Let G = (V, E) be a finite graph, and f : V -> N be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f (v) is not larger than the value...
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Let G = (V, E) be a finite graph, and f : V -> N be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f (v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity O(log n) (.) d + O(root g) (.) root n, so that we obtain a deterministic query complexity of d + O(root g) (.) root n where it is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of O(root d) + O((4)root g) (.) (4)root n log log n. Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs. (c) 2005 Elsevier B.V. All rights reserved.
The Steiner tree problem is to find a shortest subgraph that spans a given set of vertices in a graph. This problem is known to be NP-hard, and it is well known that a polynomial time 2-approximation algorithm exists....
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The Steiner tree problem is to find a shortest subgraph that spans a given set of vertices in a graph. This problem is known to be NP-hard, and it is well known that a polynomial time 2-approximation algorithm exists. In 1996, Zelikovsky suggested an approximation algorithm for the Steiner tree problem that is called the relative greedy algorithm. Until now the performance ratio of this algorithm has not been known. Zelikovsky provided 1.694 as an upper bound, and Gropl, Hougardy, Nierhoff, and Promel proved that 1.333 is a lower bound. In this article we improve the lower bound for the performance ratio of the relative greedy algorithm to 1.385. (c) 2005 Wiley Periodicals, Inc.
We present a parallel algorithm for finding a maximum weight matching in general bipartite graphs with an adjustable time complexity of O(n/omega) using O(n(max(2 omega,4+omega))) processing elements for omega >= 1...
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We present a parallel algorithm for finding a maximum weight matching in general bipartite graphs with an adjustable time complexity of O(n/omega) using O(n(max(2 omega,4+omega))) processing elements for omega >= 1. Parameter omega is not bounded. This is the fastest known strongly polynomial parallel algorithm to solve this problem. This is also the first adjustable parallel algorithm for the maximum weight bipartite matching problem in which the execution time can be reduced by an unbounded factor. We also present a general approach for finding efficient parallel algorithms for the maximum matching problem. (c) 2005 Elsevier B.V. All rights reserved.
We consider a special subgraph of a weighted directed graph: one comprising only the k heaviest edges incoming to each vertex. We show that the maximum weight branching in this subgraph closely approximates the maximu...
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We consider a special subgraph of a weighted directed graph: one comprising only the k heaviest edges incoming to each vertex. We show that the maximum weight branching in this subgraph closely approximates the maximum weight branching in the original graph. Specifically, it is within a factor of k/(k + 1). Our interest in finding branchings in this subgraph is motivated by a data compression application in which calculating edge weights is expensive but estimating which are the heaviest k incoming edges is easy. An additional benefit is that since algorithms for finding branchings run in time linear in the number of edges our results imply faster algorithms although we sacrifice optimality by a small factor. We also extend our results to the case of edge-disjoint branchings of maximum weight and to maximum weight spanning forests. (c) 2006 Elsevier B.V. All rights reserved.
We present an algorithm to find a proper fraction in simplest reduced terms between two reduced proper fractions. A proper fraction is a rational number m/n with m 1. A fraction m/n is simpler than p/q if m <= p a...
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We present an algorithm to find a proper fraction in simplest reduced terms between two reduced proper fractions. A proper fraction is a rational number m/n with m < n and n > 1. A fraction m/n is simpler than p/q if m <= p and n < q, with at least one inequality strict. The algorithm operates by walking a Farey tree in maximum steps down each branch. Through Monte Carlo simulation, we find that the present algorithm finds a simpler interpolation of two fractions than using the Euclidean-Convergent [D.W. Matula, P. Kornerup, Foundations of finite precision rational arithmetic, Computing 2 (Suppl.) (1980) 85-111] walk of a Farey tree and terminating at the first fraction satisfying the bound. Analysis shows that the new algorithms, with very high probability, will find an interpolation that is simpler than at least one of the bounds, and thus take less storage space than at least one of the bounds. (c) 2005 Elsevier B.V. All rights reserved.
We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let gamma : N -> Q(+) be any density function, i.e., gamma is computable in po...
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We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let gamma : N -> Q(+) be any density function, i.e., gamma is computable in polynomial time and satisfies gamma(k) <= k - 1 for all k is an element of N. Then gamma-CLUSTER is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices that has average degree at least gamma(k). For gamma(k) = k - 1, this problem is the same as the well-known CLIQUE problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for gamma(k) = 2. We ask for the possible functions gamma such that gamma-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: gamma CLUSTER is NP-complete if gamma = 2 + Omega(1/k(1-epsilon)) for some epsilon > 0 and has a polynomial-time algorithm for gamma = 2 + O(1/k). The algorithm also shows that for Y = 2 + O(1/k(1-o(1))), gamma-CLUSTER is solvable in subexponential time 2(no(1)). (c) 2006 Elsevier B.V. All fights reserved.
We present a hybrid approach to the scheduling of jobs in a distributed system where the critical response is the bandwidth to access stored data. Our approach supports the master-worker scheme, but could be applied t...
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We present a hybrid approach to the scheduling of jobs in a distributed system where the critical response is the bandwidth to access stored data. Our approach supports the master-worker scheme, but could be applied to other cases of parallel computation over stored data. We tested our new approach under various circumstances and measured it performance by means of several metrics. We made comparisons of our approach with respect to other scheduling policies;it performed significantly better than the majority of cases, and in worst cases, it was as good as the best of the others.
Sandwich problems generalize graph recognition problems with respect to a property Pi. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for...
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Sandwich problems generalize graph recognition problems with respect to a property Pi. A recognition problem has a graph as input, whereas a sandwich problem has two graphs as input. In a sandwich problem, we look for a third graph, whose edge set lies between the edge sets of two given graphs. This third graph is required to satisfy a property H. We present sandwich results corresponding to the polynomial recognition problems: clique cutset, star cutset, and a generalization k-star cutset. We note these graph cutset problems are of interest with respect to sandwich problems. We propose an O(n(3))-time polynomial algorithm for star cutset sandwich problem, and an O(n(2+k))-time polynomial algorithm for the k-star cutset sandwich problem. We propose an NP-completeness transformation from 1-in-3 3SAT (without negative literals) to clique cutset sandwich problem. (c) 2006 Elsevier B.V. All rights reserved.
We present an efficient algorithm for recognizing unit circular-arc (UCA) graphs, based on a characterization theorem for UCA graphs proved by Tucker in the seventies. Given a proper circular-arc (PCA) graph G, the al...
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We present an efficient algorithm for recognizing unit circular-arc (UCA) graphs, based on a characterization theorem for UCA graphs proved by Tucker in the seventies. Given a proper circular-arc (PCA) graph G, the algorithm starts from a PCA model for G, removes all its circle-covering pairs of arcs and determines whether G is a UCA graph. We also give an O(N) time bound for Tucker's 3/2-approximation algorithm for coloring circular-arc graphs with N vertices, when a circular-arc model is given. (C) 2004 Elsevier Inc. All rights reserved.
We give a new proof of a theorem of Erdos, Rubin, and Taylor. Our proof yields the first linear time algorithm to Delta-list-color any graph containing no (Delta +1)-clique, and containing no odd cycle if Delta = 2. W...
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We give a new proof of a theorem of Erdos, Rubin, and Taylor. Our proof yields the first linear time algorithm to Delta-list-color any graph containing no (Delta +1)-clique, and containing no odd cycle if Delta = 2. Without change.. our algorithm can also be used to, A-color such graphs. It has the same resource bound as, but is simpler than, the current known algorithm of Lovasz for Delta-coloring such graphs. (c) 2005 Elsevier B.V. All rights reserved.
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