Loana Tito Nogueira

On the hardness of the minimum height decision tree problem

Given a set of objects O and a set of tests T, the abstract decision tree problem (DTP) is to construct a tree with minimum height that completely identifies the objects of O, by using the tests of T. No algorithm with a good approximation ratio is known to solve this problem. We give a theoretical support for this fact by showing that DTP does not admit an o(log n)-approximation algorithm unless P = NP.

Characterizing (k,l)–partitionable Cographs

Abstract We consider the problem of partitioning a graph into k independent sets and l cliques, known as the ( k , l ) -partition problem, which was introduced by Brandstadt in [A. Bransdstadt, Partitions of graphs into one or two independent sets and cliques, Discrete Mathematics 152 (1996) 47–54], and generalized by Feder et al. in [T. Feder, P. Hell, S. Klein, and R. Motwani, Complexity of graph partition problems, in: Thirty First Annual ACM Symposium on Theory of Computing (1999), Plenum Press, New York, 1999, 464–472] as the M-partition problem. Brandstadt proved that given a graph G, it is NP-complete to decide if G is a ( k , l ) -graph for k ≥ 3 or l ≥ 3 . Since then, a lot of work have been done in order to solve the ( k , l ) -partition problem for many subclasses of perfect graphs. In this work, we consider a subclass of perfect graphs: the cographs, which correspond to graphs without paths with size 4. More precisely, we provide a characterization of cographs that are ( k , l ) -graphs by forbidden configurations, that is, a cograph G is a ( k , l ) -graph if and only if it does not contain any of the forbbiden configurations.

Packing r-Cliques in Weighted Chordal Graphs

In 1, we have previously observed that, in a chordal graph G, the maximum number of independent r-cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr, with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r-cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r-clique in G, we seek a set of independent r-cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r-cliques have the same weight, we simplify the algorithms reported in 1, although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.

Cycle transversals in perfect graphs and cographs

A cycle transversal (or feedback vertex set) of a graph G is a subset T@?V(G) such that T@?V(C) 0@? for every cycle C of G. This work considers the problem of finding special cycle transversals in perfect graphs and cographs. We prove that finding a minimum cycle transversal T in a perfect graph G is NP-hard, even for bipartite graphs with maximum degree four. Since G-T is acyclic, this result implies that finding a maximum acyclic induced subgraph of a perfect graph is also NP-hard. Other special properties of T are considered. A clique (stable, respectively) cycle transversal, or simply cct (sct, respectively) is a cycle transversal which is a clique (stable set, respectively). Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known, even for perfect graphs. We characterize cographs with cct in terms of forbidden induced subgraphs and describe their structure. This leads to linear time recognition of cographs with cct. We also prove that deciding whether a perfect graph admits an sct is NP-complete. We characterize cographs with sct in terms of forbidden induced subgraphs; this characterization also leads to linear time recognition.

Characterization and recognition of P 4 -sparse graphs partitionable into k independent sets and ℓ cliques

In this work, we focus on the class of P 4 -sparse graphs, which generalizes the well-known class of cographs. We consider the problem of verifying whether a P 4 -sparse graph is a ( k , ? ) -graph, that is, a graph that can be partitioned into k independent sets and ? cliques. First, we describe in detail the family of forbidden induced subgraphs for a cograph to be a ( k , ? ) -graph. Next, we show that the same forbidden structures suffice to characterize P 4 -sparse graphs which are ( k , ? ) -graphs. Finally, we describe how to recognize ( k , ? ) - P 4 -sparse graphs in linear time by using special auxiliary cographs.

Solving the Discretizable Molecular Distance Geometry Problem by Multiple Realization Trees

The discretizable molecular distance geometry problem (DMDGP) is a subclass of the MDGP, where instances can be solved using a discrete algorithm called branch-and-prune (BP). We present an initial study showing that the BP algorithm can be used differently from its original form, where the initial atoms are fixed and the branches of the BP tree are generated until the last atom is reached. Particularly, we show that the use of multiple BP trees may explore the search space faster than the original BP.

On (k,ℓ)-Graph Sandwich Problems

In this work we consider the Golumbic, Kaplan and Shamir graph sandwich decision problem for property Π, where given two graphs G 1 = (V,E 1) and G 2 = (V,E 2), the question is to know whether there exists a graph G = (V,E) such that E 1 ⊆ E ⊆ E 2 and G satisfies property Π. The main property Π we are interested in this paper is “being a (k,l)-graph”. We say that a graph G = (V,E) is (k,l) if there is a partition of the vertex set of G into at most k independent sets and at most l cliques. We prove that the strongly chordal-(2,l) graph sandwich problem is NP-complete, for l ≥ 1, and that the chordal-(k,l) graph sandwich problem is NP-complete, for k ≥ 2 , l ≥ 1. We also introduce in this paper a new work proposal related to graph sandwich problems: the graph sandwich problem with boundary conditions. Our goal is to redefine well-known NP-complete graph sandwich problems by cleverly assigning properties to its input graphs so that the redefined problems are polynomially solvable. Let poly-color(k) denote an infinite family of graphs G for which deciding whether G is k-colorable can be done in O(p(n)) time, where p is a polynomial and n = |V(G)|. In order to illustrate how boundary conditions can change the complexity status of a graph sandwich problem, we present here a polynomial-time solution for the (k,l)-graph sandwich problem for all k,l, when beforehand we know that G 1 belongs to poly-color(k) and G 2 has a polynomial number maximal of cliques.

A note on the size of minimal covers

For the class of monotone boolean functions f:{0,1}^n->{0,1} where all 1-certificates have size 2, we prove the tight bound n=<(@l+2)^2/4, where @l is the size of the largest 0-certificate of f. This result can be translated to graph language as follows: for every graph G=(V,E) the inequality |V|=<(@l+2)^2/4 holds, where @l is the size of the largest minimal vertex cover of G. In addition, there are infinitely many graphs for which this inequality is tight.

Clique cycle transversals in distance-hereditary graphs

Abstract A cycle transversal of a graph G is a subset T ⊆ V ( G ) such that T ∩ V ( C ) ≠ ∅ for every cycle C of G. A clique cycle transversal, or cct for short, is a cycle transversal which is a clique. Recognizing graphs which admit a cct can be done in polynomial time; however, no structural characterization of such graphs is known. We characterize distance-hereditary graphs admitting a cct in terms of forbidden induced subgraphs. This extends similar results for chordal graphs and cographs.

List Partitions of Chordal Graphs

In an earlier paper we gave efficient algorithms for partitioning chordal graphs into k independent sets and ell cliques. This is a natural generalization of the problem of recognizing split graphs, and is NP-complete for graphs in general, unless k ≤ 2 and ell ≤ 2. (Split graphs have k = ell = 1.)