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39
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 51 (4 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.
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.
Logspace Versions of the Theorems of Bodlaender and Courcelle
, 2010
"... Bodlaender’s Theorem states that for every k there is a lineartime algorithm that decides whether an input graph has tree width k and, if so, computes a widthk tree composition. Courcelle’s Theorem builds on Bodlaender’s Theorem and states that for every monadic secondorder formula φ and for eve ..."
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Cited by 20 (2 self)
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Bodlaender’s Theorem states that for every k there is a lineartime algorithm that decides whether an input graph has tree width k and, if so, computes a widthk tree composition. Courcelle’s Theorem builds on Bodlaender’s Theorem and states that for every monadic secondorder formula φ and for every k there is a lineartime algorithm that decides whether a given logical structure A of tree width at most k satisfies φ. We prove that both theorems still hold when “linear time ” is replaced by “logarithmic space.” The transfer of the powerful theoretical framework of monadic secondorder logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the logspace world.
Booleanwidth of graphs
, 2009
"... We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divideandconquer approach. Booleanwidth is similar to rankwidth, which is related to ..."
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Cited by 12 (7 self)
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We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divideandconquer approach. Booleanwidth is similar to rankwidth, which is related to the number of GF [2]sums (1+1=0) of neighborhoods instead of the Booleansums (1+1=1) used for booleanwidth. For an nvertex graph G given with a decomposition tree of booleanwidth k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n+ 23kk)). We show that for any graph the square root of its booleanwidth is never more than its rankwidth. We also exhibit a class of graphs, the Hsugrids, having the property that a Hsugrid on Θ(n2) vertices has booleanwidth Θ(log n) and treewidth, branchwidth, cliquewidth and rankwidth Θ(n). Moreover, any optimal rankdecomposition of such a graph will have booleanwidth Θ(n) , i.e. exponential in the optimal booleanwidth.
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 11 (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
Hjoin decomposable graphs and algorithms with runtime single exponential in rankwidth
"... We introduce Hjoin decompositions of graphs, indexed by a fixed bipartite graph H. These decompositions are based on a graph operation that we call Hjoin, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertice ..."
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Cited by 7 (2 self)
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We introduce Hjoin decompositions of graphs, indexed by a fixed bipartite graph H. These decompositions are based on a graph operation that we call Hjoin, which adds edges between two given graphs by taking partitions of their two vertex sets, identifying the classes of the partitions with vertices of H, and connecting classes by the pattern H. Hjoin decompositions are related to modular, split and rank decompositions. Given an Hjoin decomposition of an nvertex medge graph G we solve the Maximum Independent Set and Minimum Dominating Set problems on G in time O(n(m+2O(ρ(H) 2))), and the qColoring problem in time O(n(m + 2O(qρ(H) 2))) , where ρ(H) is the rank of the adjacency matrix of H over GF(2). Rankwidth is a graph parameter introduced by Oum and Seymour, based on ranks of adjacency matrices over GF(2). For any positive integer k we define a bipartite graph Rk and show that the graphs of rankwidth at most k are exactly the graphs having an Rkjoin decomposition, thereby giving an alternative graphtheoretic definition of rankwidth that does not use linear algebra. Combining our results we get algorithms that, for a graph G of rankwidth k given with its width k rankdecomposition, solves the Maximum Independent Set problem in time
Complexity and algorithms for wellstructured kSAT instances
 In SAT’08
, 2008
"... This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization prob ..."
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Cited by 6 (2 self)
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This paper initiates the study of SAT instances of bounded diameter. The diameter of an ordered CNF formula is defined as the maximum difference between the index of the first and the last occurrence of a variable. We study the complexity of the satisfiability, the counting and the maximization problems for formulas of bounded diameter. We investigate the relation between the diameter of a formula, and the treewidth and the pathwidth of its corresponding incidence graph, and show that under highly parallel and efficient transformations, diameter and pathwidth are equal up to a constant factor. Our main technical contribution is that the computational complexity of SAT, MaxSAT, #SAT grows smoothly with the diameter (as a function of the number of variables). Our main focus is in providing space efficient and highly parallel algorithms, while the running time of our algorithms matches previously known results. Among others, we show NLcompleteness of SAT and NC2 algorithms for MaxSAT,#SAT when diameter is O(log n). Given the tree decomposition of a formula, we further improve on the space efficiency to decide SAT as asked by Alekhnovich and Razborov [1]. 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 6 (1 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]).
On the booleanwidth of a graph: Structure and applications
 Proceedings of the 36th International Workshop on GraphTheoretic Concepts in Computer Science, WG 2010
, 2010
"... Abstract Booleanwidth is a recently introduced graph invariant. Similar to treewidth, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of booleanwidth k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time ..."
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Cited by 4 (2 self)
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Abstract Booleanwidth is a recently introduced graph invariant. Similar to treewidth, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of booleanwidth k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O∗(2O(k 2)). We relate the booleanwidth of a graph to its branchwidth and to the booleanwidth of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rankwidth by Oum (JGT 2008). For a random graph on n vertices we show that almost surely its booleanwidth is Θ(log2 n) – setting booleanwidth apart from other graph invariants – and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasipolynomial time O∗(2O(log 4 n)). 1