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Reflections on multivariate algorithmics and problem parameterization
 In Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS ’10), volume 5 of LIPIcs
"... 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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Cited by 24 (19 self)
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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.
Combinatorial Optimization on Graphs of Bounded Treewidth
, 2007
"... There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees an ..."
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Cited by 21 (1 self)
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There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees and seriesparallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.
Cliquewidth: On the Price of Generality
, 2009
"... Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreo ..."
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Cited by 7 (1 self)
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Many hard problems can be solved efficiently when the input is restricted to graphs of bounded treewidth. By the celebrated result of Courcelle, every decision problem expressible in monadic second order logic is fixed parameter tractable when parameterized by the treewidth of the input graph. Moreover, for every fixed k ≥ 0, such problems can be solved in linear time on graphs of treewidth at most k. In particular, this implies that basic problems like Dominating Set, Graph Coloring, Clique, and Hamiltonian Cycle are solvable in linear time on graphs of bounded treewidth. A significant amount of research in graph algorithms has been devoted to extending this result to larger classes of graphs. It was shown that some of the algorithmic metatheorems for treewidth can be carried over to graphs of bounded cliquewidth. Courcelle, Makowsky, and Rotics proved that the analogue of Courcelle’s result holds for graphs of bounded cliquewidth when the logical formulas do not use edge set quantifications. Despite of its generality, this does not resolve the parameterized complexity of many basic problems concerning edge subsets (like Edge Dominating Set), vertex
private communication
, 2001
"... Spanning tree congestion is a relatively new graph parameter, which has been studied intensively. This paper studies the complexity of the problem to determine the spanning tree congestion for nonsparse graph classes, while it was investigated for some sparse graph classes before. We prove that the ..."
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Cited by 3 (0 self)
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Spanning tree congestion is a relatively new graph parameter, which has been studied intensively. This paper studies the complexity of the problem to determine the spanning tree congestion for nonsparse graph classes, while it was investigated for some sparse graph classes before. We prove that the problem is NPhard even for chain graphs and split graphs. To cope with the hardness of the problem, we present a fast (exponentialtime) exact algorithm that runs in O ∗ (2 n) time, where n denotes the number of vertices. Additionally, we present simple combinatorial lemmas, which yield a constantfactor approximation algorithm for cographs, and a lineartime algorithm for chordal cographs. Submitted:
Some probabilistic results on width measures of graphs. available at http://arxiv.org/abs/0908.1772
"... Fixed parameter tractable (FPT) algorithms run in time f(p(x))poly(x), where f is an arbitrary function of some parameter p of the input x and poly is some polynomial function. Treewidth, branchwidth, cliquewidth, NLCwidth, rankwidth, and booleanwidth are parameters often used in the design and a ..."
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Cited by 1 (0 self)
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Fixed parameter tractable (FPT) algorithms run in time f(p(x))poly(x), where f is an arbitrary function of some parameter p of the input x and poly is some polynomial function. Treewidth, branchwidth, cliquewidth, NLCwidth, rankwidth, and booleanwidth are parameters often used in the design and analysis of such algorithms for problems on graphs. We show asymptotically almost surely (aas), booleanwidth βw(G) is O(rw(G) logrw(G)), where rw is rankwidth. More importantly, we show aas Ω(n) lower bounds on the treewidth, branchwidth, cliquewidth, NLCwidth, and rankwidth of graphs drawn from a simple random model. This raises important questions about the generality of FPT algorithms using the corresponding decompositions. 1
INTRACTABILITY OF CLIQUEWIDTH PARAMETERIZATIONS
, 2009
"... We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]hard parameterized by cliquewidth. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the cliquewidth, that is, solvable i ..."
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Cited by 1 (0 self)
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We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are W [1]hard parameterized by cliquewidth. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the cliquewidth, that is, solvable in time g(k) · nO(1) on nvertex graphs of cliquewidth k, where g is some function of k only. Our results imply that the running time O(nf(k) ) of many cliquewidth based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely F P T ̸ = W [1]).
Reflections on Multivariate Algorithmics and . . .
 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 ..."
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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.
the Spanning Tree Congestion Problem
, 2010
"... We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear ti ..."
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We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NPcomplete for any fixed k≥10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NPhard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NPhard in general, but solvable in linear time for fixed k. 1
EXPLOITING STRUCTURE IN INTEGER PROGRAMS
, 2011
"... This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear eq ..."
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This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear equalities and inequalities. The state of the art in solvers for this problem is the “branch and bound ” approach. The performance of such solvers depends crucially on four types of inbuilt heuristics: primal, improvement, branching, and cutseparation or, more generally, bounding heuristics. Such heuristics in generalpurpose solvers have not, until recently, exploited structure in integer linear programs beyond the recognition of certain types of singlerow constraints. Many alternative approaches to integer linear programming can be cast in the following, novel framework. “Structure” in any integer linear program
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 ..."
Abstract
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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.