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 consider the problem of scheduling jobs that are given as groups of non-intersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t 1, a positive weight, w j , and two jobs are in conflict if any of their segments intersect. Such jobs show up in a wide range of applications, including the transmission of continuous-media data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology/geometry. The objective is to schedule a subset of non-conflicting jobs of maximum total weight.

. A d-track interval is a union of d intervals, one each from d parallel lines. The intersection graphs of d-track intervals are the unions of d interval graphs. The multitrack interval number or simply track number of a graph G is the minimum number of interval graphs whose union is G. The track number for Km;n is determined by proving that the arboricity of Km;n equals its "caterpillar arboricity". Recognition of graphs with track number 2 is shown to be NP-complete. 1. Introduction Combinatorial properties of interval systems were investigated as early as the 1930's by Tibor Gallai. His two beautiful unpublished remarks are now phrased as saying that interval graphs and their complements are perfect graphs. In 1968, Gallai suggested considering more general set systems consisting of unions of d intervals, one each from d parallel lines. Such set systems have been called separated d-intervals, since the d parallel lines can be viewed as d disjoint host intervals on a single li...

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more then one interval associated with it. We initiate the study of optimization problems in multiple-interval graphs by considering three classical problems: Minimum Vertex Cover, Minimum Dominating Set, and Maximum Clique. We describe applications for each one of these problems, and then proceed to discuss approximation algorithms for them. Our results can be summarized as follows: Let t be the number of intervals associated with each vertex in a given multiple-interval graph. For Minimum Vertex Cover, we give a (2 − 1/t)-approximation algorithm which also works when a t-interval representation of our given graph is absent. Following this, we give a t 2-approximation algorithm for Minimum Dominating Set which adapts well to more general variants of the problem. We then proceed to prove that Maximum Clique is NP-hard already for 3-interval graphs, and provide a (t 2 −t+ 1)/2-approximation algorithm for general values of t ≥ 2, using bounds proven for the so-called transversal number of t-interval families.

A multiple-interval representation of a simple graph G assigns each vertex aunion of disjoint real intervals, such that vertices are adjacent if and only if their assigned sets intersect. The total interval number I(G) isthe minimum of the total number of intervals used in any such representation of G. For triangle-free graphs, I(G) = m + t(G), where m is the number of edges in G and t(G) isthe minimum number of pairwise edge-disjoint trails such that every edge of G has an endpoint in at least one of the trails. This yields the NP-completeness of testing I(G) = m + 1, even for triangle-free 3-regular planar graphs, and an alternative proof that HAMILTONIAN CYCLE is NP-complete for line graphs. It also yields a linear-time algorithm to compute I(G) for trees and a characterization of the trees requiring m + t intervals, for fixed t. Further corollaries include the Aigner/Andreae bound of I(G) ≤⎣(5n − 3)/4 ⎦ for n-vertex trees (achieved bysubdividing every edge of a star), a characterization of the extremal trees, and a shorter proof of the extremal bound ⎣(5m + 2)/4 ⎦ for connected graphs. Ke ywords: intersection graphs, intervals, trees, extremal problem

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Abstract. We consider various optimization problems in t-subtree graphs, the intersection graphs of t-subtrees, where a t-subtree is the union of t disjoint subtrees of some tree. This graph class generalizes both the class of chordal graphs and the class of t-interval graphs, a generalization of interval graphs that has recently been studied from a combinatorial optimization point of view. We present approximation