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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 24 (19 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.
Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules
, 2010
"... To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This direc ..."
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Cited by 13 (0 self)
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To make a joint decision, agents (or voters) are often required to provide their preferences as linear orders. To determine a winner, the given linear orders can be aggregated according to a voting protocol. However, in realistic settings, the voters may often only provide partial orders. This directly leads to the POSSIBLE WINNER problem that asks, given a set of partial votes, whether a distinguished candidate can still become a winner. In this work, we consider the computational complexity of POSSIBLE WINNER for the broad class of voting protocols defined by scoring rules. A scoring rule provides a score value for every position which a candidate can have in a linear order. Prominent examples include plurality, kapproval, and Borda. Generalizing previous NPhardness results for some special cases, we settle the computational complexity for all but one scoring rule. More precisely, for an unbounded number of candidates and unweighted voters, we show that POSSIBLE WINNER is NPcomplete for all pure scoring rules except plurality, veto, and the scoring rule defined by the scoring vector (2, 1,...,1, 0), while it is solvable in polynomial time for plurality and veto.
On the Parameterized Complexity of Consensus Clustering ⋆
"... Abstract. Given a collection C of partitions of a base set S, the NPhard Consensus Clustering problem asks for a partition of S which has a total Mirkin distance of at most t to the partitions in C, where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with ..."
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Cited by 1 (1 self)
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Abstract. Given a collection C of partitions of a base set S, the NPhard Consensus Clustering problem asks for a partition of S which has a total Mirkin distance of at most t to the partitions in C, where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with running time O(4.24 k ·k 3 +C·S  2), where k: = t/C is the average Mirkin distance of the solution partition to the partitions of C. Furthermore, we strengthen previous hardness results for Consensus Clustering, showing that Consensus Clustering remains NPhard even when all input partitions contain at most two subsets. Finally, we study a local search variant of Consensus Clustering, showing W[1]hardness for the parameter “radius of the Mirkindistance neighborhood”. In the process, we also consider a local search variant of the related Cluster Editing problem, showing W[1]hardness for the parameter “radius of the edge modification neighborhood”. 1
Partial Kernelization for Rank Aggregation: Theory and Experiments
"... Abstract. RANK AGGREGATION is important in many areas ranging from web search over databases to bioinformatics. The underlying decision problem KEMENY SCORE is NPcomplete even in case of four input rankings to be aggregated into a “median ranking”. We study efficient polynomialtime data reduction ..."
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Abstract. RANK AGGREGATION is important in many areas ranging from web search over databases to bioinformatics. The underlying decision problem KEMENY SCORE is NPcomplete even in case of four input rankings to be aggregated into a “median ranking”. We study efficient polynomialtime data reduction rules that allow us to find optimal median rankings. On the theoretical side, we improve a result for a “partial problem kernel ” from quadratic to linear size. On the practical side, we provide encouraging experimental results with data based on web search and sport competitions, e.g., computing optimal median rankings for realworld instances with more than 100 candidates within milliseconds. 1
Intractability; FixedParameter Tractability
"... Abstract. 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 investiga ..."
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Abstract. 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.
Studies in Computational Aspects of Voting — a Parameterized Complexity Perspective ⋆
"... Abstract. We review NPhard voting problems together with their status in terms of parameterized complexity results. In addition, we survey standard techniques for achieving fixedparameter (in)tractability results in voting. 1 ..."
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Abstract. We review NPhard voting problems together with their status in terms of parameterized complexity results. In addition, we survey standard techniques for achieving fixedparameter (in)tractability results in voting. 1
Partially Polynomial Kernels for Set Cover and Test Cover ∗
"... In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S ′ ⊆ S of size at most k satisfying some desired properties. If S ′ is required to contain all the e ..."
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In a typical covering problem we are given a universe U of size n, a family S (S could be given implicitly) of size m and an integer k and the objective is to check whether there exists a subfamily S ′ ⊆ S of size at most k satisfying some desired properties. If S ′ is required to contain all the elements of U then it corresponds to the classical Set Cover problem. On the other hand if we require S ′ to satisfy the property that for every pair of elements x, y ∈ U there exists a set S ∈ S ′ such that S ∩ {x, y}  = 1 then it corresponds to the Test Cover problem. In this paper we consider a natural parameterization of Set Cover and Test Cover. More precisely, we study the (n − k)Set Cover and (n − k)Test Cover problems, where the objective is to find a subfamily S ′ of size at most n − k satisfying the respective properties, from the kernelization perspective. It is known in the literature that both (n−k)Set Cover and (n−k)Test Cover do not admit polynomial kernels (under some well known complexity theoretic assumptions). However, in this paper we show that they do admit “partially polynomial kernels”. More precisely, we give polynomial time algorithms that take as input an instance (U, S, k) of (n − k)Set Cover ((n − k)Test Cover) and return an equivalent instance ( Ũ, ˜ S, ˜ k) of (n − k)