Relationship between matching and assignment problems
Let (<i>R</i><sub><i>ik</i></sub><i></i>)<sup><i>n</i></sup><sub><i>i,k</i>=1</sub> and (<i>J</i><sub><i>ik</i></sub><i></i>)<sup><i>n</i></sup><sub><i>i,k</i>=1</sub> be preference matrices in the stable matching problem and let (<i>J</i><sub><i>ik</i></sub><i></i>)<sup><i>n</i></sup><sub><i>i,k</i>=1</sub> be the measure of the mutual antipathy in the assignment problem. In this paper we describe all functions <i>f</i> such that if <i>H</i><sub><i>i,k</i></sub><i></i>=<i>f</i>(<i>R</i><sub><i>ik</i></sub><i>,J</i><sub><i>ik</i></sub><i></i>) then for any matrices <i>R</i> and <i>J</i> solution sets in stable matching and assignment problems (partly) coincide. Thus we answer the question about the relationship between these problems stated by D.Knuth. The results are analogous to the Arrow theorem, and the proof techniques are close to those used in the group choice theory.