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We consider colorful bin packing games in which selfish players control a set of items which are to be packed into a minimum number of unit capacity bins. Each item has one of m ≥ 2 colors and cannot be packed next to an item of the same color. All bins have the same unitary cost which is shared among the items it contains, so that players are interested in selecting a bin of minimum shared cost. We adopt two standard cost sharing functions: the egalitarian cost function which equally shares the cost of a bin among the items it contains, and the proportional cost function which shares the cost of a bin among the items it contains proportionally to their sizes. Although, under both cost functions, colorful bin packing games do not converge in general to a (pure) Nash equilibrium, we show that Nash equilibria are guaranteed to exist and we design an algorithm for computing a Nash equilibrium whose running time is polynomial under the egalitarian cost function and pseudo-polynomial for a constant number of colors under the proportional one. We also provide a complete characterization of the efficiency of Nash equilibria under both cost functions for general games, by showing that the prices of anarchy and stability are unbounded when m ≥ 3 while they are equal to 3 for black and white games, where m = 2. We finally focus on games with uniform sizes (i.e., all items have the same size) for which the two cost functions coincide. We show again a tight characterization of the efficiency of Nash equilibria and design an algorithm which returns Nash equilibria with best achievable performance.

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In this paper we consider a generalization of the well-known budgeted maximum coverage problem. We are given a ground set of elements and a set of bins. The goal is to find a subset of elements along with an associated set of bins, such that the overall cost is at most a given budget, and the profit is maximized. Each bin has its own cost and the cost of each element depends on its associated bin. The profit is measured by a monotone submodular function over the elements. We first present an algorithm that guarantees an approximation factor of 1 2 1 − 1 e α , where α ≤ 1 is the approximation factor of an algorithm for a sub-problem. We give two polynomialtime algorithms to solve this sub-problem. The first one gives us α = 1 − if the costs satisfies a specific condition, which is fulfilled in several relevant cases, including the unitary costs case and the problem of maximizing a monotone submodular function under a knapsack constraint. The second one guarantees α = 1 − 1 e − for the general case. The gap between our approximation guarantees and the known inapproximability bounds is 1 2. We extend our algorithm to a bi-criterion approximation algorithm in which we are allowed to spend an extra budget up to a factor β ≥ 1 to guarantee a 1 2 1 − 1 e αβ-approximation. If we set β = 1 α ln 1 2 , the algorithm achieves an approximation factor of 1 2 − , for any arbitrarily small > 0.