F.R. McMorris
http://repub.eur.nl/ppl/3921/
List of Publicationsenhttp://repub.eur.nl/eur_signature.png
http://repub.eur.nl/
RePub, Erasmus University RepositoryAn ABC-Problem for Location and Consensus
Functions on Graphs
http://repub.eur.nl/pub/78320/
Mon, 08 Jun 2015 00:00:01 GMT<div>F.R. McMorris</div><div>B. Novick</div><div>H.M. Mulder</div><div>R.C. Powers</div>
__Abstract__
A location problem can often be phrased as a consensus problem or a voting problem. We use these three perspectives, namely location, consensus and voting to initiate the study of several questions. The median function Med is a location/consensus function on a connected graph G that has the finite sequences of vertices of G as input. For each such sequence, Med returns the set of vertices that minimize the distance sum to the elements of the sequence. The median function satisfies three intuitively clear axioms: (A) Anonymity, (B) Betweenness and (C) Consistency. In [13] it was shown that on median graphs these three axioms actually characterize Med. This result raises a number of questions: (i) On what other classes of graphs is Med characterized by (A), (B) and (C)? (ii) If some class of graphs has other ABC-functions besides Med, then determine additional axioms that are needed to characterize Med. (iii) In the latter case, can we find characterizations of other functions that satisfy (A), (B) and (C)? We call these questions, and related questions, the ABC-Problem for location/consensus functions on graphs. In this paper we present first results. For the first question we use consensus terminology. We construct a non-trivial class different from the median graphs, on which the median function is the unique “ABC function”. For the second and third question voting terminology is most apt for our approach. On K_n with n > 2 we construct various non-trivial ABC-voting procedures. For some nice families, we present a full axiomatic characterization. We also construct an infinite family of ABC-functions on K_3.Five axioms for location functions on median graphs
http://repub.eur.nl/pub/51344/
Thu, 15 May 2014 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>B. Novick</div><div>R.C. Powers</div>
__Abstract__
In previous work, two axiomatic characterizations were given for the median function on median graphs: one involving the three simple and natural axioms anonymity, betweenness and consistency; the other involving faithfulness, consistency and ½-Condorcet. To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here non-trivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout.An axiomatic approach to location functions on finite metric spaces
http://repub.eur.nl/pub/41360/
Mon, 09 Sep 2013 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>B. Novick</div><div>R.C. Powers</div>
A location function on a finite metric space (X, d) is a function on the set, X*, of all finite sequences of elements of X, to 2X\θ, which minimizes some criteria of remoteness. Axiomatic characterizations of these functions have, for the most part, been established only for very special cases. While McMorris, Mulder and Powers [F.R. McMorris, H.M. Mulder, R.C. Powers, "The median function on median graphs and semilattices," Discrete Appl. Math., 101, (2000), 221-230] were able to characterize the median function on median graphs with three axioms, one of their axioms was very specific to the structure of median graphs. Recently, however, Mulder and Novick [H.M. Mulder, B.A. Novick, "A tight axiomatization of the median procedure on median graphs," Discrete Appl. Math., 161, (2013), 838-846] characterized the median function for all median graphs using only three very natural axioms. These three axioms are meaningful in the more general context of finite metric spaces. In this work, we establish that these same three axioms are indeed independent and then we settle completely the question of interdependence among the collection of axioms involved in the above mentioned two characterizations, giving examples for all logically relevant cases. We introduce several new location functions and pose some questions. The ℓ
p-function on trees
http://repub.eur.nl/pub/34915/
Sat, 01 Sep 2012 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
A p-value of a sequence π = (x1, x2,⋯, xk) of elements of a finite metric space (X, d) is an element x for which ∑i=1kdp(x,xi) is minimum. The function ℓpwith domain the set of all finite sequences defined by ℓp(π) = {x: x is a p-value of π} is called the ℓp-function on X. The ℓp-functions with p = 1 and p = 2 are the well-studied median and mean functions respectively. In this article, the ℓp-function on finite trees is characterized axiomatically. The Lp-function on trees
http://repub.eur.nl/pub/20773/
Thu, 23 Sep 2010 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
Axiomatic characterization of the mean function on trees
http://repub.eur.nl/pub/37668/
Wed, 01 Sep 2010 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
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 note, the mean function on finite trees is characterized axiomatically.
Axiomatic Characterization of the Mean Function on Trees
http://repub.eur.nl/pub/18261/
Tue, 23 Feb 2010 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>O. Ortega</div>
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
http://repub.eur.nl/pub/19259/
Fri, 01 Dec 2006 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>R.C. Powers</div>
A median 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 M with domain the set of all finite sequences on X and defined by M(π)={x:x is a median of π} is called the median function on X, and is one of the most studied consensus functions. Based on previous characterizations of median sets M(π), 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.The t-median function on graphs
http://repub.eur.nl/pub/6916/
Tue, 23 Aug 2005 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>R.C. Powers</div>
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.Graphs with only caterpillars as spanning trees
http://repub.eur.nl/pub/72992/
Tue, 28 Oct 2003 00:00:01 GMT<div>R.E. Jamison</div><div>F.R. McMorris</div><div>H.M. Mulder</div>
A connected graph G is caterpillar-pure if each spanning tree of G is a caterpillar. The caterpillar-pure graphs are fully characterized. Loosely speaking they are strings or necklaces of so-called pearls, except for a number of small exceptional cases. An upper bound for the number of edges in terms of the order is given for caterpillar-pure graphs, and those which attain the upper bound are characterized.The median function on distributive semilattices
http://repub.eur.nl/pub/66679/
Tue, 15 Apr 2003 00:00:01 GMT<div>F.R. McMorris</div><div>H.M. Mulder</div><div>R.C. Powers</div>
A median of a k-tuple π=(x 1,...,x k) of elements of a finite metric space (X,d) is an element x for which ∑ i=1 kd(x,x i) is minimum. The function m with domain the set of all k-tuples with k0 and defined by m(π)={x: x is a median of π} is called the median function on X. Continuing with the program of characterizing m on various metric spaces, this paper presents a characterization of the median function on distributive semilattices endowed with the standard lattice metric.