11 research outputs found
Computing Dense and Sparse Subgraphs of Weakly Closed Graphs
A graph is weakly -closed if every induced subgraph of
contains one vertex such that for each non-neighbor of it holds
that . The weak closure of a graph,
recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number
such that is weakly -closed. This graph parameter is never larger
than the degeneracy (plus one) and can be significantly smaller. Extending the
work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that
several problems related to finding dense subgraphs, such as the enumeration of
bicliques and -plexes, are fixed-parameter tractable with respect to
. Moreover, we show that the problem of determining whether a weakly
-closed graph has a subgraph on at least vertices that belongs
to a graph class which is closed under taking subgraphs admits a
kernel with at most vertices. Finally, we provide fixed-parameter
algorithms for Independent Dominating Set and Dominating Clique when
parameterized by where is the solution size.Comment: Appeared in ISAAC '2
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
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum