Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NP-complete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the k-consecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.

The graph sandwich problem for property \Pi is defined as follows: Given two graphs G ) such that E ` E , is there a graph G = (V; E) such that E which satisfies property \Pi? Such problems generalize recognition problems and arise in various applications. Concentrating mainly on properties characterizing subfamilies of perfect graphs, we give polynomial algorithms for several properties and prove the NP-completeness of others. We describe

The Physical Mapping Problem is to reconstruct the relative position of fragments (clones) of DNA along the genome from information on their pairwise overlaps. We show that two simplified versions of the problem belong to the class of NP-complete problems, which are conjectured to be computationally intractable. In one version all clones have equal length, and in another, clone lengths may be arbitrary. The proof uses tools from graph theory and complexity.

An input to the betweenness problem contains m constraints over n real variables. Each constraint consists of three variables, where one of the variables is specified to lie inside the interval defined by the other two. The order of the other two variables (which one is the largest and which one is the smallest) is not specified. This problem comes up in questions related to physical mapping in computational molecular biology. In 1979, Opatrny has shown that the problem of deciding whether the n variables can be totally ordered while satisfying the m betweenness constraints is NP-- complete. Furthermore, the problem is MAX SNP complete. Therefore, there is some ffl ? 0 such that finding a total order which satisfies at least m(1 \Gamma ffl) of the constraints (even if they are all satisfiable) is NP--hard. It is easy to find an ordering of the variables which satisfies 1=3 of the m constraints (e.g. by choosing the ordering at random). In this work we present a polynomial time algorithm which either determines that there is no feasible solution, or finds a total order which satisfies at least 1=2 of the m constraints. Our algorithm translates the problem into a set of quadratic inequalities, and solves a semidefinite relaxation of them in R n . The n solution points are then projected on a random line through the origin. Using simple geometric properties of the SDP solution, we prove the claimed performance guarantee.

Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the current knowledge about tractability in the interval algebra is complete, that is, this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NP-complete. We obtain this result by giving a new uniform description of the known maximal tractable subalgebras and then systematically using an algebraic technique for identifying maximal subalgebras with a given property.

The determination of the sequence of all nucleotide base-pairs in a DNA molecule, from restriction-fragment data, is a complex task and can be posed as the problem of finding the optima of a multi-modal function. A genetic algorithm that uses multi-niche crowding permits us to do this. Performance of this algorithm is first tested using a standard suite of test functions. The algorithm is next tested using two data sets obtained from the Human Genome Project at the Lawrence Livermore National Laboratory. The new method holds promise in automating the sequencing computations.

A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig (Algorithmica 13:1/2, 52--76, 1995) first considered a maximum-likelihood model of the problem that is equivalent to finding an ordering of the probes that minimizes a weighted sum of errors, and developed several effective heuristics. We show that by exploiting information about the end-probes of clones, this model can be formulated as a weighted Betweenness Problem. This affords the significant advantage of allowing the well-developed tools of integer linear-programming and branch-and-cut algorithms to be brought to bear on physical mapping, enabling us for the first time to solve small mapping instances to optimality even in the presence of high error. We also show that by combining the optimal solution of many small overlapping Betweenness...

A fundamental problem in computational biology is the construction of physical maps of chromosomes from hybridization experiments between unique probes and clones of chromosome fragments in the presence of error. Alizadeh, Karp, Weisser and Zweig [AKWZ94] first considered a maximumlikelihood model of the problem that is equivalent to finding an ordering of the probes that minimizes a weighted sum of errors, and developed several effective heuristics. We show that by exploiting information about the endprobes of clones, this model can be formulated as a weighted Betweenness Problem. This affords the significant advantage of allowing the well-developed tools of integer linearprogramming and branch-and-cut algorithms to be brought to bear on physical mapping, enabling us for the first time to solve small mapping instances to optimality even in the presence of high error. We also show that by combining the optimal solution of many small overlapping Betweenness Problems, one can effectively...

We introduce a new notion of embedding, called minimumrelaxation ordinal embedding, parallel to the standard notion of minimum-distortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much as possible. The (multiplicative) relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop several worst-case bounds and approximation algorithms on ordinal embedding. In particular, we establish that ordinal embedding has many qualitative differences from metric embedding, and capture the ordinal behavior of ultrametrics and shortest-path metrics of unweighted trees. 1

Allen's interval algebra is one of the best established formalisms for temporal reasoning. We study those fragments of Allen's algebra that contain a basic relation distinct from the equality relation. We prove that such a fragment is either NP-complete or else contained in some already known tractable subalgebra. We obtain this result by giving a new uniform description of known maximal tractable subalgebras and then systematically using an algebraic technique for description of maximal subalgebras with a given property. This approach avoids the need for extensive computer-assisted search.