In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.

In this paper, we describe our algorithmic approach to constructing ordered restriction maps based on the data created from the images of population of individual DNA molecules (clones) digested by restriction enzymes. The goal is to devise map-making algorithms capable of producing high-resolution, high-accuracy maps rapidly and in a scalable manner. The resulting software is a key component of our optical mapping automation tools and has been used routinely to map cosmid, lambda and BAC clones. The experimental results appear highly promising. 1 Genomics and Optical Mapping Optical mapping [CAH+95, CJI+96, HRL+95, JRH+96, MBC+95, SCH+95, SLH+93, WHS95] is a single molecule methodology for the rapid production of ordered restriction maps from individual DNA molecules. Ordered restriction maps were constructed originally from yeast chromosomes by using fluorescence microscopy to visualize restriction endonuclease cutting events on individual fluorochrome-stained DNA molecules [SCH+95,...

We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper k-coloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NP-hard, and W [t]-hard for all t. We also show that the second problem is W [1]-hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.

We propose a method for visualizing a set of related metabolic pathways using 2 2 D graph drawing. Interdependent, two-dimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant di#erences among the pathways. Layouts are determined by a global layout of the union of all pathway-representing graphs using a variant of the proven Sugiyama approach for layered graph drawing that allows edges to cross if they appear in di#erent graphs.

) Thomas Anantharaman, Bud Mishra and David Schwartz 1 Abstract In this paper, we describe our algorithmic approach to constructing an alignment of (con- tiging) a set of optical maps created from the images of individual genomic DNA molecules digested by restriction enzymes. Generally, these DNA segments are sized in the range of 1--4Mb. The problem of assembling clone contig maps is a simpler special case of this contig problem and is handled by our algorithms. The goal is to devise contiging algorithms capable of producing high-quality composite maps rapidly and in a scalable manner. The resulting software is a key component of our physical mapping automation tools and has been used routinely to create composite maps of various microorganisms (E. coli, P. falciparum and D. radioduran). The experimental results appear highly promising. 1 Introduction Single molecule approaches provide a new direction for characterizing structural and functional properties of individual DNA molec...

The reconstruction of ancestral genome architectures and gene orders from homologies between extant species is a longstanding problem, considered by both cytogeneticists and bioinformaticians. A comparison of the two approaches was recently investigated and discussed in a series of papers, sometimes with diverging points of view regarding the performance of these two approaches. We describe a general methodological framework for reconstructing ancestral genome segments from conserved syntenies in extant genomes. We show that this problem, from a computational point of view, is naturally related to physical mapping of chromosomes and benefits from using combinatorial tools developed in this scope. We develop this framework into a new reconstruction method considering conserved gene clusters with similar gene content, mimicking principles used in most cytogenetic studies, although on a different kind of data. We implement and apply it to datasets of mammalian genomes. We perform intensive theoretical and experimental comparisons with other bioinformatics methods for ancestral genome segments reconstruction. We show that the method that we propose is stable and reliable: it gives convergent results using several kinds of data at different levels of resolution, and all predicted ancestral regions are well supported. The results come eventually very close to cytogenetics studies. It suggests that the comparison of methods for ancestral genome reconstruction should include the algorithmic aspects of the methods as well

) S. Muthukrishnan Laxmi Parida y Abstract We initiate the complexity study of physical mapping with the emerging technology of Optical Mapping (OM) pioneered by the team lead by David Schwartz at the W. M. Keck Laboratory for Biomolecular Imaging, Dept of Chemistry, NYU. In currently popular electrophoretic approaches, information about the relative ordering of the fragments comprising the DNA molecule is lost, thus leading to difficult computational problems of composing the fragments in to a physical map depicting their relative order. In contrast, the relative ordering of the pieces is readily obtained in OM. However, OM faces serious technological challenges as it has low resolution and is faultprone. We take a combinatorial approach to the problem of constructing physical maps from the erroneous data generated by OM. We identify two abstract problems in this context, namely, the Exclusive Binary Flip-Cut and Exclusive Weighted Flip-Cut problems. For both, we present polynom...

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(|V | + |E|) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is

Inferring phylogenetic trees is a fundamental problem in computational biology. We present a new objective criterion, the phylogenetic number, for evaluating evolutionary trees for species defined by biomolecular sequences or other qualitative characters. The phylogenetic number of a tree T is the maximum number of times that any given character state arises in T . By contrast, the classical parsimony criterion measures the total number of times that different character states arise in T . We consider the following related problems: finding the tree with minimum phylogenetic number, and computing the phylogenetic number of a given topology in which only the leaves are labeled by species. When the number of states is bounded (as is the case for biomolecular sequence characters), we can solve the second problem in polynomial time. Given the topology for an evolutionary tree, we can also compute a phylogeny with phylogenetic number 2 (when one exists) for an arbitrary number of states. Th...

Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NP-completeness is the appropriate tool for studying apparent intractability. At issue in the theory of parameterized complexity is whether a problem can be solved in time O(n ff ) for each fixed parameter value, where ff is a constant independent of the parameter. In addition to surveying this complexity framework, we describe a new result for the Longest common subsequence problem. In particular, we show that the problem is hard for W [t] for all t when parameterized by the number of strings and the size of the alphabet. Lower bounds on the complexity of this basic combinatorial problem imply lower bounds on more general sequence alignment and consensus discovery problems. We also describe a number of open problems pertaining to the parameterized complexity of pro...