In this work we add a graph theoretical perspective to a classical problemof 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 Partitionand Independent Set problems. In particular, we{\sc  {\sc   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  .
We consider the fair allocation of indivisible items to several agents and add a graph theoretical perspective to this classical problem. Thereby 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{\sc Partition and{\sc Independent Set . In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can 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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In this work we
We consider the fair allocation of indivisible items to several agents and
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a classical problemof fairly allocating indivisible items to several agents. Agents have different profit valuations of items and we allow
this classical problem. Thereby we introduce
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Hence, every feasible
Every subset of items assigned to one agent has to form an independent set in this graph. Thus, the
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, 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 Partitionand Independent Set problems. In particular, we
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
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Partition
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and
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Independent Set
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. In our contribution we derive complexity and algorithmic results depending on the properties of the given graph. We can
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NP-complete in the classes of
NP-hard for
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in
for
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cocomparability graphs and
chordal graphs, cocomparability graphs,
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, and graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also be turned into a fully polynomial approximation scheme (FPTAS)