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22
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 58 (23 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Linear kernels and singleexponential algorithms via protrusion decompositions.
 In Proc. of the 40th International Colloquium on Automata, Languages and Programming (ICALP),
, 2013
"... Abstract A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G − X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been ex ..."
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Abstract A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G − X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or implicitly used for obtaining polynomial kernels Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and is treewidthbounding admits a linear kernel on the class of Htopologicalminorfree graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a ttreewidthmodulator of size O(k), for some constant t. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus Our second application concerns the PlanarFDeletion problem. Let F be a fixed finite family of graphs containing at least one planar graph. Given an nvertex graph G and a nonnegative integer k, PlanarFDeletion asks whether G has a set X ⊆ V (G) such that X k and G − X is Hminorfree for every H ∈ F. This problem encompasses a number of wellstudied parameterized problems such as Vertex Cover, Feedback Vertex Set, and Treewidtht Vertex Deletion. Very recently, an algorithm for PlanarFDeletion with running time 2 O(k) · n log 2 n (such an algorithm is called singleexponential) has been presented in
MSO decidability of MultiPushdown Systems via SplitWidth
, 2012
"... Multithreaded programs with recursion are naturally modeled as multipushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW ca ..."
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Cited by 15 (4 self)
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Multithreaded programs with recursion are naturally modeled as multipushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW can be decomposed by two basic and natural operations: shuffle of two sequences of factors and merge of consecutive factors of a sequence. We say that the splitwidth of an MNW is k if it admits a decomposition where the number of factors in each sequence is at most k. The MSO theory of MNWs with splitwidth k is decidable. We introduce two very general classes of MNWs that strictly generalize known decidable classes and prove their MSO decidability via their splitwidth and obtain comparable or better bounds of treewidth of known classes.
Deciding firstorder properties of nowhere dense graphs
 In Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC
, 2014
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On the Parameterised Intractability of Monadic SecondOrder Logic
"... Abstract. One of Courcelle’s celebrated results states that if C is a class of graphs of bounded treewidth, then modelchecking for monadic second order logic (MSO2) is fixedparameter tractable (fpt) on C by linear time parameterised algorithms. An immediate question is whether this is best possib ..."
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Cited by 8 (4 self)
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Abstract. One of Courcelle’s celebrated results states that if C is a class of graphs of bounded treewidth, then modelchecking for monadic second order logic (MSO2) is fixedparameter tractable (fpt) on C by linear time parameterised algorithms. An immediate question is whether this is best possible or whether the result can be extended to classes of unbounded treewidth. In this paper we show that in terms of treewidth, the theorem can not be extended much further. More specifically, we show that if C is a class of graphs which is closed under colourings and satisfies certain constructibility conditions such that the treewidth of C is not bounded by log 16 n then MSO2model checking is not fpt unless SAT can be solved in subexponential time. If the treewidth of C is not polylog. bounded, then MSO2model checking is not fpt unless all problems in the polynomialtime hierarchy can be solved in subexponential time. 1
Algorithmic meta theorems for circuit classes of constant and logarithmic depth
, 2011
"... An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic secondorder (mso) definable problems are lineartime solvable on gr ..."
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Cited by 5 (1 self)
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An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic secondorder (mso) definable problems are lineartime solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that msodefinable problems are (a) solvable by uniform constantdepth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmicdepth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1. Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width3 tree decompositions of trees in TC 0, which encapsulates most of the technical difficulties surrounding earlier results connecting tree automata and NC 1.
Lower Bounds for the Complexity of Monadic SecondOrder Logic
"... Abstract—Courcelle’s famous theorem from 1990 states that any property of graphs definable in monadic secondorder logic (MSO2) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO2 is fixedparameter tractable in linear time on any such class of graphs. ..."
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Cited by 5 (2 self)
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Abstract—Courcelle’s famous theorem from 1990 states that any property of graphs definable in monadic secondorder logic (MSO2) can be decided in linear time on any class of graphs of bounded treewidth, or in other words, MSO2 is fixedparameter tractable in linear time on any such class of graphs. From a logical perspective, Courcelle’s theorem establishes a sufficient condition, or an upper bound, for tractability of MSO2model checking. Whereas such upper bounds on the complexity of logics have received significant attention in the literature, almost nothing is known about corresponding lower bounds. In this paper we estbalish a strong lower bound for the complexity of monadic secondorder logic. In particular, we show that if C is any class of graphs which is closed under taking subgraphs and whose treewidth is not bounded by a polylogarithmic function (in fact, log c n for some small c suffices) then MSO2model checking is intractable on C (under a suitable assumption from complexity theory). I.
Faster Deciding MSO Properties of Trees of Fixed Height, and Some Consequences
"... We prove, in the universe of trees of bounded height, that for any MSO formula with m variables there exists a set of kernels such that the size of each of these kernels can be bounded by an elementary function of m. This yields a faster MSO model checking algorithm for trees of bounded height than ..."
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Cited by 3 (0 self)
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We prove, in the universe of trees of bounded height, that for any MSO formula with m variables there exists a set of kernels such that the size of each of these kernels can be bounded by an elementary function of m. This yields a faster MSO model checking algorithm for trees of bounded height than the one for general trees. From that we obtain, by means of interpretation, corresponding results for the classes of graphs of bounded treedepth (MSO2) and shrubdepth (MSO1), and thus we give wide generalizations of Lampis ’ (ESA 2010) and Ganian’s (IPEC 2011) results. In the second part of the paper we use this kernel structure to show that FO has the same expressive power as MSO1 on the graph classes of bounded shrubdepth. This makes bounded shrubdepth a good candidate for characterization of the hereditary classes of graphs on which FO and MSO1 coincide, a problem recently posed by Elberfeld, Grohe, and Tantau (LICS 2012).