具有冲突图的不可分项目的公平分配
Fair allocation of indivisible items with conflict graphs
摘要：
我们考虑公平分配的不可分割的项目，几个代理商，并添加一个图论的角度来看，这个经典的问题。也就是说，我们引入了以冲突图描述的项对之间的不兼容关系。分配给一个代理的每个项目子集必须在该图中形成一个独立集。因此，将项目分配给代理对应于冲突图的部分着色。每个代理人对每一个项目都有自己的利润估值。针对公平分配，我们的目标是最大化的最低总利润的项目分配给任何一个代理。作为特殊情况，由此产生的优化问题包含分区和独立集。在我们的贡献，我们得出的复杂性和算法的结果取决于给定的图形的属性。我们证明了这个问题对于二部图和它们的线图是强NP-难的，对于弦图、余可比图、双凸二部图和有界树宽图类是伪多项式时间可解的.每个伪多项式算法也可以变成一个完全多项式近似方案（FPTAS）。
英文摘要：
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).
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