4 research outputs found
Densities in graphs and matroids
Certain graphs can be described by the distribution of the edges in its subgraphs.
For example, a cycle C is a graph that satisfies |E(H)|
|V (H)| < |E(C)|
|V (C)| = 1 for all non-trivial
subgraphs of C. Similarly, a tree T is a graph that satisfies |E(H)|
|V (H)|−1 ≤ |E(T)|
|V (T)|−1 = 1
for all non-trivial subgraphs of T. In general, a balanced graph G is a graph such
that |E(H)|
|V (H)| ≤ |E(G)|
|V (G)| and a 1-balanced graph is a graph such that |E(H)|
|V (H)|−1 ≤ |E(G)|
|V (G)|−1
for all non-trivial subgraphs of G. Apart from these, for integers k and l, graphs G
that satisfy the property |E(H)| ≤ k|V (H)| − l for all non-trivial subgraphs H of G
play important roles in defining rigid structures.
This dissertation is a formal study of a class of density functions that extends the
above mentioned ideas. For a rational number r ≤ 1, a graph G is said to be r-balanced
if and only if for each non-trivial subgraph H of G, we have |E(H)|
|V (H)|−r ≤ |E(G)|
|V (G)|−r . For
r > 1, similar definitions are given. Weaker forms of r-balanced graphs are defined
and the existence of these graphs is discussed. We also define a class of vulnerability
measures on graphs similar to the edge-connectivity of graphs and show how it is
related to r-balanced graphs. All these definitions are matroidal and the definitions
of r-balanced matroids naturally extend the definitions of r-balanced graphs.
The vulnerability measures in graphs that we define are ranked and are lesser
than the edge-connectivity. Due to the relationship of the r-balanced graphs with
the vulnerability measures defined in the dissertation, identifying r-balanced graphs
and calculating the vulnerability measures in graphs prove to be useful in the area of network survivability. Relationships between the various classes of r-balanced
matroids and their weak forms are discussed. For r ∈ {0, 1}, we give a method to
construct big r-balanced graphs from small r-balanced graphs. This construction is a
generalization of the construction of Cartesian product of two graphs. We present an
algorithmic solution of the problem of transforming any given graph into a 1-balanced
graph on the same number of vertices and edges as the given graph. This result is
extended to a density function defined on the power set of any set E via a pair of
matroid rank functions defined on the power set of E. Many interesting results may
be derived in the future by choosing suitable pairs of matroid rank functions and
applying the above result
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