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Reflections on multivariate algorithmics and problem parameterization
 PROC. 27TH STACS
, 2010
"... Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and e ..."
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Cited by 37 (21 self)
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Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space” of computationally hard problems.
Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
, 2009
"... The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the ..."
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Cited by 6 (1 self)
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The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]hard. Giving two fixedparameter reductions, one from the W[1]complete problem Multicolored Clique and one to the W[1]complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]complete with respect to the parameter k, which in particular means that there is no hope for fixedparameter tractability with respect to the parameter k. Our reductions show also the W[1]completeness of the more general problem Set Cover on instances that “almost have the consecutiveones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k · n O(1) time, we show that Rectangle Stabbing is fixedparameter tractable in the still NPhard case where both these restrictions apply.
The parameterized complexity of some geometric problems in unbounded dimension
 In Proc. 4th IWPEC, volume 5917 of LNCS
"... We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d: i) Given n points in R d, compute their minimum enclosing cylinder. ii) Given two npoint sets in R d, decide whether they can be separated by two hyperplanes. iii) Given a system o ..."
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Cited by 6 (3 self)
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We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d: i) Given n points in R d, compute their minimum enclosing cylinder. ii) Given two npoint sets in R d, decide whether they can be separated by two hyperplanes. iii) Given a system of n linear inequalities with d variables, find a maximumsize feasible subsystem. We show that (the decision versions of) all these problems are W[1]hard when parameterized by the dimension d. Our reductions also give a n Ω(d)time lower bound (under the Exponential Time Hypothesis).
The parameterized complexity of the rectangle stabbing problem and its variants
 In Proc. 2nd FAW, volume 5059 of LNCS
, 2008
"... Abstract. We study an NPcomplete geometric covering problem called dDimensional Rectangle Stabbing, where, given a set of axisparallel ddimensional hyperrectangles, a set of axisparallel (d − 1)dimensional hyperplanes and a positive integer k, the question is whether one can select at most k o ..."
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Cited by 3 (2 self)
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Abstract. We study an NPcomplete geometric covering problem called dDimensional Rectangle Stabbing, where, given a set of axisparallel ddimensional hyperrectangles, a set of axisparallel (d − 1)dimensional hyperplanes and a positive integer k, the question is whether one can select at most k of the hyperplanes such that every hyperrectangle is intersected by at least one of these hyperplanes. This problem is wellstudied from the approximation point of view, while its parameterized complexity remained unexplored so far. Here we show, by giving a nontrivial reduction from a problem called Multicolored Clique, that for d ≥ 3 the problem is W[1]hard with respect to the parameter k. For the case d = 2, whose parameterized complexity is still open, we consider several natural restrictions and show them to be fixedparameter tractable. 1
Constrained kcenter and Movement to Independence
"... We obtain hardness results and approximation algorithms for two related geometric problems involving movement. The first is a constrained variant of the kcenter problem, arising from a geometric clientserver problem. The second is the problem of moving points towards an independent set. 1 1 ..."
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We obtain hardness results and approximation algorithms for two related geometric problems involving movement. The first is a constrained variant of the kcenter problem, arising from a geometric clientserver problem. The second is the problem of moving points towards an independent set. 1 1
Multivariate Algorithmics for NPHard String Problems
, 2014
"... String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NPhard. This motivates the search for (fixedparameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NPha ..."
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String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NPhard. This motivates the search for (fixedparameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NPhard string problems and identify challenges for future research.
Reduction rules deliver efficient fptalgorithms for covering points with lines
 ACM Journal of Experimental Algorithmics
"... We present efficient algorithms to solve the Line Cover problem exactly. In this NPcomplete problem, the inputs are n points in the plane and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines. Our approach is based on fixedparameter tractability, ..."
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We present efficient algorithms to solve the Line Cover problem exactly. In this NPcomplete problem, the inputs are n points in the plane and a positive integer k, and we are asked to answer if we can cover these n points with at most k lines. Our approach is based on fixedparameter tractability, and in particular, kernelization. We propose several reduction rules to transform instances of Line Cover into equivalent smaller instances. Once instances are no longer susceptible to these reduction rules, we obtain a problem kernel whose size is bounded by a polynomial function of the parameter k and does not depend on the size n of the input. Our algorithms provide exact solutions and are easy to implement. We also describe the design of algorithms to solve the corresponding optimization problem exactly. We experimentally evaluated ten variants of the algorithms to determine the impact and tradeoffs of several reduction rules. We show that our approach provides tractability for a larger range of values of the parameter and larger inputs, improving the execution time by several orders of magnitude with respect to previously known algorithms.
On a problem of Danzer and Grünbaum
, 2014
"... We say that a family of subsets C of a universe U has the (l; k)property, if every subfamily C0 C of cardinality at most l is khittable. In this paper we revisit a conjecture of Danzer and Grünbaum which inquires about the existence of a function g(k; d) such that if any family of objects C in Rd ..."
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We say that a family of subsets C of a universe U has the (l; k)property, if every subfamily C0 C of cardinality at most l is khittable. In this paper we revisit a conjecture of Danzer and Grünbaum which inquires about the existence of a function g(k; d) such that if any family of objects C in Rd has the (g(k; d); k)property, then C is kpiercable (hittable). Danzer and Grünbaum showed that g(k; d) is infinite for families of boxes and translates of centrally symmetric convex hexagons. Negative results for other families of objects have been shown. Very few positive results are known and only for small values of k. In this paper, we show that, for a family of subsets C of a given universe U, with the property that any subfamily of p+1 distinct subsets in C intersect in at most l elements, if C has the ((k(klp+1)+ 1; k)property then C is khittable. We use the above result to show Danzer Grünbaum type results for pseudolines and halfplanes in Rd. These are the first positive results for arbitrary k in a general setting. We also pose a relaxed version of the above question which we call the khelly problem: For every positive integer k, determine the smallest f(k; d) such that if any family of convex objects C in Rd has the (g(k; d); k)property for some g(k; d), then C is f(k; d)piercable. We show that any family of boxes in Rd with the (k2d; k)property is 2dkpiercable. We finally show the existence of polynomial sized kernels for the corresponding hitting set and covering problems in the setting described above.
On a Linear Program for MinimumWeight Triangulation (Extended Abstract)
"... Minimumweight triangulation (MWT) is NPhard. It has a polynomialtime constantfactor approximation algorithm, and a variety of effective polynomialtime heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are wellstudied, but previously no connection was kno ..."
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Minimumweight triangulation (MWT) is NPhard. It has a polynomialtime constantfactor approximation algorithm, and a variety of effective polynomialtime heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are wellstudied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP.