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Document

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326, author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath}, title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {326--337}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326}, URN = {urn:nbn:de:0030-drops-34185}, doi = {10.4230/LIPIcs.STACS.2012.326}, annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity} }

Document

**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence.

Robert Ganian, Petr Hlineny, and Jan Obdrzalek. Clique-width: When Hard Does Not Mean Impossible. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 404-415, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{ganian_et_al:LIPIcs.STACS.2011.404, author = {Ganian, Robert and Hlineny, Petr and Obdrzalek, Jan}, title = {{Clique-width: When Hard Does Not Mean Impossible}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {404--415}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.404}, URN = {urn:nbn:de:0030-drops-30309}, doi = {10.4230/LIPIcs.STACS.2011.404}, annote = {Keywords: clique-width, bi-rank-width, minimum leaf out-branching, minimum leaf spanning tree, edge-disjoint paths} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known --e.g. [Fischer, Makowsky, and Ravve]-- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

Robert Ganian, Petr Hlinený, and Jan Obdrzálek. Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 73-83, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{ganian_et_al:LIPIcs.FSTTCS.2010.73, author = {Ganian, Robert and Hlinen\'{y}, Petr and Obdrz\'{a}lek, Jan}, title = {{Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {73--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.73}, URN = {urn:nbn:de:0030-drops-28541}, doi = {10.4230/LIPIcs.FSTTCS.2010.73}, annote = {Keywords: propositional model counting; satisfiability; rank-width; clique-width ; parameterized complexity} }

Document

**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint `${>}0$' or `${=}1$'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in $\textbf{NP} \cap \textbf{co-NP}$. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.

Tomas Brazdil, Vaclav Brozek, Antonin Kucera, and Jan Obdrzalek. Qualitative Reachability in Stochastic BPA Games. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 207-218, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brazdil_et_al:LIPIcs.STACS.2009.1837, author = {Brazdil, Tomas and Brozek, Vaclav and Kucera, Antonin and Obdrzalek, Jan}, title = {{Qualitative Reachability in Stochastic BPA Games}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {207--218}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1837}, URN = {urn:nbn:de:0030-drops-18375}, doi = {10.4230/LIPIcs.STACS.2009.1837}, annote = {Keywords: Stochastic games, Reachability, Pushdown automata} }

Document

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326, author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath}, title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {326--337}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326}, URN = {urn:nbn:de:0030-drops-34185}, doi = {10.4230/LIPIcs.STACS.2012.326}, annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity} }

Document

**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence.

Robert Ganian, Petr Hlineny, and Jan Obdrzalek. Clique-width: When Hard Does Not Mean Impossible. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 404-415, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.STACS.2011.404, author = {Ganian, Robert and Hlineny, Petr and Obdrzalek, Jan}, title = {{Clique-width: When Hard Does Not Mean Impossible}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {404--415}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.404}, URN = {urn:nbn:de:0030-drops-30309}, doi = {10.4230/LIPIcs.STACS.2011.404}, annote = {Keywords: clique-width, bi-rank-width, minimum leaf out-branching, minimum leaf spanning tree, edge-disjoint paths} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known --e.g. [Fischer, Makowsky, and Ravve]-- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

Robert Ganian, Petr Hlinený, and Jan Obdrzálek. Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 73-83, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.FSTTCS.2010.73, author = {Ganian, Robert and Hlinen\'{y}, Petr and Obdrz\'{a}lek, Jan}, title = {{Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {73--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.73}, URN = {urn:nbn:de:0030-drops-28541}, doi = {10.4230/LIPIcs.FSTTCS.2010.73}, annote = {Keywords: propositional model counting; satisfiability; rank-width; clique-width ; parameterized complexity} }

Document

**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint `${>}0$' or `${=}1$'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in $\textbf{NP} \cap \textbf{co-NP}$. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.

Tomas Brazdil, Vaclav Brozek, Antonin Kucera, and Jan Obdrzalek. Qualitative Reachability in Stochastic BPA Games. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 207-218, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

Copy BibTex To Clipboard

@InProceedings{brazdil_et_al:LIPIcs.STACS.2009.1837, author = {Brazdil, Tomas and Brozek, Vaclav and Kucera, Antonin and Obdrzalek, Jan}, title = {{Qualitative Reachability in Stochastic BPA Games}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {207--218}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1837}, URN = {urn:nbn:de:0030-drops-18375}, doi = {10.4230/LIPIcs.STACS.2009.1837}, annote = {Keywords: Stochastic games, Reachability, Pushdown automata} }

Document

**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

One of the most important algorithmic meta-theorems is a famous result
by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH).
In this paper we present a closely related result. We show that
even MSO1 model-checking with a fixed set of vertex labels,
but without edge-set quantifications, is not in XP wrt the formula
size as parameter for graph classes which are subgraph-closed and
whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no
subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

Robert Ganian, Petr Hlineny, Alexander Langer, Jan Obdržálek, Peter Rossmanith, and Somnath Sikdar. Lower Bounds on the Complexity of MSO_1 Model-Checking. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 326-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.STACS.2012.326, author = {Ganian, Robert and Hlineny, Petr and Langer, Alexander and Obdr\v{z}\'{a}lek, Jan and Rossmanith, Peter and Sikdar, Somnath}, title = {{Lower Bounds on the Complexity of MSO\underline1 Model-Checking}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {326--337}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.326}, URN = {urn:nbn:de:0030-drops-34185}, doi = {10.4230/LIPIcs.STACS.2012.326}, annote = {Keywords: Monadic Second-Order Logic, Treewidth, Lower Bounds, Exponential Time Hypothesis, Parameterized Complexity} }

Document

**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence.

Robert Ganian, Petr Hlineny, and Jan Obdrzalek. Clique-width: When Hard Does Not Mean Impossible. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 404-415, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.STACS.2011.404, author = {Ganian, Robert and Hlineny, Petr and Obdrzalek, Jan}, title = {{Clique-width: When Hard Does Not Mean Impossible}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {404--415}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.404}, URN = {urn:nbn:de:0030-drops-30309}, doi = {10.4230/LIPIcs.STACS.2011.404}, annote = {Keywords: clique-width, bi-rank-width, minimum leaf out-branching, minimum leaf spanning tree, edge-disjoint paths} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula. Previously, analogous algorithms have been known --e.g. [Fischer, Makowsky, and Ravve]-- with a single-exponential dependency on the clique-width of a formula. Our algorithm thus presents an exponential runtime improvement (since clique-width reaches up to exponentially higher values than rank-width), and can be of practical interest for small values of rank-width. We also provide an algorithm for the MAX-SAT problem along the same lines.

Robert Ganian, Petr Hlinený, and Jan Obdrzálek. Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 73-83, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

Copy BibTex To Clipboard

@InProceedings{ganian_et_al:LIPIcs.FSTTCS.2010.73, author = {Ganian, Robert and Hlinen\'{y}, Petr and Obdrz\'{a}lek, Jan}, title = {{Better Algorithms for Satisfiability Problems for Formulas of Bounded Rank-width}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {73--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.73}, URN = {urn:nbn:de:0030-drops-28541}, doi = {10.4230/LIPIcs.FSTTCS.2010.73}, annote = {Keywords: propositional model counting; satisfiability; rank-width; clique-width ; parameterized complexity} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

We consider a class of infinite-state stochastic games generated by stateless pushdown automata (or, equivalently, 1-exit recursive state machines), where the winning objective is specified by a regular set of target configurations and a qualitative probability constraint `${>}0$' or `${=}1$'. The goal of one player is to maximize the probability of reaching the target set so that the constraint is satisfied, while the other player aims at the opposite. We show that the winner in such games can be determined in $\textbf{NP} \cap \textbf{co-NP}$. Further, we prove that the winning regions for both players are regular, and we design algorithms which compute the associated finite-state automata. Finally, we show that winning strategies can be synthesized effectively.

Tomas Brazdil, Vaclav Brozek, Antonin Kucera, and Jan Obdrzalek. Qualitative Reachability in Stochastic BPA Games. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 207-218, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{brazdil_et_al:LIPIcs.STACS.2009.1837, author = {Brazdil, Tomas and Brozek, Vaclav and Kucera, Antonin and Obdrzalek, Jan}, title = {{Qualitative Reachability in Stochastic BPA Games}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {207--218}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1837}, URN = {urn:nbn:de:0030-drops-18375}, doi = {10.4230/LIPIcs.STACS.2009.1837}, annote = {Keywords: Stochastic games, Reachability, Pushdown automata} }

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