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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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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.
Fixedparameter algorithms for Cochromatic Number and Disjoint Rectangle Stabbing
"... Given a permutation π of {1,..., n} and a positive integer k, we give an algorithm with that decides whether π can be partitioned into at most k running time 2O(k2 log k) O(1) n increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed par ..."
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Given a permutation π of {1,..., n} and a positive integer k, we give an algorithm with that decides whether π can be partitioned into at most k running time 2O(k2 log k) O(1) n increasing or decreasing subsequences. Thus we resolve affirmatively the open question of whether the problem is fixed parameter tractable. This NPcomplete problem is equivalent to deciding whether the cochromatic number, partitioning into the minimum number of cliques or independent sets, of a given permutation graph on n vertices is at most k. In fact, we give a more general result: within the mentioned running time, one can decide whether the cochromatic number of a given perfect graph on n vertices is at most k. To obtain our result we use a combination of two wellknown techniques within parameterized algorithms, namely greedy localization and iterative compression. We further demonstrate the power of this combination by giving a 2O(k2 log k) n log n time algorithm for deciding whether a given set of n nonoverlapping axisparallel rectangles can be stabbed by at most k of a given set of horizontal and vertical lines. Whether such an algorithm exists was mentioned as an open question in several papers.
The Approximability and Integrality Gap of Interval Stabbing and Independence Problems
"... Abstract Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of dintervals and dunionintervals. We obtain the following: (1) constructions yielding asymptotically tight lower bounds on the integrality g ..."
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Abstract Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of dintervals and dunionintervals. We obtain the following: (1) constructions yielding asymptotically tight lower bounds on the integrality gaps of the associated natural linear programming relaxations; (2) an LPrelative dapproximation for the hitting set problem on dintervals; (3) a proof that the approximation ratios for independent set on families of 2intervals and 2unionintervals can be improved to match tight duality gap lower bounds obtained via topological arguments, if one has access to an oracle for a PPADcomplete problem related to finding BorsukUlam fixedpoints.
Partial Multicovering and the dconsecutive Ones Property
, 2011
"... A dinterval is the union of d disjoint intervals on the real line. In the dinterval stabbing problem (dis) we are given a set of dintervals and a set of points, each dinterval I has a stabbing requirement r(I) and each point has a weight, and the goal is to find a minimum weight multiset of poi ..."
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A dinterval is the union of d disjoint intervals on the real line. In the dinterval stabbing problem (dis) we are given a set of dintervals and a set of points, each dinterval I has a stabbing requirement r(I) and each point has a weight, and the goal is to find a minimum weight multiset of points that stabs each dinterval I at least r(I) times. In practice there is a tradeoff between fulfilling requirements and cost, and therefore it is interesting to study problems in which one is required to fulfill only a subset of the requirements. In this paper we study variants of dis in which a feasible solution is a multiset of points that may satisfy only a subset of the stabbing requirements. In partial dis we are given an integer t, and the sum of requirements satisfied by the computed solution must be at least t. In prize collecting dis each dinterval has a penalty that must be paid for every unit of unsatisfied requirement. We also consider a maximization version of prize collecting dis in which each dinterval has a prize that is gained for every time, up to r(I), it is stabbed. Our study is motivated by several resource allocation and geometric facility location problems. We present a (ρ+d−1 ρ)approximation algorithm for prize collecting dis, where ρ = minI r(I), and an O(d)approximation algorithm for partial dis. We obtain the latter result by designing a general framework for approximating partial multicovering problems that extends the framework for approximating partial covering problems from [21]. We also show that maximum prize collecting dis is at least as hard to approximate as maximum independent set, even for d = 2, and present a dapproximation algorithm for maximum prize collecting ddimensional rectangle stabbing.
Stabbing Polygonal Chains with Rays is Hard to Approximate
"... We study a geometric hitting set problem involving unirectional rays and curves in the plane. We show that this problem is hard to approximate within a logarithmic factor even when the curves are convex polygonal xmonotone chains. Additionally, it is hard to approximate within a factor of 76 even ..."
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We study a geometric hitting set problem involving unirectional rays and curves in the plane. We show that this problem is hard to approximate within a logarithmic factor even when the curves are convex polygonal xmonotone chains. Additionally, it is hard to approximate within a factor of 76 even when the curves are line segments with bounded slopes. Lastly, we demonstrate that the problem is W [2]complete when the curves are convex polygonal xmonotone chains and is W [1]hard when the curves are line segments. 1