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).

翻译：我们考虑带有额外冲突约束的多个智能体间不可分割物品的公平分配问题。冲突约束由冲突图表示，其中每个物品对应图中的一个顶点，图中的边代表不应分配给同一智能体的不相容物品对。该设置结合了划分问题和独立集问题，可视为冲突图的部分着色。在产生的优化问题中，每个智能体对物品收益拥有自己的估值函数。我们旨在最大化任意智能体获得的最低总收益。此前论文已证明该问题对若干著名图类（如二分图及其线图）是强NP难的。另一方面，对于弦图、共可比图、双凸二分图及有界树宽图类，已证明存在伪多项式时间算法。本文通过为凸二分冲突图、有界团宽图及有界树独立数图类开发伪多项式时间算法来拓展这一研究方向。这些算法基于动态规划，并允许完全多项式时间近似方案（FPTAS）。