Set covering with almost consecutive ones property

In this paper, we consider set covering problems with a coefficient matrix almost having the consecutive ones property, i.e., in most rows of the coefficient matrix, the ones appear consecutively and only a few blocks of consecutive ones appear in the remaining rows. If this property holds for all rows it is well known that the set covering problem can be solved efficiently. For our case of almost consecutive ones we present a reformulation exploiting the consecutive ones structure to develop bounds and a branching scheme. Our approach has been tested on real-world data as well as on theoretical problem instances.     

Hamiltonicity of regular graphs and blocks of consecutive ones in symmetric matrices

We show that the Hamiltonicity of a regular graph G can be fully characterized by the numbers of blocks of consecutive ones in the binary matrix A+I, where A is the adjacency matrix of G, I is the unit matrix, and the blocks can be either linear or circular. Concretely, a k-regular graph G with girth g(G)?5 has a Hamiltonian circuit if and only if the matrix A+I can be permuted on rows such that each column has at most (or exactly) k-1 circular blocks of consecutive ones; and if the graph G is k-regular except for two (k-1)-degree vertices a and b, then there is a Hamiltonian path from a to b if and only if the matrix A+I can be permuted on rows to have at most (or exactly) k-1 linear blocks per column.Then we turn to the problem of determining whether a given matrix can have at most k blocks of consecutive ones per column by some row permutation. For this problem, Booth and Lueker gave a linear algorithm for k=1 [Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, 1975, pp. 255-265]; Flammini et al. showed its NP-completeness for general k [Algorithmica 16 (1996) 549-568]; and Goldberg et al. proved the same for every fixed k?2 [J. Comput. Biol. 2 (1) (1995) 139-152]. In this paper, we strengthen their result by proving that the problem remains NP-complete for every constant k?2 even if the matrix is restricted to (1) symmetric, or (2) having at most three blocks per row.     

Minimising the number of gap-zeros in binary matrices

We study a problem of minimising the total number of zeros in the gaps between blocks of consecutive ones in the columns of a binary matrix by permuting its rows. The problem is referred to as the Consecutive Ones Matrix Augmentation Problem, and is known to be NP-hard. An analysis of the structure of an optimal solution allows us to focus on a restricted solution space, and to use an implicit representation for searching the space. We develop an exact solution algorithm, which is linear-time in the number of rows if the number of columns is constant, and two constructive heuristics to tackle instances with an arbitrary number of columns. The heuristics use a novel solution representation based upon row sequencing. In our computational study, all heuristic solutions are either optimal or close to an optimum. One of the heuristics is particularly effective, especially for problems with a large number of rows.     

The weighted consecutive ones problem for a fixed number of rows or columns

The NP-hard weighted consecutive ones problem consists of converting a given 0/1-matrix to a consecutive ones matrix by switching entries with a minimum total cost. Here we consider this problem when the number of rows or columns is fixed. We show that in this case the problem can be solved in polynomial time.     

Possible numbers of ones in 0-1 matrices with a given rank

We determine the possible numbers of ones in a 0-1 matrix with given rank in the generic case and in the symmetric case. There are some unexpected phenomena. The rank 2 symmetric case is subtle.     

A characterization of triangle-free tolerance graphs

We prove that a triangle-free graph G is a tolerance graph if and only if there exists a set of consecutively ordered stars that partition the edges of G. Since tolerance graphs are weakly chordal, a tolerance graph is bipartite if and only if it is triangle-free. We, therefore, characterize those tolerance graphs that are also bipartite. We use this result to show that in general, the class of interval bigraphs properly contains tolerance graphs that are triangle-free (and hence bipartite).     

On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries

We consider the set Σ(R,C) of all m×n matrices having 0-1 entries and prescribed row sums R=(r₁,…,rm) and column sums C=(c₁,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions.     

A structural characterization of real k-potent matrices

Some well-known characterizations of nonnegative k-potent matrices have been obtained by Flor [P. Flor, On groups of nonnegative matrices, Compositio Math. 21 (1969), pp. 376–382.] and Jeter and Pye [M. Jeter and W. Pye, Nonnegative (s,?t)-potent matrices, Linear Algebra Appl. 45 (1982), pp. 109–121.]. In this article, we obtain a structural characterization of a real k-potent matrix A, provided that (sgn(A)) ^k+1 is unambiguously defined, regardless of whether A is nonnegative or not.     

A characterization of even doubly-stochastic matrices

Even doubly-stochastic matrices are characterized with the aid of the minima of functionals defined by the even diagonals contained in the matrix.     

A new efficient approach to the characterization of D‐stable matrices

The concept of D‐stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D‐stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D‐stable matrix was solved for low‐order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D‐stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.     

On Balanced Graphs

Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are characterized by 0–1 matrices and can be recognized in polynomial time. Furthermore, we propose polynomial time combinatorial algorithms for the problems of stable set, clique-independent set and clique-transversal for one of these subclasses of balanced graphs. Finally, we analyse the behavior of balanced graphs and these four subclasses under the clique graph operator. Received: April, 2004     

A new infinite family of minimally nonideal matrices

Minimally nonideal matrices are a key to understanding when the set covering problem can be solved using linear programming. The complete classification of minimally nonideal matrices is an open problem. One of the most important results on these matrices comes from a theorem of Lehman, which gives a property of the core of a minimally nonideal matrix. Cornuéjols and Novick gave a conjecture on the possible cores of minimally nonideal matrices. This paper disproves their conjecture by constructing a new infinite family of square minimally nonideal matrices. In particular, we show that there exists a minimally nonideal matrix with r ones in each row and column for any r?3.     

‘Duality’ of the Nikodym property and the Hahn property: Densities defined by sequences of matrices

The authors investigated in Boos and Leiger (2008) [5] the ‘duality’ of the Nikodym property (NP) of the set of all null sets of the density defined by any nonnegative matrix and the Hahn property (HP) of the strong null domain of it. In this paper, the investigation of the intimated duality is continued by considering densities defined by sequences of nonnegative matrices. These considerations are motivated by the known result that the ideal of the null sets of the uniform density has NP. In this context the general notion of S-convergence of double sequences (cf. Drewnowski, 2002 [8]) containing Pringsheim convergence, Hardy convergence and uniform convergence of double sequences is used.     

Constructions of infinite graphs with Ramsey property

For every infinite cardinal λ and 2 ≤ n < ω there is a directed graph D of size λ such that D does not contain directed circuits of length ≤n and if its vertices are colored with <λ colors, then there is a monochromatic directed circuit of length n + 1. For every infinite cardinal λ and finite graph X there is a λ-sized graph Y such that if the vertices of Y are colored with <λ colors, then there is a monochromatic induced copy of X. Further, Y does not contain larger cliques or shorter odd circuits than X. The constructions are using variants of Specker-type graphs.     

On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B₁,B₂ of a certain ten element matroid. Given that B₁,B₂ are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Testing balancedness and perfection of linear matrices

A (0, 1) matrix is linear if it does not contain a 2×2 submatrix of all ones. In this paper we give polynomial algorithms to test whether a linear matrix is balanced or perfect. The algorithms are based on decomposition results previously obtained by the authors.Partial support under NSF Grants DMS 8606188, DDM8800281 and DDM9001705.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).