1,746 research outputs found
Axioms for consensus functions on the n-cube
An elementary general result is proved that allows for simple
characterizations of well-known location/consensus functions (median, mean and
center) on the n-cube. In addition, alternate new characterizations are given
for the median and anti-median functions on the n-cube.Comment: 12 page
Axiomatic characterization of the absolute median on cube-free median networks
In Vohra, European J. Operational Research 90 (1996) 78 – 84, a characterization of the absolute median of a tree network using three simple axioms is presented. This note extends that result from tree networks to cube-free median networks. A special case of such networks is the grid structure of roads found in cities equipped with the Manhattan metric.
A simple axiomatization of the median procedure on median graphs
A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequence
of vertices of G, with repetitions allowed. A median x of is a vertex for which
the sum of the distances from x to the vertices in the profile is minimum. The
median function finds the set of all medians of a profile. Medians are important in
location theory and consensus theory. A median graph is a graph for which every
profile of length 3 has a unique median. Median graphs are well studied. They
arise in many arenas, and have many applications.
We establish a succinct axiomatic characterization of the median procedure on
median graphs. This is a simplification of the characterization given by McMorris,
Mulder and Roberts [17] in 1998. We show that the median procedure can be characterized
on the class of all median graphs with only three simple and intuitively
appealing axioms: anonymity, betweenness and consistency. We also extend a key
result of the same paper, characterizing the median function for profiles of even
length on median graphs
Axiomatic Characterization of the Mean Function on Trees
A mean of a sequence π = (x1, x2, . . . , xk) of elements of a finite metric space (X, d) is an element x for which is minimum. The function Mean whose domain is the set of all finite sequences on X and is defined by Mean(π) = { x | x is a mean of π } is called the mean function on X. In this paper the mean function on finite trees is characterized axiomatically
The t-median function on graphs
A median of a sequence pi = x1, x2, … , xk of elements of a finite metric space (X, d ) is an element x for which ∑ k, i=1 d(x, xi) is minimum. The function M with domain the set of all finite sequences on X and defined by M(pi) = {x: x is a median of pi} is called the median function on X, and is one of the most studied consensus functions. Based on previous characterizations of median sets M(pi), a generalization of the median function is introduced and studied on various graphs and ordered sets. In addition, new results are presented for median graphs
Axiomatic characterization of the absolute median on cube-free median networks
In Vohra, European J. Operational Research 90 (1996) 78 – 84, a characterization of the absolute median of a tree network using three simple axioms is presented. This note extends that result from tree networks to cube-free median networks. A special case of such networks is the grid structure of roads found in cities equipped with the Manhattan metric
Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes
An antimedian of a profile of vertices of a graph is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian is
well-behaved: paths and hypercubes
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