Fundamentals of Parameterized Complexity

Downey, Rodney G.; Fellows, Michael R.

doi:10.1007/978-1-4471-5559-1

Cited by 1,174 publications

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“…In contrast, problems which are NP-hard even for constant values of the parameter are said to be Para-NP-hard with respect to the parameter. For further information, we point the readers to textbooks on parameterized complexity theory [9,15,22].…”

Section: Preliminariesmentioning

confidence: 99%

Elections with Few Candidates: Prices, Weights, and Covering Problems

Bredereck

Faliszewski

Niedermeier

et al. 2015

Algorithmic Decision Theory

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Abstract. We show that a number of election-related problems with prices (such as, for example, bribery) are fixed-parameter tractable (in FPT) when parameterized by the number of candidates. For bribery, this resolves a nearly 10-year old family of open problems. Our results follow by a general technique that formulates voting problems as covering problems and extends the classic approach of using integer linear programming and the algorithm of Lenstra [19]. In this context, our central result is that WEIGHTED SET MULTICOVER parameterized by the universe size is fixed-parameter tractable. Our approach is also applicable to weighted electoral control for Approval voting. We improve previously known XP-memberships to FPT-memberships. Our preliminary experiments on real-world-based data show the practical usefulness of our approach for instances with few candidates.

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Section: Preliminariesmentioning

confidence: 99%

Elections with Few Candidates: Prices, Weights, and Covering Problems

Bredereck

Faliszewski

Niedermeier

et al. 2015

Algorithmic Decision Theory

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“…Moreover, in the setting of parameterized complexity analysis (Downey & Fellows, 2013;Flum & Grohe, 2006;Niedermeier, 2006), Christian et al showed that it is W[2]-complete with respect to the parameter "number k of rows to modify", that is, even if only a small number of voters shall be influenced the problem is computationally intractable. In this work, we provide a significantly more refined view on the parameterized and multivariate computational complexity (Fellows, Jansen, & Rosamond, 2013;Niedermeier, 2010) of Lobbying.…”

Section: Lobbyingmentioning

confidence: 99%

“…To this end, we employ classical complexity classes such as P (polynomial time) and NP (nondeterministic polynomial time) (Garey & Johnson, 1979) as well as the class LOGSNP of limited nondeterminism (lying between P and NP) (Papadimitriou & Yannakakis, 1996;Goldsmith et al, 1996) and parameterized complexity classes such as FPT (fixed-parameter tractability), W[2] (second level of the "weft hierarchy" of presumable parameterized intractability), and XP (Downey & Fellows, 2013;Flum & Grohe, 2006;Niedermeier, 2006). Throughout this paper, we denote by log the logarithm to base two.…”

Section: Preliminariesmentioning

confidence: 99%

“…The concept of parameterized complexity was pioneered by Downey and Fellows (2013) (see also further textbooks: Flum & Grohe, 2006;Niedermeier, 2006). The fundamental goal is to find out whether the seemingly unavoidable combinatorial explosion occurring in algorithms to solve NP-hard problems can be confined to certain problem-specific parameters.…”

Section: Fixed-parameter Tractability and Xpmentioning

confidence: 99%

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A Multivariate Complexity Analysis of Lobbying in Multiple Referenda

Bredereck

Hartung

Kratsch

et al. 2014

jair

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Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n×m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m ≤ 4. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g ≥ 1, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

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“…Parameterized complexity [13] analyzes a problem in two dimensions: the size of the instance |I| and a fixed parameter k. A problem is fixed-parameter tractable if it is solvable in f (k) · |I| p time, where f is a computable function that depends only on k and p is a constant. The class FPT contains fixed-parameter tractable problems.…”

Section: Parameterized Complexitymentioning

confidence: 99%