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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.

We study the parameterized complexity of three NP-hard graph completion problems. The MINIMUM FILL-IN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The PROPER

We show that there is an O(n^3) algorithm to approximate the bandwidth of an AT-free graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n log n) algorithm to approximate the bandwidth of an AT-free graph within a factor 4 and an O(n+ e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid graphs we obtain O(n log² n) algorithms with worst case performance ratio 2. For cocomparability graphs we obtain an O(n + e) algorithm with worst case performance ratio 3. Finally, we show that there is an O(n² log² n) algorithm to compute the exact bandwidth of chain graphs.

In this paper, we consider the complexity of a number of combinatorial problems; namely, Intervalizing Colored Graphs (DNA physical mapping), Triangulating Colored Graphs (perfect phylogeny), (Directed) (Modified) Colored Cutwidth, Feasible Register Assignment and Module Allocation for graphs of bounded pathwidth. Each of these problems has as a characteristic a uniform upper bound on the tree or path width of the graphs in "yes"-instances. For all of these problems with the exceptions of Feasible Register Assignment and Module Allocation, a vertex or edge coloring is given as part of the input. Our main results are that the parameterized variant of each of the considered problems is hard for the complexity classes W [t] for all t 2 N. We also show that Intervalizing Colored Graphs, Triangulating Colored Graphs, and Colored Cutwidth are NP-Complete. 1 Introduction This paper focuses on a number of graph decision problems which share the characteristic that all have a uniform upper bo...

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...

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

This document is mainly based on "A Compendium of Parameterized Complexity Results", version 2.0 (May 22, 1996), by Michael T. Hallett and H. Todd Wareham, and on Downey and Fellows' book [53]. However, this document includes several new results that have been published in the last few years

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.