Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover th...
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Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like SET COVER, VERTEX COVER, DOMINATING SET and FACILITY LOCATION to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems has been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems. (C) 2011 Elsevier Inc. All rights reserved.
A diversity of tracking problems exists in which cohorts of densely packed particles move in an organized fashion, however the stability of individual particles within the cohort is low. Moreover, the flows of cohorts...
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A diversity of tracking problems exists in which cohorts of densely packed particles move in an organized fashion, however the stability of individual particles within the cohort is low. Moreover, the flows of cohorts can regionally overlap. Together, these conditions yield a complex tracking scenario that cannot be addressed by optical flow techniques that assume piecewise coherent flows, or by multi-particle tracking techniques that suffer from the local ambiguity in particle assignment. Here, we propose a graph-based assignment of particles in three consecutive frames to recover from image sequences the instantaneous organized motion of groups of particles, i.e. flows. The algorithm makes no a priori assumptions on the fraction of particles participating in organized movement, as this number continuously alters with the evolution of the flow fields in time. graph-based assignment methods generally maximize the number of acceptable particles assignments between consecutive frames and only then minimize the association cost. In dense and unstable particle flow fields this approach produces many false positives. The here proposed approach avoids this via solution of a multi-objective optimization problem in which the number of assignments is maximized while their total association cost is minimized. The method is validated on standard benchmark data for particle tracking. In addition, we demonstrate its application to live cell microscopy where several large molecular populations with different behaviors are tracked. (C) 2011 Elsevier Inc. All rights reserved.
We show that the exact computation of a minimum or a maximum cut of a given graph G IS out of reach for any one-pass streaming algorithm, that is, for any algorithm that runs over the input stream of G's edges onl...
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We show that the exact computation of a minimum or a maximum cut of a given graph G IS out of reach for any one-pass streaming algorithm, that is, for any algorithm that runs over the input stream of G's edges only once and has a working memory of o(n(2)) bits. This holds even if randomization is allowed. (C) 2010 Elsevier B.V. All rights reserved.
The negative cost cycle detection (NCCD) problem in weighted directed graphs is a fundamental problems in theoretical computer science with applications in a wide range of domains ranging from maximum flows to image s...
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The negative cost cycle detection (NCCD) problem in weighted directed graphs is a fundamental problems in theoretical computer science with applications in a wide range of domains ranging from maximum flows to image segmentation. From the perspective of program verification, this problem is identical to the problem of checking the satisfiability of a conjunction of difference constraints. There exist a number of approaches in the literature for NCCD with each approach having its own set of advantages. Recently, a greedy, space-efficient algorithm called the stressing algorithm was proposed for this problem. In this paper, we present a novel proof of the Stressing algorithm and its verification using the Prototype Verification System (PVS) theorem prover. This example is part of a larger research program to verify the soundness and completeness of a core set of decision procedures. (C) 2010 Elsevier B.V. All rights reserved.
We show that, for every 0 <= p <= 1, there is an O(n(2.575-p/(7.4-2.3p)))-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every p...
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We show that, for every 0 <= p <= 1, there is an O(n(2.575-p/(7.4-2.3p)))-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u, v in the graph to within an additive error delta(p)(u, v), where delta(u, v) is the exact length of the shortest path between u and v. This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p <= 1. Previously the only way to "beat" the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [2002] that approximates the shortest path distances within a multiplicative error of (1 + epsilon). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + epsilon) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when epsilon << 1 and the graph has small integer weights.
We show that the 3-colorability problem can be solved in O(1.296(n)) time on any n-vertex graph with minimum degree at least 15. This algorithm is obtained by constructing a dominating set of the graph greedily, enume...
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We show that the 3-colorability problem can be solved in O(1.296(n)) time on any n-vertex graph with minimum degree at least 15. This algorithm is obtained by constructing a dominating set of the graph greedily, enumerating all possible 3-colorings of the dominating set, and then solving the resulting 2-list coloring instances in polynomial time. We also show that a 3-coloring can be obtained in 2(o(n)) time for graphs having minimum degree at least w(n) where w(n) is any function which goes to infinity. We also show that if the lower bound on minimum degree is replaced by a constant (however large it may be), then neither a 2(o(n)) time nor a 2(o(m)) time algorithm is possible (m denotes the number of edges) for 3-colorability unless Exponential Time Hypothesis (ETH) fails. We also describe an algorithm which obtains a 4-coloring of a 3-colorable graph in O(1.2535(n)) time. (C) 2010 Elsevier B.V. All rights reserved.
In a recent paper Soleimanfallah and Yeo proposed a kernelization algorithm for vertex cover which, for any fixed constant c, produces a kernel of order 2k - c in polynomial time. In this paper we show how their techn...
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In a recent paper Soleimanfallah and Yeo proposed a kernelization algorithm for vertex cover which, for any fixed constant c, produces a kernel of order 2k - c in polynomial time. In this paper we show how their techniques can be extended to improve the produced kernel to order 2k - c log k, for any fixed constant c. (C) 2011 Elsevier B.V. All rights reserved.
In a balloon drawing of a tree, all the children under the same parent are placed on the circumference of the circle centered at their parent, and the radius of the circle centered at each node along any path from the...
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In a balloon drawing of a tree, all the children under the same parent are placed on the circumference of the circle centered at their parent, and the radius of the circle centered at each node along any path from the root reflects the number of descendants associated with the node. Among various styles of tree drawings reported in the literature, the balloon drawing enjoys a desirable feature of displaying tree structures in a rather balanced fashion. For each internal node in a balloon drawing, the ray from the node to each of its children divides the wedge accommodating the subtree rooted at the child into two sub-wedges. Depending on whether the two sub-wedge angles are required to be identical or not, a balloon drawing can further be divided into two types: even sub-wedge and uneven sub-wedge types. In the most general case, for any internal node in the tree there are two dimensions of freedom that affect the quality of a balloon drawing: (1) altering the order in which the children of the node appear in the drawing, and (2) for the subtree rooted at each child of the node, flipping the two sub-wedges of the subtree. In this paper, we give a comprehensive complexity analysis for optimizing balloon drawings of rooted trees with respect to angular resolution, aspect ratio and standard deviation of angles under various drawing cases depending on whether the tree is of even or uneven sub-wedge type and whether (1) and (2) above are allowed. It turns out that some are NP-complete while others can be solved in polynomial time. We also derive approximation algorithms for those that are intractable in general. (c) 2010 Elsevier B.V. All rights reserved.
In this work we confront from a computational viewpoint the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 20...
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In this work we confront from a computational viewpoint the Multiple Domination problem, introduced by Harary and Haynes in 2000 among other variations of domination, with the Limited Packing problem, introduced in 2009. In particular, we prove that the Limited Packing problem is NP-complete for split graphs and for bipartite graphs, two graph classes for which the Multiple Domination problem is also NP-complete (Liao and Chang, 2003). For a fixed capacity, we prove that these two problems are polynomial time solvable in quasi-spiders. Furthermore, by analyzing the combinatorial numbers that are involved in their definitions applied to the join and the union of graphs, we show that both problems can be solved in polynomial time for P(4)-tidy graphs. From this result, we derive that they are polynomial time solvable in P(4)-lite graphs, giving in this way an answer to a question stated by Liao and Chang on the domination side. (C) 2011 Elsevier B.V. All rights reserved.
The square H(2) of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. Lau and Cornell [Recognizing powers of proper interval, split and chordal graphs, SIAM J. Discre...
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The square H(2) of a graph H is obtained from H by adding new edges between every two vertices having distance two in H. Lau and Cornell [Recognizing powers of proper interval, split and chordal graphs, SIAM J. Discrete Math. 18 (2004) 83-102] proved that recognizing squares of split graphs is an NP-complete problem. In contrast, we show that squares of strongly chordal split graphs can be recognized in quadratic-time by giving a structural characterization of these graph class. (C) 2010 Elsevier B.V. All rights reserved.
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