Preference aggregation and DEA: An analysis of the methods proposed to discriminate efficient candidates

There are different ways to allow the voters to express their preferences on a set of candidates. In ranked voting systems, each voter selects a subset of the candidates and ranks them in order of preference. A well-known class of these voting systems are scoring rules, where fixed scores are assigned to the different ranks and the candidates with the highest score are the winners. One of the most important issues in this context is the choice of the scoring vector, since the winning candidate can vary according to the scores used. To avoid this problem, Cook and Kress [W.D. Cook, M. Kress, A data envelopment model for aggregating preference rankings, Management Science 36 (11) (1990) 1302–1310], using a DEA/AR model, proposed to assess each candidate with the most favorable scoring vector for him/her. However, the use of this procedure often causes several candidates to be efficient, i.e., they achieve the maximum score. For this reason, several methods to discriminate among efficient candidates have been proposed. The aim of this paper is to analyze and show some drawbacks of these methods.     

Preference voting and project ranking using DEA and cross-evaluation

Cook and Kress (1990), using Data Envelopment Analysis (DEA) as their starting point, proposed a procedure to rank order the candidates in a preferential election. Notionally, each candidate is permitted to choose the most favourable weights to be applied to his/her standings (first place, second place, etc. votes) in the usual DEA manner with the additional ‘assurance region’ restriction that the weight for a j place vote should be more than that for a j +1 amount. We consider that this freedom to choose weights is essentially illusory when maximum discrimination between the candidates is sought, in which case the weights used to evaluate and rank the candidates are as if imposed externally at the outset. To avoid this, we present an alternative procedure which retains Cook and Kress' central idea but where, as well as using each candidate's rating of him/herself, we now make use of each candidate's ratings of all the candidates. We regard the so-called cross-evaluation matrix as the summary of a self- and peer-rating process in which the candidates seek to interpret the voters preferences as favourably for themselves, relative to the other candidates, as possible. The problem then becomes one of establishing an overall rating for each candidate from these individual ratings. For this, for each candidate, we use a weighted average of all the candidates ratings of that candidate, where the weights themselves are in proportion to each candidate's overall rating. The overall ratings are therefore proportional to the components of the principal (left-hand) eigenvector of the cross-evaluation matrix. These ideas are then applied to the selection of R & D projects to comprise an R & D program, thus indicating thier wider applicability.     

Sums and k-sums in abelian groups of order k

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a₁,…,ar that does not have 0 as a k-sum is attained at the sequence b₁,…,br−k+1,0,…,0, where b₁,…,br−k+1 is chosen to minimise the number of sums without 0 being a sum. Equivalently, to minimise the number of k-sums one should repeat some value k−1 times. This proves a conjecture of Bollobás and Leader, and extends results of Gao and of Bollobás and Leader.     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Arbitrarily vertex decomposable suns with few rays

A graph G of order n is called arbitrarily vertex decomposable if for each sequence (n₁,…,n_k) of positive integers with n₁+?+n_k=n, there exists a partition (V₁,…,V_k) of the vertex set of G such that V_i induces a connected subgraph of order n_i, for all i=1,…,k. A sun with r rays is a unicyclic graph obtained by adding r hanging edges to r distinct vertices of a cycle. We characterize all arbitrarily vertex decomposable suns with at most three rays. We also provide a list of all on-line arbitrarily vertex decomposable suns with any number of rays.     

Steiner intervals, geodesic intervals, and betweenness

The concept of the k-Steiner interval is a natural generalization of the geodesic (binary) interval. It is defined as a mapping S:V×?×V?2^V such that S(u₁,…,u_k) consists of all vertices in G that lie on some Steiner tree with respect to a multiset W={u₁,…,u_k} of vertices from G. In this paper we obtain, for each k, a characterization of the class of graphs in which every k-Steiner interval S has the so-called union property, which says that S(u₁,…,u_k) coincides with the union of geodesic intervals I(u_i,u_j) between all pairs from W. It turns out that, as soon as k>3, this class coincides with the class of graphs in which the k-Steiner interval enjoys the monotone axiom (m), respectively (b2) axiom, the conditions from betweenness theory. Notably, S satisfies (m), if x₁,…,x_k∈S(u₁,…,u_k) implies S(x₁,…,x_k)⊆S(u₁,…,u_k), and S satisfies (b2) if x∈S(u₁,u₂,…,u_k) implies S(x,u₂,…,u_k)⊆S(u₁,…,u_k). In the case k=3, these three classes are different, and we give structural characterizations of graphs for which their Steiner interval S satisfies the union property as well as the monotone axiom (m). We also prove several partial observations on the class of graphs in which the 3-Steiner interval satisfies (b2), which lead to the conjecture that these are precisely the graphs in which every block is a geodetic graph with diameter at most two.     

On the total positivity of restricted Stirling numbers

This note shows that the matrix whose (n,k) entry is the number of set partitions of {1,…,n} into k blocks with size at most m is never totally positive for m≥3; thus answering a question posed in [H. Han, S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, European J. Combin. 29 (2008) 1544-1554].     

Tournament immersion and cutwidth

A (loopless) digraph H is strongly immersed in a digraph G if the vertices of H are mapped to distinct vertices of G, and the edges of H are mapped to directed paths joining the corresponding pairs of vertices of G, in such a way that the paths used are pairwise edge-disjoint, and do not pass through vertices of G that are images of vertices of H. A digraph has cutwidth at most k if its vertices can be ordered {v₁,…,vn} in such a way that for each j, there are at most k edges uv such that u∈{v₁,…,vj−1} and v∈{vj,…,vn}.We prove that for every set S of tournaments, the following are equivalent:

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there is a digraph H such that H cannot be strongly immersed in any member of S,

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there exists k such that every member of S has cutwidth at most k,

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there exists k such that every vertex of every member of S belongs to at most k edge-disjoint directed cycles.

This is a key lemma towards two results that will be presented in later papers: first, that strong immersion is a well-quasi-order for tournaments, and second, that there is a polynomial time algorithm for the k edge-disjoint directed paths problem (for fixed k) in a tournament.     

Maximum vertex and face degree of oblique graphs

Let G=(V,E,F) be a 3-connected simple graph imbedded into a surface S with vertex set V, edge set E and face set F. A face α is an 〈a₁,a₂,…,a_k〉-face if α is a k-gon and the degrees of the vertices incident with α in the cyclic order are a₁,a₂,…,a_k. The lexicographic minimum 〈b₁,b₂,…,b_k〉 such that α is a 〈b₁,b₂,…,b_k〉-face is called the type of α.Let z be an integer. We consider z-oblique graphs, i.e. such graphs that the number of faces of each type is at most z. We show an upper bound for the maximum vertex degree of any z-oblique graph imbedded into a given surface. Moreover, an upper bound for the maximum face degree is presented. We also show that there are only finitely many oblique graphs imbedded into non-orientable surfaces.     

Two equivalent measures on weighted hypergraphs

Let H=(N,E,w) be a hypergraph with a node set N={0,1,…,n-1}, a hyperedge set E⊆2^N, and real edge-weights w(e) for e∈E. Given a convex n-gon P in the plane with vertices x₀,x₁,…,x_n-1 which are arranged in this order clockwisely, let each node i∈N correspond to the vertex x_i and define the area A_P(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0?i<j<k?n-1, a convex three-cut C(i,j,k) of N is {{i,…,j-1}, {j,…,k-1}, {k,…,n-1,0,…,i-1}} and its size c_H(i,j,k) in H is defined as the sum of weights of edges e∈E such that e contains at least one node from each of {i,…,j-1}, {j,…,k-1} and {k,…,n-1,0,…,i-1}. We show that the following two conditions are equivalent:

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A_P(H)?A_P(H^′) for all convex n-gons P.

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c_H(i,j,k)?c_H^′(i,j,k) for all convex three-cuts C(i,j,k).

From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H^′ satisfy “A_P(H)?A_P(H^′) for all convex n-gons P” is immediately obtained.     

On chains and Sperner k-families in ranked posets,II

A ranked poset P satisfies condition S if for all k the set of elements of the k largest ranks in P is a Sperner k-family. It satisfies condition T if for all k there exist disjoint chains in P which each meet the k largest ranks and which cover the kth largest rank. Griggs employed the theory of saturated partitions to prove that if P satisfies S it also satisfies T, and that the converse is true for posets with unimodal Whitney numbers. Here we present new proofs of these facts which do not require the existence of saturated partitions. The first result is proven with a simple network flow argument and the second is proven directly.     

On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture

We address conjectures of P. Erd?s and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpiński number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k,k²,k³,…,kr are simultaneously Sierpiński numbers.     

Testing the equality of several covariance matrices with fewer observations than the dimension

For normally distributed data from the k populations with m×m covariance matrices Σ₁,…,Σ_k, we test the hypothesis H:Σ₁=?=Σ_k vs the alternative A≠H when the number of observations N_i, i=1,…,k from each population are less than or equal to the dimension m, N_i≤m, i=1,…,k. Two tests are proposed and compared with two other tests proposed in the literature. These tests, however, do not require that N_i≤m, and thus can be used in all situations, including when the likelihood ratio test is available. The asymptotic distributions of the test statistics are given, and the power compared by simulations with other test statistics proposed in the literature. The proposed tests perform well and better in several cases than the other two tests available in the literature.     

On chains and Sperner k-families in ranked posets

A ranked poset P has the Sperner property if the sizes of the largest rank and of the largest antichain in P are equal. A natural strengthening of the Sperner property is condition S: For all k, the set of elements of the k largest ranks in P is a Sperner k-family. P satisfies condition T if for all k there exist disjoint chains in P each of which meets the k largest ranks and which covers the kth largest rank. It is proven here that if P satisfies S, it also satisfies T, and that the converse, although in general false, is true for posets with unimodal Whitney numbers. Conditions S and T and the Sperner property are compared here with two other conditions on posets concerning the existence of certain partitions of P into chains.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

An adaptive memory algorithm for the k-coloring problem

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. The adaptive memory algorithm is a hybrid evolutionary heuristic that uses a central memory. At each iteration, the information contained in the central memory is used for producing an offspring solution which is then possibly improved using a local search algorithm. The so obtained solution is finally used to update the central memory. We describe in this paper an adaptive memory algorithm for the k-coloring problem. Computational experiments give evidence that this new algorithm is competitive with, and simpler and more flexible than, the best known graph coloring algorithms.     

Preservers of pairs of bivectors with bounded distance

We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all n×n alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, rank(A-B)?2k if and only if rank(T(A)-T(B))?2k, where k is a fixed positive integer such that n?2k+2 and k?2.     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.     

Unextendible product bases and 1-factorization of complete graphs

Let f(k₁,…,k_m) be the minimal value of size of all possible unextendible product bases in the tensor product space . We have trivial lower bounds and upper bound k₁?k_m. Alon and Lovász determined all cases such that f(k₁,…,k_m)=n(k₁,…,k_m). In this paper we determine all cases such that f(k₁,…,k_m)=k₁?k_m by presenting a sharper upper bound. We also determine several cases such that f(k₁,…,k_m)=n(k₁,…,k_m)+1 by using a result on 1-factorization of complete graphs.     

The Eulerian distribution on involutions is indeed unimodal

Let In,k (respectively Jn,k) be the number of involutions (respectively fixed-point free involutions) of {1,…,n} with k descents. Motivated by Brenti's conjecture which states that the sequence In,0,In,1,…,In,n−1 is log-concave, we prove that the two sequences In,k and J2n,k are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers an,k such that