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

A min-max relation for the partial q- colourings of a graph. Part II: Box perfection

AbstractThis paper examines extensions of a min-max equality (stated in C Berge, Part I) for the maximum number of nodes in a perfect graph which can be q-coloured.A system L of linear inequalities in the variables x is called TDI if for every linear function cx such that c is all integers, the dual of the linear program: maximize {cx: x satisfies L} has an integer-valued optimum solution or no optimum solution. A system L is called box TDI if L together with any inequalities l⩽x⩽u is TDI. It is a corollary of work of Fulkerson and Lov́asz that: where A is a 0–1 matrix with no all-0 column and with the 1-columns of any row not a proper subset of the 1-columns of any other row, the system L(G) = {Ax⩽1, x⩾0} is TDI if and only if A is the matrix of maximal cliques (rows) versus nodes (columns) of a perfect graph. Here we will describe a class of graphs in a graph-theoretic way, and characterize them as the graphs G for which the system L(G) is box TDI. Thus we call these graphs box perfect. We also describe some classes of box perfect graphs

Structure Analysis of Some Generalizations of Matchings and Matroids under Algorithmic Aspects of Matchings and Matroids Under Algorithmic Aspects

Combinatorial optimization problems whose underlying structures are matchings or matroids are well-known to be solvable with efficient algorithms. Matroids can even be characterized by a simple greedy algorithm. In the first part of this thesis, some generalizations of matroids which allow the ground set to be partially ordered are considered. In particular, it will be shown that a special type of lattice polyhedra, for which Dietrich and Hoffman recently established a dual greedy algorithm, can be reduced to ordinary polymatroids. Moreover, strong exchange structures, Gauss greedoids and Delta-matroids will be extended from Boolean lattices to general distributive lattices, and the resulting structures will be characterized by certain greedy-type algorithms. While a matching of maximal size can be determined by a polynomial algorithm, the dual problem of finding a vertex cover of minimal size in general graphs is one of the hardest problems in combinatorial optimization. However, in case the graph belongs to the class of K\"onig-Egerv\'ary graphs, a maximum matching can be used to construct a minimum vertex cover. Lovasz and Korach characterized König-Egervary graphs by the exclusion of forbidden subgraphs. In the second part of this dissertation, the structure of König-Egervary graphs and the more general Red/Blue-split graphs will be analyzed. Red/Blue-split graphs have red and blue colored edges and the vertices of which can be split into two stable sets with respect to the red and blue edges, respectively. An algorithm that either determines a feasible partition of the vertices, or returns a red-blue colored subgraph (called ``flower'') characterizing non-Red/Blue-split graphs will be presented. This characterization allows the deduction of Lovasz and Korach's characterizations of König-Egerv\'ary graphs in case the red edges of the flower form a maximum matching. Furthermore, weighted Red/Blue-split graphs which model integrally solvable simple systems are introduced. A simple system is an inequality system where the sum of absolute values in each row of the integral matrix does not exceed the value two. A shortest-path algorithm and the presented Red/Blue-split algorithm will be used to find an integral solution of a simple system. These two algorithms lead to a characterization of weighted Red/Blue-split graphs by forbidden weighted subgraphs

On resolutions of ideals associated to subspace arrangements and the algebraic matroid of the determinantal variety

THIS THESIS DESCRIBES THE GRADED MINIMAL FREE RESOLUTION OF A PRODUCT OF IDEALS, EACH GENERATED BY LINEAR FORMS. IT ALSO STUDIES A PHENOMENON OF LINEARIZATION OF THE RESOLUTION OF AN ARBITRARY IDEAL, UPON MULTIPLICATION BY SUFFICIENTLY MANY IDEALS OF GENERIC POINTS IN PROJECTIVE SPACE. FURTHER, IT PROVIDES A CLASS OF BASE SETS OF THE ALGEBRAIC MATROID OF THE DETERMINANTAL VARIETY AND CONJECTURES THAT THESE COMPLETELY CHARACTERIZE THE MATROID. FINALLY, IT PROVIDES DETERMINANTAL CONDITIONS FOR HOMOMORPHIC SENSING, A PROBLEM THAT STUDIES THE UNIQUENESS OF IMAGES OF POINTS IN A VECTOR SUBSPACE UNDER A FINITE SET OF LINEAR TRANSFORMATIONS