on the occasion of his 60 th birthday

this paper we give a simplified model for Physical Mapping with probes that tend to occur very rarely along the DNA and show that the problem is NP-complete even for sparse matrices. Moreover, we show that Physical Mapping with chimeric clones (a clone is chimeric if it stems from a concatenation of several fragments of the DNA) is NPcomplete even for sparse matrices. Both problems are modeled as variants of the Consecutive Ones Problem which makes our results interesting for other application areas. 1 Supported by the Applied Mathematical Sciences program, U.S. Dept. of Energy, Office of Energy Research, and the work was performed at Sandia National Labs, operated for the U.S. DOE under contract No. DE-AC04-76DP00789. Preprint submitted to Elsevier Preprint 19 January 1996 1 Introduction In order to study a long DNA molecule it is necessary to break several copies of the molecule into smaller fragments. For further investigation copies

We give an algorithm with runtime O(k 2k n 3 m) for the NP-complete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a long-standing open problem [13, 5, 19, 11]. The problem has applications in Physical Mapping of DNA [9] and in Profile Minimization for Sparse Matrix Computations [7, 20]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.

We propose a method for visualizing a set of related metabolic pathways across organisms using 2 1/2 dimensional graph visualization. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant differences among the pathways. The (dis)similarities between pathways are expressed by the Hamming distances of the underlying graphs which are used to compute a stacking order for the pathways. Layouts are determined by a global layout of the union of all pathway graphs using a variant of the proven Sugiyama approach for layered graph drawing. Our variant layout approach allows edges to cross if they appear in different graphs.

Abstract. We present a general framework for approximating several NP-hard problems that have two underlying properties in common. First, the problems we consider can be formulated as integer covering programs, possibly with additional side constraints. Second, the number of covering options is restricted in some sense, although this property may be well hidden. Our method is a natural extension of the threshold rounding technique. 1

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n − 1 + k, considering k as the parameter; this is a parameterization above guaranteed value, since n − 1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. 1

For an interval graph with some additional order constraints between pairs of non-intersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the order constraints. Previous algorithms for this problem (known also as seriation with side constraints) required quadratic time. This problem contains as subproblems interval graph and interval order recognition. On the other hand, it is a special case of the interval satisfiability problem, which is concerned with the realizability of a set of intervals along a line, subject to precedence and intersection constraints. We study such problems for all possible restrictions on the types of constraints, when all intervals must have the same length. We give efficient algorithms for several restrictions of the problem, and show the NP-completeness of another restriction. 1 Introduction Two intervals x; y on the real line may either intersect or one of them is completely to the left of the othe...

The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter.

We survey the consecutive-ones property of binary matrices. Herein, a binary matrix has the consecutive-ones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.

The problems of Interval Sandwich (IS) and Intervalizing Colored Graphs (ICG) have received a lot of attention recently, due to their applicability to DNA physical mapping problems with ambiguous data. Most of the results obtained so far on the problems were hardness results. Here we study the problems under assumptions of sparseness, which hold in the biological context. We prove that both problems are polynomial when either (1) the input graph degree and the solution graph clique size are bounded, or (2) the solution graph degree is bounded. In particular, this implies that ICG is polynomial on bounded degree graphs for every fixed number of colors, in contrast with the recent result of Bodlaender and de Fluiter. 1. Introduction A graph is an interval graph if one can assign an interval on the real line to each vertex so that two vertices are adjacent if and only if their intervals have a nonempty intersection. Consider the following problem: INTERVAL SANDWICH (IS): INSTANCE: A t...