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25
Certifying LexBFS recognition algorithms for proper inteval graphs and proper interval bigraphs
 SIAM J. Discrete Math
"... Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the r ..."
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Cited by 16 (3 self)
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Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m + n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n), the certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). 1
Localized and compact datastructure for comparability graphs
, 2009
"... We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure i ..."
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Cited by 12 (5 self)
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We show that every comparability graph of any twodimensional poset over n elements (a.k.a. permutation graph) can be preprocessed in O(n) time, if two linear extensions of the poset are given, to produce an O(n) space datastructure supporting distance queries in constant time. The datastructure is localized and given as a distance labeling, that is each vertex receives a label of O(log n) bits so that distance queries between any two vertices are answered by inspecting their labels only. This result improves the previous scheme due to Katz, Katz and Peleg [M. Katz, N.A. Katz, D. Peleg, Distance labeling schemes for wellseparated graph classes, Discrete Applied Mathematics 145 (2005) 384–402] by a log n factor. As a byproduct, our datastructure supports allpair shortestpath queries in O(d) time for distanced pairs, and so identifies in constant time the first edge along a shortest path between any source and destination. More fundamentally, we show that this optimal space and time datastructure cannot be extended for higher dimension posets. More precisely, we prove that for comparability graphs of threedimensional posets, every distance labeling scheme requires Ω(n 1/3) bit labels.
Certifying Algorithms
, 2010
"... A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manual ..."
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Cited by 11 (2 self)
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A certifying algorithm is an algorithm that produces, with each output, a certificate or witness (easytoverify proof) that the particular output has not been compromised by a bug. A user of a certifying algorithm inputs x, receives the output y and the certificate w, and then checks, either manually or by use of a program, that w proves that y is a correct output for input x. In this way, he/she can be sure of the correctness of the output without having to trust the algorithm. We put forward the thesis that certifying algorithms are much superior to noncertifying algorithms, and that for complex algorithmic tasks, only certifying algorithms are satisfactory. Acceptance of this thesis would lead to a change of how algorithms are taught and how algorithms are researched. The widespread use of certifying algorithms would greatly enhance the reliability of algorithmic software. We survey the state of the art in certifying algorithms and add to it. In particular, we start a
A Finegrained Analysis of a Simple Independent Set Algorithm
 LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... We present a simple exact algorithm for the INDEPENDENT SET problem with a runtime bounded by O(1.2132 npoly(n)). This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computeraided case analysis. The new branching rule uses the concept of satellites and ha ..."
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Cited by 10 (0 self)
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We present a simple exact algorithm for the INDEPENDENT SET problem with a runtime bounded by O(1.2132 npoly(n)). This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computeraided case analysis. The new branching rule uses the concept of satellites and has previously only been used in an algorithm for sparse graphs. The computeraided case analysis allows us to capture the behavior of our algorithm in more detail than in a traditional analysis. The main purpose of this paper is to demonstrate how a very simple algorithm can outperform more complicated ones if the right analysis of its running time is performed.
A simple 3sweep LBFS algorithm for the recognition of unit interval graphs
 DISCRETE APPLIED MATHEMATICS
, 2003
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Interval completion with few edges
 In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first p ..."
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Cited by 9 (1 self)
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We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) · n O(1). The problem has applications in Physical Mapping of DNA [11] and in Profile Minimization for Sparse Matrix Computations [9, 25]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively.
Approximation and FixedParameter Algorithms for Consecutive Ones Submatrix Problems
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
"... We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard ..."
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Cited by 9 (0 self)
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We develop an algorithmically useful refinement of a forbidden submatrix characterization of 0/1matrices fulfilling the Consecutive Ones Property (C1P). This characterization finds applications in new polynomialtime approximation algorithms and fixedparameter tractability results for the NPhard problem to delete a minimum number of rows or columns from a 0/1matrix such that the remaining submatrix has the C1P.
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones 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 ..."
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Cited by 7 (1 self)
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones 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.
Interval Completion is Fixed Parameter Tractable
"... We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [9, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 7, 25, 14], first p ..."
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Cited by 4 (1 self)
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We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [9, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 7, 25, 14], first posed by Kaplan, Shamir and Tarjan, of whether this problem was fixed parameter tractable. The problem has applications in Profile Minimization for Sparse Matrix Computations [10, 26], and our results show tractability for the case of a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search.