Results 1  10
of
13
Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... 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 ..."
Abstract

Cited by 86 (11 self)
 Add to MetaCart
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 NPcomplete. 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 NPcomplete. This problem is als...
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... 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 NPcomplete ..."
Abstract

Cited by 55 (8 self)
 Add to MetaCart
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 NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive 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.
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show th ..."
Abstract

Cited by 29 (6 self)
 Add to MetaCart
We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NPhard, and W [t]hard for all t. We also show that the second problem is W [1]hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.
Probe Interval Graphs and Its Applications to Physical Mapping of DNA. manuscript
, 1994
"... A new class of graph called the probe interval graph has been introduced. It is an extension of the interval graph. It requires only partial overlap information. An enhanced probe interval graph has been derived from the probe interval graph. Its fundamental properties (such as that the enhanced pro ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
A new class of graph called the probe interval graph has been introduced. It is an extension of the interval graph. It requires only partial overlap information. An enhanced probe interval graph has been derived from the probe interval graph. Its fundamental properties (such as that the enhanced probe interval graph is triangulated) are investigated. The structure of the probe interval graph
Inferring Ordered Trees from Local Constraints
"... We consider a problem of inferring an ordered tree from a set of local constraints on its leaves which we term as the ordered local consensus tree problem. Using our efficient decremental interval union algorithm, we show that the ordered local consensus tree problem for m constraints on n leaves ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
We consider a problem of inferring an ordered tree from a set of local constraints on its leaves which we term as the ordered local consensus tree problem. Using our efficient decremental interval union algorithm, we show that the ordered local consensus tree problem for m constraints on n leaves can be deterministically solved in time O((m+n) log n): We also show that the related optimization problem of constructing an ordered local consensus tree for the maximum number of 3leaf constraints is solvable in cubic time.
Satisfiability Problems on Intervals and Unit Intervals
 Theoretical Computer Science
, 1997
"... For an interval graph with some additional order constraints between pairs of nonintersecting 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 constrain ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
For an interval graph with some additional order constraints between pairs of nonintersecting 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 NPcompleteness 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...
Interval Graphs with Side (and Size) Constraints
 In Proc. of the Third Annual European Symp. on Algorithms, (ESA 95) Corfu, Greece
, 1995
"... . 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.). ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
. 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...
Realizing Interval Graphs With Size And Distance Constraints
 SIAM Journal on Discrete Mathematics
, 1997
"... . 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. 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 NPcomplete. Hence, problems (1) and (2) are als...
A Simple Efficient Parallel Algorithms to Recognize Chordal Graphs as Interval Graphs
 Discrete Applied Mathematics
, 1995
"... We present an efficient parallel algorithm that recognizes interval graphs, provided it is known that the given graph is chordal and a representation as a collection of subtrees of a tree is known. The running time is logarithmic and the processor number is linear. 0 Introduction Interval graph ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We present an efficient parallel algorithm that recognizes interval graphs, provided it is known that the given graph is chordal and a representation as a collection of subtrees of a tree is known. The running time is logarithmic and the processor number is linear. 0 Introduction Interval graphs are intersection graphs of intervals in the real line, i.e. each vertex of an interval can be associated as an interval and two vertices are joined by an edge iff their associated intervals have a nonempty intersection. Interval graphs have a couple of applications as seriation in archeology, consecutive information retrieval, and gate layout problems (see for example [12, 16]). Linear time sequential algorithms to recognize interval graphs are quiet well known [3, 15]. Still there is a development to simplify the recognition of interval graphs. A parallel algorithms with a linear number of processors and a time bound of O(log 2 n) is due to Klein and Reif [14]. In this algorithm the cli...