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Fair packing of independent sets
arXiv - CS - Data Structures and Algorithms Pub Date : 2020-03-25 , DOI: arxiv-2003.11313
Nina Chiarelli, Matja\v{z} Krnc, Martin Milani\v{c}, Ulrich Pferschy, Nevena Piva\v{c}, Joachim Schauer

In this work we add a graph theoretical perspective to a classical problem of fairly allocating indivisible items to several agents. Agents have different profit valuations of items and we allow an incompatibility relation between pairs of items described in terms of a conflict graph. Hence, every feasible allocation of items to the agents corresponds to a partial coloring, that is, a collection of pairwise disjoint independent sets. The sum of profits of vertices/items assigned to one color/agent should be optimized in a maxi-min sense. We derive complexity and algorithmic results for this problem, which is a generalization of the classical Partition and Independent Set problems. In particular, we show that the problem is strongly NP-complete in the classes of bipartite graphs and their line graphs, and solvable in pseudo-polynomial time in the classes of cocomparability graphs and biconvex bipartite graphs.

中文翻译:

独立套装的公平包装

在这项工作中,我们为将不可分割的项目公平分配给多个代理的经典问题添加了图论观点。代理对物品有不同的利润估值,我们允许根据冲突图描述的物品对之间存在不兼容关系。因此,给代理的每一个可行的项目分配都对应于一个部分着色,即成对不相交的独立集合的集合。分配给一种颜色/代理的顶点/项目的利润总和应该在最大最小意义上进行优化。我们推导出这个问题的复杂性和算法结果,这是经典分区和独立集问题的推广。特别是,我们证明了该问题在二部图及其折线图的类别中是强 NP 完全的,
更新日期:2020-03-26
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