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We consider the fair allocation of indivisible items to several agents with additional conflict constraints. These are represented by a conflict graph where each item corresponds to a vertex of the graph and edges in the graph represent incompatible pairs of items which should not be allocated to the same agent. This setting combines the issues of Partition and Independent Set and can be seen as a partial coloring of the conflict graph. In the resulting optimization problem, each agent has its own valuation function for the profits of the items. We aim at maximizing the lowest total profit obtained by any of the agents. In a previous paper, this problem was shown to be strongly NP-hard for several well-known graph classes, e.g., bipartite graphs and their line graphs. On the other hand, it was shown that pseudo-polynomial time algorithms exist for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. In this contribution, we extend this line of research by developing pseudo-polynomial time algorithms that solve the problem for the class of convex bipartite conflict graphs, graphs of bounded clique-width, and graphs of bounded tree-independence number. The algorithms are based on dynamic programming and also permit fully polynomial-time approximation schemes (FPTAS). [ABSTRACT FROM AUTHOR]

We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Namely, we introduce an incompatibility relation between pairs of items described in terms of a conflict graph. Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the allocation of items to the agents corresponds to a partial coloring of the conflict graph. Every agent has its own profit valuation for every item. Aiming at a fair allocation, our goal is the maximization of the lowest total profit of items allocated to any one of the agents. The resulting optimization problem contains, as special cases, both Partition and Independent Set. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We show that the problem is strongly NP-hard for bipartite graphs and their line graphs, and solvable in pseudo-polynomial time for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS). [ABSTRACT FROM AUTHOR]

Stackelberg pricing games are pricing problems over a set of items. One player, the leader , sets prices for the items and the second player, the follower , buys a subset of items at minimal total cost subject to feasibility constraints. The constraints are determined by an optimization problem. For example, the Stackelberg shortest path game is used to optimize the income from road tolls (cf. [21]). The items are edges in a network graph and the follower buys a subset of edges forming a path. The Stackelberg pricing games studied in the literature are formulated on top of covering problems. In this paper, we introduce pricing games based on packing problems. We are interested in the complexity of computing leader-optimal prices depending on different types of follower-constraints. We show that optimal prices can be computed in polynomial time if the follower-constraints are determined by the well-known interval scheduling problem. This problem is equivalent to the independent set problem on interval graphs. In case the follower-constraints are based on the independent set problem on perfect graphs or on the bipartite matching problem, we prove APX-hardness for the pricing problem. On a more general note, we prove Σ 2 p -completeness if the follower-constraints are given by a packing problem that is NP-complete, i.e., the leader's pricing problem is hard even if she has an NP-oracle at hand. [ABSTRACT FROM AUTHOR]