are permutations of players representing cyclic donations. We show that the problems to decide whether a given permutation is not (weakly) Pareto optimal are NP-complete.
For a nontrivial connected graph G, let c : V (G) → N be a vertex coloring of G where adjacent vertices may be colored the same. For a vertex v ∈ V (G), the neighborhood color set NC(v) is the set of colors of the neighbors of v. The coloring c is called a set coloring if NC(u) 6= NC(v) for every pair u, v of adjacent vertices of G. The minimum number of colors required of such a coloring is called the set chromatic number χs(G). We show that the decision variant of determining χs(G) is NP-complete in the general case, and show that χs(G) can be efficiently calculated when G is a threshold graph. We study the difference χ(G) − χs(G), presenting new bounds that are sharp for all graphs G satisfying χ(G) = ω(G). We finally present results of the Nordhaus-Gaddum type, giving sharp bounds on the sum and product of χs(G) and χs(G).