24 research outputs found
Consensus Strategies for Signed Profiles on Graphs
The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,-}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes.median;consensus function;median graph;majority rule;plurality strategy;Graph theory;Hamming graph;Johnson graph;halfcube;scarcity strategy;Discrete location and assignment;Distance in graphs
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
Consensus Strategies for Signed Profiles on Graphs
The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+,-}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes
Axiomatic Characterization of the Median and Antimedian Function on a Complete Graph minus a Matching
__Abstract__
A median (antimedian) of a profile of vertices on a graph G is a vertex that minimizes (maximizes) the sum of the distances to the elements in the profile. The median (antimedian) function has as output the set of medians (antimedians) of a profile. It is one of the basic models for the location of a desirable (obnoxious) facility in a network. 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 an axiomatic characterization is obtained for the median and antimedian function on complete graphs minus a matching
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
Fair Sets of Some Class of Graphs
Given a non empty set of vertices of a graph, the partiality of a vertex
with respect to is the difference between maximum and minimum of the
distances of the vertex to the vertices of . The vertices with minimum
partiality constitute the fair center of the set. Any vertex set which is the
fair center of some set of vertices is called a fair set. In this paper we
prove that the induced subgraph of any fair set is connected in the case of
trees and characterise block graphs as the class of chordal graphs for which
the induced subgraph of all fair sets are connected. The fair sets of ,
, , wheel graphs, odd cycles and symmetric even graphs are
identified. The fair sets of the Cartesian product graphs are also discussed.Comment: 14 pages, 4 figure
Medians in median graphs and their cube complexes in linear time
The median of a set of vertices of a graph is the set of all vertices
of minimizing the sum of distances from to all vertices of . In
this paper, we present a linear time algorithm to compute medians in median
graphs, improving over the existing quadratic time algorithm. We also present a
linear time algorithm to compute medians in the -cube complexes
associated with median graphs. Median graphs constitute the principal class of
graphs investigated in metric graph theory and have a rich geometric and
combinatorial structure, due to their bijections with CAT(0) cube complexes and
domains of event structures. Our algorithm is based on the majority rule
characterization of medians in median graphs and on a fast computation of
parallelism classes of edges (-classes or hyperplanes) via
Lexicographic Breadth First Search (LexBFS). To prove the correctness of our
algorithm, we show that any LexBFS ordering of the vertices of satisfies
the following fellow traveler property of independent interest: the parents of
any two adjacent vertices of are also adjacent. Using the fast computation
of the -classes, we also compute the Wiener index (total distance) of
in linear time and the distance matrix in optimal quadratic time