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Packing and covering δhyperbolic spaces by balls
 In APPROXRANDOM 2007
"... Abstract. We consider the problem of covering and packing subsets of δhyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic gr ..."
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Abstract. We consider the problem of covering and packing subsets of δhyperbolic metric spaces and graphs by balls. These spaces, defined via a combinatorial Gromov condition, have recently become of interest in several domains of computer science. Specifically, given a subset S of a δhyperbolic graph G and a positive number R, let γ(S, R) be the minimum number of balls of radius R covering S. It is known that computing γ(S, R) or approximating this number within a constant factor is hard even for 2hyperbolic graphs. In this paper, using a primaldual approach, we show how to construct in polynomial time a covering of S with at most γ(S, R) balls of (slightly larger) radius R + δ. This result is established in the general framework of δhyperbolic geodesic metric spaces and is extended to some other set families derived from balls. This covering algorithm is used to design better than in general case approximation algorithms for the augmentation problem of δhyperbolic graphs with diameter constraints and slackness δ and for the kcenter problem in δhyperbolic graphs. 1
Using fractional primaldual to schedule split intervals with demands
 IN 13TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS
, 2005
"... Given a limited resource (such as bandwidth or memory) we consider the problem of scheduling requests from clients that are given as groups of nonintersecting time intervals. Each request j is associated with a demand (the amount of resource required), dj, a tinterval, which consistsof up to t s ..."
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Given a limited resource (such as bandwidth or memory) we consider the problem of scheduling requests from clients that are given as groups of nonintersecting time intervals. Each request j is associated with a demand (the amount of resource required), dj, a tinterval, which consistsof up to t segments, for some t> = 1, and a weight, w(j). A schedule is a collection of requests. Itis feasible if for every time instance s, the total demand of scheduled requests whose tintervalcontains s does not exceed 1, the amount of resource available. Our goal is to find a feasibleschedule that maximizes the total weight of scheduled requests. This problem generalizes many problems from the literature, and show up in a wide range of applications.We present a 6 tapproximation algorithm for this problem that uses a novel extension of theprimaldual schema we call fractional primaldual. A fractional primaldual algorithm produces a primal solution x, and a dual solution y, whose value divided by r, the approximation ratio,bounds the weight of x. However, y is not a solution of the dual of an LP relaxation of theproblem. The algorithm induces a new LP that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. yis a feasible solution of the dual of the new LP. x is rapproximate, since some optimal solutionof an LP relaxation of the original problem is in the feasible set of this new LP. We present a fractional local ratio interpretation of our 6tapproximation algorithm. We alsodiscuss the connection between fractional primaldual and the fractional local ratio technique. Specifically, we show that the former is the primaldual manifestation of latter.
Approximation Hardness of Optimization Problems in Intersection Graphs of ddimensional Boxes
 proceedings of the 16th Annual ACMSIAM Symposium on Discrete Algorithms
, 2005
"... Abstract The Maximum Independent Set problem in dbox graphs, i.e., in the intersection graphs of axisparallel rectangles in R d , is a challenge open problem. For any fixed d ≥ 2 the problem is NPhard and no approximation algorithm with ratio o(log d−1 n) is known. In some restricted cases, e.g. ..."
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Abstract The Maximum Independent Set problem in dbox graphs, i.e., in the intersection graphs of axisparallel rectangles in R d , is a challenge open problem. For any fixed d ≥ 2 the problem is NPhard and no approximation algorithm with ratio o(log d−1 n) is known. In some restricted cases, e.g., for dboxes with bounded aspect ratio, a PTAS exists Introduction The intersection graph of a family of sets S v , v ∈ V , is a graph with vertex set V such that u is adjacent to v if and only if S u ∩ S v = ∅. The family {S v , v ∈ V } is an intersection representation of this graph. Geometrical models of intersection graphs deal with families of subsets of R d with some geometric properties. In this paper we are mainly interested in families of axis For convenience, terms an interval and a rectangle are used for 1box and 2box, respectively. In this paper we describe a generic approach to approximation hardness results for many graph optimization problems as, e.g., Minimum Vertex Cover,
Approximation Algorithms for Intersection Graphs
, 2009
"... We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which ..."
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We introduce three new complexity parameters that in some sense measure how chordallike a graph is. The similarity to chordal graphs is used to construct simple polynomialtime approximation algorithms with constant approximation ratio for many NPhard problems, when restricted to graphs for which at least one of our new complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.
Satisfying complex data needs using pullbased online monitoring of volatile data sources
 in Proceedings of the IEEE CS International Conference on Data Engineering
, 2008
"... Abstract — Emerging applications on the Web require better management of volatile data in pullbased environments. In a pull based setting, data may be periodically removed from the server. Data may also become obsolete, no longer serving client needs. In both cases, we consider such data to be vola ..."
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Abstract — Emerging applications on the Web require better management of volatile data in pullbased environments. In a pull based setting, data may be periodically removed from the server. Data may also become obsolete, no longer serving client needs. In both cases, we consider such data to be volatile. To model such constraints on data usability, and support complex user needs we define profiles to specify which data sources are to be monitored and when. Using a novel abstraction of execution intervals we model complex profiles that access simultaneously several servers to gain from the used data. Given some budgetary constraints (e.g., bandwidth), the paper formalizes the problem of maximizing completeness. I.
P.J.: Separating points by axisparallel lines
 International Journal of Computational Geometry & Applications
, 2005
"... PengJun Wan ¢ We study the problem of separating £ points in the plane, no two of which have the same ¤ or ¥coordinate using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. We prove that this problem and some v ..."
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PengJun Wan ¢ We study the problem of separating £ points in the plane, no two of which have the same ¤ or ¥coordinate using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. We prove that this problem and some variants of it are NPcomplete. We give an approximation algorithm with ratio ¦ for the planar problem, and a ratio § approximation algorithm for the §dimensional variant, in which the points are to be separated using axisparallel hyperplanes. We reduce the problem to the rectangle stabbing problem studied by Gaur et al [5]. Their approximation algorithm uses LProunding. Our algorithm presents an alternative LProunding procedure which also works for the rectangle stabbing problem. We also discuss some dual problems suggested by the linear programs used to solve the separation problem. 1
Maximal Strip Recovery Problem with Gaps: Hardness and Approximation Algorithms
"... Given two comparative maps, that is two sequences of markers each representing a genome, the Maximal Strip Recovery problem (MSR) asks to extract a largest sequence of markers from each map such that the two extracted sequences are decomposable into nonoverlapping strips (or synteny blocks). This ..."
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Given two comparative maps, that is two sequences of markers each representing a genome, the Maximal Strip Recovery problem (MSR) asks to extract a largest sequence of markers from each map such that the two extracted sequences are decomposable into nonoverlapping strips (or synteny blocks). This aims at de ning a robust set of synteny blocks between di erent species, which is a key to understand the evolution process since their last common ancestor. In this paper, we add a fundamental constraint to the initial problem, which expresses the biologically sustained need to bound the number of intermediate (nonselected) markers between two consecutive markers in a strip. We therefore introduce the problem δgapMSR, where δ is a (usually small) nonnegative integer that upper bounds the number of nonselected markers between two consecutive markers in a strip. Depending on the nature of the comparative maps (i.e., with or without duplicates), we show that δgapMSR is NPcomplete for any δ ≥ 1, and even APXhard for any δ ≥ 2. We also provide two approximation algorithms, with ratio 1.8 for δ = 1, and ratio 4 for δ ≥ 2.
On the tractability of maximal strip recovery
 Erratum in Journal of Computational Biology, 18:129, 2011. Cited
"... Abstract. Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery ..."
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Abstract. Given two genomic maps G and H represented by a sequence of n gene markers, a strip (syntenic block) is a sequence of distinct markers of length at least two which appear as subsequences in the input maps, either directly or in reversed and negated form. The problem Maximal Strip Recovery (MSR) is to find two subsequences G ′ and H ′ of G and H, respectively, such that the total length of disjoint strips in G ′ and H ′ is maximized (or, conversely, the number of markers hence deleted, is minimized). Previously, besides some heuristic solutions, a factor4 polynomialtime approximation is known for the MSR problem; moreover, several close variants of MSR, MSRd (with d>2 input maps), MSRDU (with marker duplications) and MSRWT (with markers weighted) are all shown to be NPcomplete. Before this work, the complexity of the original MSR problem was left open. In this paper, we solve the open problem by showing that MSR is NPcomplete, using a polynomial time reduction from OneinThree 3SAT. We also solve the MSR problem and its variants exactly with FPT algorithms, i.e., showing that MSR is fixedparameter tractable. Let k be the minimum number of markers deleted in various versions of MSR, the running time of our
New results for the 2interval pattern problem
 IN PROCEEDINGS OF THE 15TH ANNUAL SYMPOSIUM ON COMBINATORIAL PATTERN MATCHING (CPM
, 2004
"... We present new results concerning the problem of nding a constrained pattern in a set of 2intervals. Given a set of n 2intervals D and a model R describing if two disjoint 2intervals can be in precedence order (<), be allowed to nest (@) and/or be allowed to cross (G), the problem asks to nd ..."
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Cited by 6 (3 self)
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We present new results concerning the problem of nding a constrained pattern in a set of 2intervals. Given a set of n 2intervals D and a model R describing if two disjoint 2intervals can be in precedence order (<), be allowed to nest (@) and/or be allowed to cross (G), the problem asks to nd a maximum cardinality subset D ′ ⊆ D such that any two 2intervals in D ′ agree with R. We improve the time complexity of the best known algorithm for R = {@} by giving an optimal O(n log n) time algorithm. Also, we give a graphlike relaxation for R =