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23
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 ..."
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Cited by 86 (11 self)
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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...
A spectral algorithm for seriation and the consecutive ones problem
 SIAM Journal on Computing
, 1998
"... Abstract. In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire ..."
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Cited by 46 (0 self)
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Abstract. In applications ranging from DNA sequencing through archeological dating to sparse matrix reordering, a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other. That is, given a correlation function f reflecting the desire for each pair of elements to be near each other, find all permutations π with the property that if π(i) < π(j) < π(k) then f(i, j) ≥ f(i, k) and f(j, k) ≥ f(i, k). This seriation problem is a generalization of the wellstudied consecutive ones problem. We present a spectral algorithm for this problem that has a number of interesting features. Whereas most previous applications of spectral techniques provide only bounds or heuristics, our result is an algorithm that correctly solves a nontrivial combinatorial problem. In addition, spectral methods are being successfully applied as heuristics to a variety of sequencing problems, and our result helps explain and justify these applications.
New Approximation Techniques for Some Ordering Problems
 IN 9TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1998
"... We describe logarithmic times optimal approximation algorithms for the NPhard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storagetime product. This improves on the best previous approximation bounds of Even, Naor, Rao, and Schieber for ..."
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Cited by 40 (1 self)
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We describe logarithmic times optimal approximation algorithms for the NPhard graph optimization problems of minimum linear arrangement, minimum containing interval graph, and minimum storagetime product. This improves on the best previous approximation bounds of Even, Naor, Rao, and Schieber for these problems by an \Omega\Gamma/15 log n) factor. Even, Naor, Rao, and Schieber defined "spreading metrics" for each of the ordering problems above (and to other problems); for each of these problems, they provided a spreading metric of volume W , such that W is a lower bound on the cost of a solution to the problem. They used this spreading metric to find a solution of cost O(W log n log log n) (for simplicity, assume that all tasks have unit processing time in the minimum storagetime product problem). In this paper, we show how to find a solution within a logarithmic factor times W for these problems. We develop a recursion where at each level we identify cost which, if incurred, yi...
Fixedparameter complexity of minimum profile problems
 In Proceedings IWPEC 2006
, 2006
"... The profile of a graph is an integervalued 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 NPhard problem, we consider parameterized versions of the problem. ..."
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Cited by 12 (7 self)
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The profile of a graph is an integervalued 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 NPhard 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 fixedparameter 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
The consecutive ones submatrix problem for sparse matrices
 Algorithmica
, 2004
"... A 01 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 01 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Su ..."
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Cited by 11 (0 self)
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A 01 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 01 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Such a problem finds application in physical mapping with hybridization data in genome sequencing. Let (a, b)matrices be the 01 matrices in which there are at most a 1’s in each column and at most b 1’s in each row. This paper proves that the C1S problem remains NPhard for i) (2, 3)matrices and ii) (3, 2)matrices. This solves an open problem posed in a recent paper of Hajiaghayi and Ganjali [1]. We further prove that the C1S problem is polynomialtime 0.8approximatable for (2, 3)matrices in which no two columns are identical and 0.5approximatable for (2, ∞)matrices in general. we also show that the C1S problem is polynomialtime 0.5approximatable for (3, 2)matrices. However, there exists an ɛ> 0 such that approximating the C1S problem for (∞, 2)matrices within a factor of n ɛ (where n is the number of columns of the input matrix) is NPhard. Keywords: NPhardness, approximation algorithm, consecutive ones property, consecutive ones submatrix, caterpillar spanning tree 1
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 ..."
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Cited by 5 (1 self)
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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...
Convergence Theorems for Some Layout Measures on Random Lattice and Random Geometric Graphs
"... This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection ..."
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Cited by 5 (0 self)
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This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. Specifically, we consider the following problems: Minimum Linear Arrangement, Cutwidth, Sum Cut, Vertex Separation, Edge Bisection and Vertex Bisection. For full square lattices, we give optimal layouts for the problems still open. For arbitrary lattice graphs, we present best possible bounds disregarding a constant factor. We apply percolation theory to the study of lattice graphs in a probabilistic setting. In particular, we deal with the subcritical regime that this class of graphs exhibits and characterize the behavior of several layout measures in this space of probability. We extend the results on random lattice graphs to random geometric graphs, which are graphs whose nodes are spread at random in the unit square and whose edges connect pairs of points which are within a given distance. We also characterize the behavior of several layout measures on random geometric graphs in their subcritical regime. Our main results are convergence theorems that can be viewed as an analog of the Beardwood, Halton and Hammersley theorem for the Euclidean TSP on random points in the unit square.
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.). ..."
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Cited by 3 (1 self)
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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. 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...
Seriation in the presence of errors: NPhardness of l∞fitting Robinson structures to dissimilarity matrices
"... In this paper, we establish that the following fitting problem is NPhard: given a finite set X and a dissimilarity measure d on X (d is a symmetric function d from X 2 to the nonnegative real numbers and vanishing on the diagonal), we wish to find a Robinsonian dissimilarity dR on X minimizing the ..."
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Cited by 2 (2 self)
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In this paper, we establish that the following fitting problem is NPhard: given a finite set X and a dissimilarity measure d on X (d is a symmetric function d from X 2 to the nonnegative real numbers and vanishing on the diagonal), we wish to find a Robinsonian dissimilarity dR on X minimizing the l∞error d − dR ∞ = maxx,y∈X{d(x, y) − dR(x, y)} between d and dR. Recall that a dissimilarity dR on X is called monotone (or Robinsonian) if there exists a total order ≺ on X such that x ≺ z ≺ y implies that d(x, y) ≥ max{d(x, z), d(z, y)}. The Robinsonian dissimilarities appear in seriation and clustering problems, in sparse matrix ordering and DNA sequencing.