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A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
Abstract

Cited by 59 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 14 (7 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
Approximating the spanning star forest problem and its applications to genomic sequence alignment
 In SODA
, 2007
"... Abstract. This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomialtime approximation scheme for planar graphs; (2) there is a polynomialtime 3approximation algorithm for graphs; (3) it is NPhard to approxi5 mate the ..."
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Cited by 5 (2 self)
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Abstract. This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomialtime approximation scheme for planar graphs; (2) there is a polynomialtime 3approximation algorithm for graphs; (3) it is NPhard to approxi5 mate the problem within ratio 259 + ɛ for graphs; (4) there is a lineartime algorithm to compute the 260 maximum star forest of a weighted tree; (5) there is a polynomialtime 1approximation algorithm 2 for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region. Key words. Dominating set, spanning star forest, approximation algorithm, genomic sequence alignment AMS subject classifications. 68Q17, 68Q25, 68R10, 68W25 1. Introduction. A
Packing paths in digraphs
"... f ~P1g, or f ~P1; ~P2g, the Gpacking problem is NPcomplete. When G = f ~P1g, the Gpacking problem is simply the matching problem. We treat in detail the one remaining case, G = f ~P1; ~P2g. We give in this case a polynomial algorithm for the packing problem. We also give the following positive re ..."
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Cited by 1 (0 self)
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f ~P1g, or f ~P1; ~P2g, the Gpacking problem is NPcomplete. When G = f ~P1g, the Gpacking problem is simply the matching problem. We treat in detail the one remaining case, G = f ~P1; ~P2g. We give in this case a polynomial algorithm for the packing problem. We also give the following positive results: a Berge type augmenting configuration theorem, a minmax characterization, and a reduction to bipartite matching. These results apply also to packings by the family G consisting of all directed paths and cycles. We also explore weighted variants of the problem and include a polyhedral analysis.
Star Forests, Dominating Sets and Ramseytype Problems
"... A star forest of a graph G is a spanning subgraph of G in which each component is a star. The minimum number of edges required to guarantee that an arbitrary graph, or a bipartite graph, has a star forest of size n is determined. Sharp lower bounds on the size of a largest star forest are also deter ..."
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A star forest of a graph G is a spanning subgraph of G in which each component is a star. The minimum number of edges required to guarantee that an arbitrary graph, or a bipartite graph, has a star forest of size n is determined. Sharp lower bounds on the size of a largest star forest are also determined. For bipartite graphs, these are used to obtain an upper bound on the domination number in terms of the number of vertices and edges in the graph, which is an improvement on a bound of Vizing. In turn, the results on bipartite graphs are used to determine the minimum number of lattice points required so that there exists a subset of n lattice points, no three of which form a right triangle with legs parallel to the coordinate axes. Key words: star forests, dominating sets, Ramsey theory, 1
Approximating the Spanning Star Forest Problem and Its Applications to Genomic Sequence Alignment
"... This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) there is a polynomialtime approximation scheme for planar unweighted graphs; (2) there is a polynomialtime algorithm with approximation ratio 3 5 for unweighted graphs; (3) it is NPh ..."
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This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) there is a polynomialtime approximation scheme for planar unweighted graphs; (2) there is a polynomialtime algorithm with approximation ratio 3 5 for unweighted graphs; (3) it is NPhard to approximate the problem within ratio 545 546 + ɛ for unweighted graphs; (4) there is a lineartime algorithm to compute the maximum star forest of a weighted tree; (5) there is a polynomialtime algorithm with approximation ratio 1 2 for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region. 1
THE SUBCHROMATIC INDEX OF GRAPHS
, 2007
"... In an edge coloring of a graph each color class forms a subgraph without a path of length two (amatching). An edge subcoloring extends this concept: each color class in an edge subcoloring forms a subgraph without path of length three. While every graph with maximum degree at most two isedge 2subco ..."
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In an edge coloring of a graph each color class forms a subgraph without a path of length two (amatching). An edge subcoloring extends this concept: each color class in an edge subcoloring forms a subgraph without path of length three. While every graph with maximum degree at most two isedge 2subcolorable, we point out in this paper that recognizing edge 2subcolorable graphs with maximum degree three is NPcomplete, even when restricted to trianglefree graphs. As byproducts, we obtain NPcompleteness results for the star index and the subchromatic number for several classes of graphs. It is also proved that recognizing edge 3subcolorable graphs is NPcomplete. Moreover, edge subcolorings and subchromatic index of various basic graph classes are studied. In particular, we show thatevery unicyclic graph is edge 3subcolorable and edge 2subcolorable unicyclic graphs have a simple structure allowing an easy linear time recognition.
Abstract A parallel algorithmic version of the Local Lemma
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
Abstract
 Add to MetaCart
The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique in several points and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for various interesting algorithmic search problems. Two representing examples are the problem of finding a directed even simple cycle in a digraph in which the maximum outdegree is at most a small exponent of the minimum indegree and the problem of exhibiting a satisfying assignment for a Boolean CNF Formula in which the maximum number of occurences of each literal is bounded by a small exponent of the minimum number of literals in a disjunction. 1