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59
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 36 (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.
Constant ratio fixedparameter approximation of the edge multicut problem
 In ESA 2009
, 2009
"... Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the ..."
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Cited by 18 (3 self)
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Abstract. The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s1, t1},..., {sm, tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1 ≤ i ≤ m. The parameterized complexity of the problem, parameterized by the maximum number k of edges that are allowed to be removed, is currently open. The main result of the paper is a parameterized 2approximation algorithm: in time f(k) · nO(1), we can either find a solution of size 2k or correctly conclude that there is no solution of size k. The proposed algorithm is based on a transformation of the Edge Multicut problem into a variant of parameterized Max2SAT problem, where the parameter is related to the number of clauses that are not satisfied. It follows from previous results that the latter problem can be 2approximated in a fixedparameter time; on the other hand, we show here that it is W[1]hard. Thus the additional contribution of the present paper is introducing the first natural W[1]hard problem that is constantratio fixedparameter approximable. 1
Approximation schemes for firstorder definable optimisation problems
 In Proc. LICS’06
, 2006
"... Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, ..."
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Cited by 17 (9 self)
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Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, if X only occurs negatively in ϕ(X), then ϕ(X) defines a maximisation problem. Many wellknown optimisation problems are firstorder definable in this sense, for example, MINIMUM DOMINATING SET or MAXIMUM INDEPENDENT SET. We prove that for each class C of graphs with excluded minors, in particular for each class of planar graphs, the restriction of a firstorder definable optimisation problem to the class C has a polynomial time approximation scheme. A crucial building block of the proof of this approximability result is a version of Gaifman’s locality theorem for formulas positive in a set variable. This result may be of independent interest. 1.
On the optimality of planar and geometric approximation schemes
"... We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a plan ..."
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Cited by 17 (5 self)
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We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a planar graph are essentially optimal: if there is a δ> 0 such that any of these problems admits a 2O((1/ǫ)1−δ) O(1) n time PTAS, then the Exponential Time Hypothesis (ETH) fails. It is known that MAXIMUM INDEPENDENT SET on unit disk graphs and the planar logic problems MPSAT, TMIN, TMAX admit nO(1/ǫ) time approximation schemes. We show that they are optimal in the sense that if there is a δ> 0 such that any of these problems admits a 2 (1/ǫ)O(1) nO((1/ǫ)1−δ) time PTAS, then ETH fails.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 15 (5 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Parameterized approximability of the disjoint cycle problem
 Proc. ICALP 2007, Lecture Notes in Computer Science
, 2007
"... Abstract. We give an fpt approximation algorithm for the directed vertex disjoint cycle problem. Given a directed graph G with n vertices and a positive integer k, the algorithm constructs a family of at least k/ρ(k) disjoint cycles of G if the graph G has a family of at least k disjoint cycles (and ..."
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Abstract. We give an fpt approximation algorithm for the directed vertex disjoint cycle problem. Given a directed graph G with n vertices and a positive integer k, the algorithm constructs a family of at least k/ρ(k) disjoint cycles of G if the graph G has a family of at least k disjoint cycles (and otherwise may still produce a solution, or just report failure). Here ρ is a computable function such that k/ρ(k) is nondecreasing and unbounded. The running time of our algorithm is polynomial. The directed vertex disjoint cycle problem is hard for the parameterized complexity class W[1], and to the best of our knowledge our algorithm is the first fpt approximation algorithm for a natural W[1]hard problem. Key words: approximation algorithms, fixedparameter tractability, parameterized complexity theory. 1
Exponentialtime approximation of weighted set cover
 Inf. Process. Lett
"... The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for ..."
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Cited by 10 (0 self)
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The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponentialtime algorithms for problems of that kind. The goal is getting the time complexity still of order O(cn), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation. In fact, we describe a kind of approximation schemes — tradeoffs between approximation factor and the time complexity. We use general transformations from exponentialtime exact algorithms to approximations that are faster but still exponentialtime. For example, we show that for any reduction rate r, one can transform any O∗(cn)time1 algorithm for Set Cover into a (1+ln r)approximation algorithm running in time O∗(cn/r). We believe that results of that kind extend the applicability of exact algorithms for NPhard problems.
Parameterized approximation scheme for the multiple knapsack problem
 IN PROCEEDINGS OF THE 20TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS, SODA 2009
, 2009
"... The multiple knapsack problem (MKP) is a wellknown generalization of the classical knapsack problem. We are given a set A of n items and set B of m bins (knapsacks) such that each item a ∈ A has a size size(a) and a profit value profit(a), and each bin b ∈ B has a capacity c(b). The goal is to find ..."
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Cited by 9 (2 self)
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The multiple knapsack problem (MKP) is a wellknown generalization of the classical knapsack problem. We are given a set A of n items and set B of m bins (knapsacks) such that each item a ∈ A has a size size(a) and a profit value profit(a), and each bin b ∈ B has a capacity c(b). The goal is to find a subset U ⊂ A of maximum total profit such that U can be packed into B without exceeding the capacities. The decision version of MKP is strongly NPcomplete, since it is a generalization of the classical knapsack and bin packing problem. Furthermore, MKP does not admit an FPTAS even if the number m of bins is two. Kellerer gave a PTAS for MKP with identical capacities and Chekuri and Khanna presented a PTAS for MKP with general capacities with running time n O(log(1/ǫ)/ǫ8). In this
Approximating Solution Structure
"... Abstract. Approximations can aim at having close to optimal value or, alternatively, they can aim at structurally resembling an optimal solution. Whereas valueapproximation has been extensively studied by complexity theorists over the last three decades, structuralapproximation has not yet been de ..."
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Cited by 7 (3 self)
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Abstract. Approximations can aim at having close to optimal value or, alternatively, they can aim at structurally resembling an optimal solution. Whereas valueapproximation has been extensively studied by complexity theorists over the last three decades, structuralapproximation has not yet been defined, let alone studied. However, structuralapproximation is theoretically no less interesting, and has important applications in cognitive science. Building on analogies with existing valueapproximation algorithms and classes, we develop a general framework for analyzing structural (in)approximability. We identify dissociations between solution value and solution structure, and generate a list of open problems that may stimulate future research.
Parameterized algorithmics for computational social choice: nine research challenges
 Tsinghua Science and Technology
, 2014
"... Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in ..."
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Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multiagent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problemspecific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context.