Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NP-complete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NP-complete. This problem is als...

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...

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.

. We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of interval orders, where the class of proper interval orders is exactly the same as the class of unit interval orders. Introduction Definition 1. An order OE of a set X is called a trapezoid order if, given two parallel lines in the plane (which we will take to be horizontal), there is, associated to each x 2 X, a trapezoid T x with one base on each of the lines with the property that x OE y if and only if T x " T y = ; and each point in T x is to the left of some point in T y . [Langley, 1993] A trapezoid order is a generalization of an interval order, in which there is an interval I x associated to each element x of the order, with x OE y if and only if each point of I x lies to the left of each point of I y . We will only deal with interval...

. We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a linear time algorithm. Extant algorithms for this problem (known also as seriation with side constraints) required quadratic time. When the constraints are bounds on distances between endpoints, and the graph admits a unique clique order, we show that the problem is polynomial. However, we show that even when the lengths of all intervals are precisely predetermined, the problem is NPcomplete. We also study unit interval satisfiability problems, which are concerned with the realizability of a set of unit intervals along a line, subject to precedence and intersection constraints. For all po...

. We investigate NeST graphs, namely, intersection tolerance graphs of neighborhood subtrees of a tree, introduced recently by Bibelnieks and Dearing. We establish a "tolerance-free" characterization of NeST graphs. This in turn yields tolerancefree characterizations of two known subclasses of NeST graphs, namely, proper NeST graphs (in which no neighborhood subtree is properly contained within another) and fixed tolerance NeST graphs (also called constant NeST graphs). Also, we introduce two new subclasses of NeST graphs, namely, fixed diameter NeST graphs and fixed distance NeST graphs. We show that the class of fixed diameter NeST graphs (equivalent to unit NeST graphs) is exactly the class of proper NeST graphs, analogous to the case for interval graphs and in contrast to the case for interval tolerance graphs. We show that the class of fixed distance NeST graphs is exactly the class of threshold tolerance graphs, introduced by Monma, Reed and Trotter. This responds to a question t...

. We study the following problem: Given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance or sizes. An important application is physical mapping, a central challenge in the human genome project. Our results are: (1) A polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO graphs). This class of graphs properly contains all prime interval graphs. (2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly equivalent to deciding if a system of difference inequalities is feasible. (3) Even if all the constraints are prescribed lengths of individual intervals, the problem is NP-complete. Hence, problems (1) and (2) are als...

Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ � � V (D)

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We study problems of determining whether a given interval graph has a realization which satisfies additional given constraints. Such problems occur frequently in applications where entities are modeled as intervals along a line (events along a time line, DNA segments along a chromosome, etc.). When the additional information is order constraints on pairs of disjoint intervals, we give a linear time algorithm. Extant 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. When the constraints are bounds on distances between endpoints, and the graph admits a unique clique order, we show that the problem is polynomial. The special case of this problem where the constraints are bounds on interval length is shown to be linearly equivalent to deciding if a system of difference inequalities is feasible. However, we show that even when the lengths of all intervals ar...