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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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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.
FixedParameter Tractability of Error Correction in Graphical Linear Systems
"... Abstract. In an overdetermined and feasible system of linear equations Ax = b, letvectorb be corrupted, in the way that at most k entries are off their true values. Assume that we can check in the restricted system given by any minimal dependent set of rows, the correctness of all corresponding valu ..."
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Abstract. In an overdetermined and feasible system of linear equations Ax = b, letvectorb be corrupted, in the way that at most k entries are off their true values. Assume that we can check in the restricted system given by any minimal dependent set of rows, the correctness of all corresponding values in b. Furthermore,A has only coefficients 0 and 1, with at most two 1s in each row. We wish to recover the correct values in b and x as much as possible. The problem arises in a certain chemical mixture inference application in molecular biology, where every observable reaction product stems from at most two candidate substances. After formalization we prove that the problem is NPhard but fixedparameter tractable in k. The FPT result relies on the small girth of certain graphs.
On the Computational Complexity of some problems from Combinatorial Geometry
"... We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]hard (and NPhard) if the dimension is part of the input by fptreductions (which are actually ptimereductions) from the dSum problem. Among others, we sh ..."
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We study several canonical decision problems that arise from the most famous theorems from combinatorial geometry. We show that these are W[1]hard (and NPhard) if the dimension is part of the input by fptreductions (which are actually ptimereductions) from the dSum problem. Among others, we show that computing the minimum size of a Caratheodory set and a Helly set and certain decision versions of the HamSandwich cut problem are W[1]hard. Our reductions also imply that the problems we consider cannot be solved in time no(d) (where n is the size of the input), unless the ExponentialTime Hypothesis (ETH) is false. Our technique of embedding dSum into a geometric setting is conceptually much simpler than direct fptreductions from purely combinatorial W[1]hard problems (like the clique problem) and has great potential to show (parameterized) hardness and (conditional) lower bounds for many other problems.
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"... is the best possible exponent for ddimensional geometric problems ..."
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Fixed Parameter Complexity and Approximability of Norm Maximization∗
, 2014
"... The problem of maximizing the pth power of a pnorm over a halfspacepresented polytope in Rd is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NPhard for all values p ∈ N, if the dimension d of the ambient sp ..."
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The problem of maximizing the pth power of a pnorm over a halfspacepresented polytope in Rd is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NPhard for all values p ∈ N, if the dimension d of the ambient space is part of the input. In this paper, we use the theory of parametrized complexity to analyze how heavily the hardness of norm maximization relies on the parameter d. More precisely, we show that for p = 1 the problem is fixed parameter tractable but that for all p ∈ N \ {1} norm maximization is W[1]hard. Concerning approximation algorithms for norm maximization, we show that for fixed accuracy, there is a straightforward approximation algorithm for norm maximization in FPT running time, but there is no FPT approximation algorithm, the running time of which depends polynomially on the accuracy. As with the NPhardness of norm maximization, the W[1]hardness immediately carries over to various radius computation tasks in Computational Convexity. 1