Binary matrix factorisations under Boolean arithmetic

For a binary matrix X, the Boolean rank br(X) is the smallest integer for which X can be factorised into the Boolean matrix product of two binary matrices A and B with inner dimension br(X). The isolation number i(X) of X is the maximum number of 1s no two of which are in a same row, column or a 2 x 2 submatrix of all 1s. In Part I. of this thesis, we continue Anna Lubiw's study of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds for all its submatrices. We show that the stronger concept of superfirmness of X is equivalent to having no odd holes in the rectangle cover graph of X, the graph in which br(X) and i(X) translate to the clique cover number and the independence number, respectively. A binary matrix is minimally non-firm if it is not firm but all of its proper submatrices are. We introduce a matrix operation that leads to generalised binary matrices and, under some conditions, preserves firmness and superfirmness. Then we use this matrix operation to derive several infinite families of minimally non-firm matrices. To the best of our knowledge, minimally non-firm matrices have not been studied before and our constructions provide the first infinite families of them. In Part II. of this thesis, we explore rank-k binary matrix factorisation (k-BMF). In k-BMF, we are given an m x n binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension m x k and k x n respectively, which minimise the distance between X and the Boolean matrix product of A and B in the squared Frobenius norm. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times a zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations

Algorithms and hardness results for the jump number problem, the joint replenishment problem, and the optimal clustering of frequency-constrained maintenance jobs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 107-110).In the first part of this thesis we present a new, geometric interpretation of the jump number problem on 2-dimensional 2-colorable (2D2C) partial order. We show that the jump number of a 2D2C poset is equivalent to the maximum cardinality of an independent set in a properly defined collection of rectangles in the plane. We then model the geometric problem as a linear program. Even though the underlying polytope may not be integral, we show that one can always find an integral optimal solution. Inspired by this result and by previous work of A. Frank, T. Jordan and L. Vegh [13, 14, 15] on set-pairs, we derive an efficient combinatorial algorithm to find the maximum independent set and its dual, the minimum hitting set, in polynomial time. The combinatorial algorithm solves the jump number problem on convex posets (a subclass of 2D2C posets) significantly faster than current methods. If n is the number of nodes in the partial order, our algorithm runs in 0((n log n)2.5) time, while previous algorithms ran in at least 0(n9 ) time. In the second part, we present a novel connection between certain sequencing problems that involve the coordination of activities and the problem of factorizing integer numbers. We use this connection to derive hardness results for three different problems: -- The Joint Replenishment Problem with General Integer Policies. -- The Joint Replenishment Problem with Correction Factor. -- The Problem of Optimal Clustering of Frequency-Constrained Maintenance Jobs. Our hardness results do not follow from a standard type of reduction (e.g., we do not prove NP-hardness), and imply that no polynomial-time algorithm exists for the problems above, unless Integer Factorization is solvable in polynomial time..by Claudio Telha Cornejo.Ph.D

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D

Finding Minimum Generators of Path-Systems

A simple algorithmic proof of a min-max theorem of E. Gyori on generators of path systems is described. The algorithm relies on Dilworth's theorem. 1. INTRODUCTION Let P = (v 0 ; j 1 ; v 1 ; j 2 ; v 2 ; : : : ; j n ; v n ) be a simple directed path where each directed edge j h has tail v h\Gamma1 and head v h . Let V := fv 0 ; : : : ; v ng and E := fj 1 ; : : : ; j n g denote the node-set and edge-set of P , respectively. Path P defines an ordering of the elements of V in which v h ! v k if 0 h ! k n. In this case we will say that node v h precedes node v k and that edge j h precedes edge j k : Let E := fuv : u; v 2 V; u ! vg be the set of all directed edges whose tail u precedes its head v. For a subpath J of P , let f(J) and l(J) denote the first and last nodes of J , respectively. Also, for a (directed) edge e = uv 2 E let f(e) := u and l(e) := v. Let P be a system of distinct subpaths of P . We use the convention that the edge-set of P i will be denoted by the same letter ..