The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure

We present a new polynomially solvable case of the Quadratic Assignment Problem in Koopmans-Beckman form $QAP(A,B)$, by showing that the identity permutation is optimal when $A$ and $B$ are respectively a Robinson similarity and dissimilarity matrix and one of $A$ or $B$ is a Toeplitz matrix. A Robinson (dis)similarity matrix is a symmetric matrix whose entries (increase) decrease monotonically along rows and columns when moving away from the diagonal, and such matrices arise in the classical seriation problem.Comment: 15 pages, 2 figure

Combinatorial algorithms for the seriation problem

In this thesis we study the seriation problem, a combinatorial problem arising in data analysis, which asks to sequence a set of objects in such a way that similar objects are ordered close to each other. We focus on the combinatorial structure and properties of Robinsonian matrices, a special class of structured matrices which best achieve the seriation goal. Our contribution is both theoretical and practical, with a particular emphasis on algorithms. In Chapter 2 we introduce basic concepts about graphs, permutations and proximity matrices used throughout the thesis. In Chapter 3 we present Robinsonian matrices, discussing their characterizations and recognition algorithms existing in the literature. In Chapter 4 we discuss Lexicographic Breadth-First search (Lex-BFS), a special graph traversal algorithm used in multisweep algorithms for the recognition of several classes of graphs. In Chapter 5 we introduce a new Lex-BFS based algorithm to recognize Robinsonian matrices, which is derived from a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. In Chapter 6 we introduce the novel Similarity-First Search algorithm (SFS), a weighted version of Lex-BFS which we use in a multisweep algorithm for the recognition of Robinsonian matrices. In Chapter 7 we model the seriation problem as an instance of Quadratic Assignment Problem (QAP) and we show that if the data has a Robinsonian structure, then one can find an optimal solution for QAP using a Robinsonian recognition algorithm. In Chapter 8 we discuss how to solve the seriation problem when the data does not have a Robinsonian structure, by finding a Robinsonian approximation of the original data. Finally, in Chapter 9 we discuss some experiments which we have carried out in order to compare the performance of the algorithms introduced in the thesis

A Lex-BFS-based recognition algorithm for Robinsonian matrices

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be re- ordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a non- negative symmetric n × n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n + m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A

A structural characterization for certifying Robinsonian matrices

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian

Entwurfskriterien fuer Mehrprozessorsysteme mit einem globalen Bus

SIGLECopy held by FIZ Karlsruhe; available from UB/TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Burnip: the implications for housing benefit and its adjudication

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be reordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a nonnegative symmetric n×n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n+m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A

A structural characterization for certifying Robinsonian matrices

A symmetric matrix is Robinsonian if its rows and columns can be simultaneously reordered in such a way that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. The adjacency matrix of a graph is Robinsonian precisely when the graph is a unit interval graph, so that Robinsonian matrices form a matrix analogue of the class of unit interval graphs. Here we provide a structural characterization for Robinsonian matrices in terms of forbidden substructures, extending the notion of asteroidal triples to weighted graphs. This implies the known characterization of unit interval graphs and leads to an efficient algorithm for certifying that a matrix is not Robinsonian