Cristina G. Fernandes

Motif Search in Graphs: Application to Metabolic Networks

The classic view of metabolism as a collection of metabolic pathways is being questioned with the currently available possibility of studying whole networks. Novel ways of decomposing the network into modules and motifs that could be considered as the building blocks of a network are being suggested. In this work, we introduce a new definition of motif in the context of metabolic networks. Unlike in previous works on (other) biochemical networks, this definition is not based only on topological features. We propose instead to use an alternative definition based on the functional nature of the components that form the motif, which we call a reaction motif. After introducing a formal framework motivated by biological considerations, we present complexity results on the problem of searching for all occurrences of a reaction motif in a network and introduce an algorithm that is fast in practice in most situations. We then show an initial application to the study of pathway evolution. Finally, we give some general features of the observed number of occurrences in order to highlight some structural features of metabolic networks

Multicuts in unweighted graphs and digraphs with bounded degree and bounded tree-width

The Multicut problem can be defined as: given a graph G and a collection of pairs of distinct vertices {si, ti} of G, find a minimum set of edges of G whose removal disconnects each si from the corresponding ti. Multicut is known to be NP-hard and Max SNP-hard even when the input graph is restricted to being a tree. The main result of the paper is a polynomial-time approximation scheme (PTAS) for Multicut in unweighted graphs with bounded degree and bounded tree-width. That is, for any e > 0, we present a polynomial-time (1 + e)-approximation algorithm. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provide some hardness results: we prove that Multicut is still NP-hard for binary trees and that it is Max SNP-hard if we drop any of the three conditions (unweighted, bounded-degree, bounded tree-width). Finally we show that some of these results extend to the vertex version of Multicut and to a directed version of Multicut.

A better approximation algorithm for finding planar subgraphs

The MAXIMUM PLANAR SUBGRAPH problem?given a graphG, find a largest planar subgraph ofG?has applications in circuit layout, facility layout, and graph drawing. No previous polynomial-time approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1/3, which is achieved simply by producing a spanning tree ofG. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4/9 instead of 1/3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

A Better Approximation Ratio for the Minimum Sizek-Edge-Connected Spanning Subgraph Problem

Consider the minimum sizek-edge-connected spanning subgraph problem: given a positive integerkand ak-edge-connected graphG, find ak-edge-connected spanning subgraph ofGwith the minimum number of edges. This problem is known to be NP-complete. Khuller and Raghavachari presented the first algorithm which, for allk, achieves a performance ratio smaller than a constant which is less than two. They proved an upper bound of 1.85 for the performance ratio of their algorithm. Currently, the best known performance ratio for the problem is 1+2/(k+1), achieved by a slower algorithm of Cheriyan and Thurimella. In this article, we improve Khuller and Raghavacharis analysis, proving that the performance ratio of their algorithm is smaller than 1.7 for large enoughk, and that it is at most 1.75 for allk. Second, we show that the minimum size 2-edge-connected spanning subgraph problem is MAX SNP-hard.

Primal-dual approximation algorithms for the Prize-Collecting Steiner Tree Problem

The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2-1/(n-1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n^3logn) time-it applies the primal-dual scheme once for each of the n vertices of the graph. We present a primal-dual algorithm that runs in O(n^2logn), as it applies this scheme only once, and achieves the slightly better ratio of (2-2/n). We also show a tight example for the analysis of the algorithm and discuss briefly a couple of other algorithms described in the literature.

Repetition-free longest common subsequence

We study the following problem. Given two sequences x and y over a finite alphabet, find a repetition-free longest common subsequence of x and y. We show several algorithmic results, a computational complexity result, and we describe a preliminary experimental study based on the proposed algorithms. We also show that this problem is APX-hard.

Intersecting longest paths

Abstract In 1966, Gallai asked whether every connected graph has a vertex that is common to all longest paths. The answer to this question is negative. We prove that the answer is positive for outerplanar graphs and 2-trees. Another related question was raised by Zamfirescu in the 1980s: Do any three longest paths in a connected graph have a vertex in common? The answer to this question is unknown. We prove that for connected graphs in which all nontrivial blocks are Hamiltonian the answer is affirmative. Finally, we state a conjecture and explain how it relates to the three longest paths question.

A 5/3-approximation for finding spanning trees with many leaves in cubic graphs

For a connected graph G, let L(G) denote the maximum number of leaves in a spanning tree in G. The problem of computing L(G) is known to be NP-hard even for cubic graphs. We improve on Lorys and Zwozniaks result presenting a 5/3-approximation for this problem on cubic graphs. This result is a consequence of new lower and upper bounds for L(G) which are interesting on their own. We also show a lower bound for L(G) that holds for graphs with minimum degree at least 3.

Reaction motifs in metabolic networks

The classic view of metabolism as a collection of metabolic pathways is being questioned with the currently available possibility of studying whole networks. Novel ways of decomposing the network into modules and motifs that could be considered as the building blocks of a network are being suggested. In this work, we introduce a new definition of motif in the context of metabolic networks. Unlike in previous works on (other) biochemical networks, this definition is not based only on topological features. We propose instead to use an alternative definition based on the functional nature of the components that form the motif. After introducing a formal framework motivated by biological considerations, we present complexity results on the problem of searching for all occurrences of a reaction motif in a network, and introduce an algorithm that is fast in practice in most situations. We then show an initial application to the study of pathway evolution.

Multicuts in Unweighted Graphs with Bounded Degree and Bounded Tree-Width

The Multicut problem is defined as follows: given a graph G and a collection of pairs of distinct vertices (s i; t i) of G, find a small- est set of edges of G whose removal disconnects each s i from the corre- sponding t i. Our main result is a polynomial-time approximation scheme for Multicut in unweighted graphs with bounded degree and bounded tree-width: for any ∈ > 0, we presented a polynomial-time algorithm with performance ratio at most 1 + ∈. In the particular case when the input is a bounded-degree tree, we have a linear-time implementation of the algorithm. We also provided some hardness results. We proved that Multicut is still NP-hard for binary trees and that, unless P = NP, no polynomial-time approximation scheme exists if we drop any of the the three conditions: unweighted, bounded-degree, bounded-tree-width. Some of these results extend to the vertex version of Multicut.