Document

**Published in:** LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Width-based automated theorem proving is a framework where counter-examples for graph theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time 2^{2^{k^{O(1)}}} on the class of graphs of treewidth at most k. In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counter-example of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form ℙ₁ → ℙ₂, for graph properties ℙ₁ and ℙ₂, where ℙ₁ is a property closed under subgraphs.
As a concrete application, we use our framework in the study of graph theoretic conjectures related to coloring triangle free graphs. In particular, our algorithm is able to show that Reed’s conjecture for triangle free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counter-examples for non-valid strengthenings of Reed’s conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.

Mateus de Oliveira Oliveira and Farhad Vadiee. State Canonization and Early Pruning in Width-Based Automated Theorem Proving. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{deoliveiraoliveira_et_al:LIPIcs.FSCD.2024.33, author = {de Oliveira Oliveira, Mateus and Vadiee, Farhad}, title = {{State Canonization and Early Pruning in Width-Based Automated Theorem Proving}}, booktitle = {9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)}, pages = {33:1--33:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-323-2}, ISSN = {1868-8969}, year = {2024}, volume = {299}, editor = {Rehof, Jakob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.33}, URN = {urn:nbn:de:0030-drops-203622}, doi = {10.4230/LIPIcs.FSCD.2024.33}, annote = {Keywords: Width-Based Automated Theorem Proving, Dynamic Programming, Parameterized Complexity} }

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**Published in:** LIPIcs, Volume 233, 20th International Symposium on Experimental Algorithms (SEA 2022)

Many important NP-hard problems, arising in a wide variety of contexts, can be reduced straightforwardly to the satisfiability problem for CSPs whose underlying graph is a grid. In this work, we push forward the study of grid CSPs by analyzing, from an experimental perspective, a symbolic parameter called smoothness.
More specifically, we implement an algorithm that provably works in polynomial time on grids of polynomial smoothness. Subsequently, we compare our algorithm with standard combinatorial optimization techniques, such as SAT-solving and integer linear programming (ILP). For this comparison, we use a class of grid-CSPs encoding the pigeonhole principle. We demonstrate, empirically, that these CSPs have polynomial smoothness and that our algorithm terminates in polynomial time.
On the other hand, as strong evidence that the grid-like encoding is not destroying the essence of the pigeonhole principle, we show that the standard propositional translation of pigeonhole CSPs remains hard for state-of-the-art SAT solvers, such as minisat and glucose, and even to state-of-the-art integer linear-programming solvers, such as Coin-OR CBC.

Vasily Alferov and Mateus de Oliveira Oliveira. On the Satisfiability of Smooth Grid CSPs. In 20th International Symposium on Experimental Algorithms (SEA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 233, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alferov_et_al:LIPIcs.SEA.2022.18, author = {Alferov, Vasily and de Oliveira Oliveira, Mateus}, title = {{On the Satisfiability of Smooth Grid CSPs}}, booktitle = {20th International Symposium on Experimental Algorithms (SEA 2022)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-251-8}, ISSN = {1868-8969}, year = {2022}, volume = {233}, editor = {Schulz, Christian and U\c{c}ar, Bora}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2022.18}, URN = {urn:nbn:de:0030-drops-165526}, doi = {10.4230/LIPIcs.SEA.2022.18}, annote = {Keywords: Grid CSPs, Smoothness, SAT Solving, Linear Programming} }

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**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

In the Intersection Non-emptiness problem, we are given a list of finite automata A_1, A_2,… , A_m over a common alphabet Σ as input, and the goal is to determine whether some string w ∈ Σ^* lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B_0 or B_{1/2} and already PSPACE-hard when all automata accept a language from the level B_1. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC^0 when restricted to level L_0; complete for L or NL, depending on the input representation, when restricted to languages in the level L_{1/2}; NP-complete when the input is given as DFAs accepting a language in L_1 or L_{3/2}; and finally, PSPACE-complete when the input automata accept languages in level L_2 or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in L_1 or L_{3/2} does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

Emmanuel Arrighi, Henning Fernau, Stefan Hoffmann, Markus Holzer, Ismaël Jecker, Mateus de Oliveira Oliveira, and Petra Wolf. On the Complexity of Intersection Non-emptiness for Star-Free Language Classes. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2021.34, author = {Arrighi, Emmanuel and Fernau, Henning and Hoffmann, Stefan and Holzer, Markus and Jecker, Isma\"{e}l and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{On the Complexity of Intersection Non-emptiness for Star-Free Language Classes}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {34:1--34:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.34}, URN = {urn:nbn:de:0030-drops-155456}, doi = {10.4230/LIPIcs.FSTTCS.2021.34}, annote = {Keywords: Intersection Non-emptiness Problem, Star-Free Languages, Straubing-Th\'{e}rien Hierarchy, dot-depth Hierarchy, Commutative Languages, Complexity} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

In this work, we consider the following order reconfiguration problem: Given a graph G together with linear orders ω and ω' of the vertices of G, can one transform ω into ω' by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most k? We show that this problem always has an affirmative answer when the input linear orders ω and ω' have cutwidth (pathwidth) at most k/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Order Reconfiguration Under Width Constraints. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.MFCS.2021.8, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Order Reconfiguration Under Width Constraints}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.8}, URN = {urn:nbn:de:0030-drops-144486}, doi = {10.4230/LIPIcs.MFCS.2021.8}, annote = {Keywords: Parameterized Complexity, Order Reconfiguration, String Rewriting Systems} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Clique-width and treewidth are two of the most important and useful graph parameters, and several problems can be solved efficiently when restricted to graphs of bounded clique-width or treewidth. Bounded treewidth implies bounded clique-width, but not vice versa. Problems like Longest Cycle, Longest Path, MaxCut, Edge Dominating Set, and Graph Coloring are fixed-parameter tractable when parameterized by the treewidth, but they cannot be solved in FPT time when parameterized by the clique-width unless FPT = W[1], as shown by Fomin, Golovach, Lokshtanov, and Saurabh [SIAM J. Comput. 2010, SIAM J. Comput. 2014]. For a given problem that is fixed-parameter tractable when parameterized by treewidth, but intractable when parameterized by clique-width, there may exist infinite families of instances of bounded clique-width and unbounded treewidth where the problem can be solved efficiently. In this work, we initiate a systematic study of the parameters co-treewidth (the treewidth of the complement of the input graph) and co-degeneracy (the degeneracy of the complement of the input graph). We show that Longest Cycle, Longest Path, and Edge Dominating Set are FPT when parameterized by co-degeneracy. On the other hand, Graph Coloring is para-NP-complete when parameterized by co-degeneracy but FPT when parameterized by the co-treewidth. Concerning MaxCut, we give an FPT algorithm parameterized by co-treewidth, while we leave open the complexity of the problem parameterized by co-degeneracy. Additionally, we show that Precoloring Extension is fixed-parameter tractable when parameterized by co-treewidth, while this problem is known to be W[1]-hard when parameterized by treewidth. These results give evidence that co-treewidth is a useful width parameter for handling dense instances of problems for which an FPT algorithm for clique-width is unlikely to exist. Finally, we develop an algorithmic framework for co-degeneracy based on the notion of Bondy-Chvátal closure.

Gabriel L. Duarte, Mateus de Oliveira Oliveira, and Uéverton S. Souza. Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 42:1-42:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{duarte_et_al:LIPIcs.MFCS.2021.42, author = {Duarte, Gabriel L. and de Oliveira Oliveira, Mateus and Souza, U\'{e}verton S.}, title = {{Co-Degeneracy and Co-Treewidth: Using the Complement to Solve Dense Instances}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {42:1--42:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.42}, URN = {urn:nbn:de:0030-drops-144828}, doi = {10.4230/LIPIcs.MFCS.2021.42}, annote = {Keywords: FPT, treewidth, degeneracy, complement graph, Bondy-Chv\'{a}tal closure} }

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**Published in:** LIPIcs, Volume 190, 19th International Symposium on Experimental Algorithms (SEA 2021)

In the Steiner tree problem, the input consists of an edge-weighted graph G together with a set S of terminal vertices. The goal is to find a minimum weight tree in G that spans all terminals. This fundamental NP-hard problem has direct applications in many subfields of combinatorial optimization, such as planning, scheduling, etc. In this work we introduce a new heuristic for the Steiner tree problem, based on a simple routine for improving the cost of sub-optimal Steiner trees: first, the sub-optimal tree is split into three connected components, and then these components are reconnected by using an algorithm that computes an optimal Steiner tree with 3-terminals (the roots of the three components). We have implemented our heuristic into a solver and compared it with several state-of-the-art solvers on well-known data sets. Our solver performs very well across all the data sets, and outperforms most of the other benchmarked solvers on very large graphs, which have been either obtained from real-world applications or from randomly generated data sets.

Emmanuel Arrighi and Mateus de Oliveira Oliveira. Three Is Enough for Steiner Trees. In 19th International Symposium on Experimental Algorithms (SEA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 190, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arrighi_et_al:LIPIcs.SEA.2021.5, author = {Arrighi, Emmanuel and de Oliveira Oliveira, Mateus}, title = {{Three Is Enough for Steiner Trees}}, booktitle = {19th International Symposium on Experimental Algorithms (SEA 2021)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-185-6}, ISSN = {1868-8969}, year = {2021}, volume = {190}, editor = {Coudert, David and Natale, Emanuele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2021.5}, URN = {urn:nbn:de:0030-drops-137775}, doi = {10.4230/LIPIcs.SEA.2021.5}, annote = {Keywords: Steiner Tree, Heuristics, 3TST} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

We are studying a weighted version of a linear extension problem, given some finite partial order ρ, called Completion of an Ordering. While this problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, or computer memory management. Each application yields a special ρ. We also relate the interval width of ρ to parameterizations such as maximum range that have been introduced earlier in these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems.

Emmanuel Arrighi, Henning Fernau, Mateus de Oliveira Oliveira, and Petra Wolf. Width Notions for Ordering-Related Problems. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arrighi_et_al:LIPIcs.FSTTCS.2020.9, author = {Arrighi, Emmanuel and Fernau, Henning and de Oliveira Oliveira, Mateus and Wolf, Petra}, title = {{Width Notions for Ordering-Related Problems}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {9:1--9:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.9}, URN = {urn:nbn:de:0030-drops-132505}, doi = {10.4230/LIPIcs.FSTTCS.2020.9}, annote = {Keywords: Parameterized algorithms, interval width, linear extension, one-sided crossing minimization, Kemeny rank aggregation, grouping by swapping} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

It can be shown that each permutation group G ⊑ 𝕊_n can be embedded, in a well defined sense, in a connected graph with O(n+|G|) vertices. Some groups, however, require much fewer vertices. For instance, 𝕊_n itself can be embedded in the n-clique K_n, a connected graph with n vertices.
In this work, we show that the minimum size of a context-free grammar generating a finite permutation group G⊑ 𝕊_n can be upper bounded by three structural parameters of connected graphs embedding G: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group G ⊑ 𝕊_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΔlogΔ)}⋅ m^{O(k)}.
The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ω(n)} lower bound on the grammar complexity of the symmetric group 𝕊_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of 𝕊_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 1 and maximum degree o(n/log n).

Lars Jaffke, Mateus de Oliveira Oliveira, and Hans Raj Tiwary. Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2020.50, author = {Jaffke, Lars and de Oliveira Oliveira, Mateus and Tiwary, Hans Raj}, title = {{Compressing Permutation Groups into Grammars and Polytopes. A Graph Embedding Approach}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {50:1--50:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.50}, URN = {urn:nbn:de:0030-drops-127161}, doi = {10.4230/LIPIcs.MFCS.2020.50}, annote = {Keywords: Permutation Groups, Context Free Grammars, Extension Complexity, Graph Embedding Complexity} }

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**Published in:** LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)

In the embedded planar diameter improvement problem (EPDI) we are given a graph G embedded in the plane and a positive integer d. The goal is to determine whether one can add edges to the planar embedding of G in such a way that planarity is preserved and in such a way that the resulting graph has diameter at most d. Using non-constructive techniques derived from Robertson and Seymour's graph minor theory, together with the effectivization by self-reduction technique introduced by Fellows and Langston, one can show that EPDI can be solved in time f(d)* |V(G)|^{O(1)} for some function f(d). The caveat is that this algorithm is not strongly uniform in the sense that the function f(d) is not known to be computable. On the other hand, even the problem of determining whether EPDI can be solved in time f_1(d)* |V(G)|^{f_2(d)} for computable functions f_1 and f_2 has been open for more than two decades [Cohen at. al. Journal of Computer and System Sciences, 2017]. In this work we settle this later problem by showing that EPDI can be solved in time f(d)* |V(G)|^{O(d)} for some computable function f. Our techniques can also be used to show that the embedded k-outerplanar diameter improvement problem (k-EOPDI), a variant of EPDI where the resulting graph is required to be k-outerplanar instead of planar, can be solved in time f(d)* |V(G)|^{O(k)} for some computable function f. This shows that for each fixed k, the problem k-EOPDI is strongly uniformly fixed parameter tractable with respect to the diameter parameter d.

Daniel Lokshtanov, Mateus de Oliveira Oliveira, and Saket Saurabh. A Strongly-Uniform Slicewise Polynomial-Time Algorithm for the Embedded Planar Diameter Improvement Problem. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lokshtanov_et_al:LIPIcs.IPEC.2018.25, author = {Lokshtanov, Daniel and de Oliveira Oliveira, Mateus and Saurabh, Saket}, title = {{A Strongly-Uniform Slicewise Polynomial-Time Algorithm for the Embedded Planar Diameter Improvement Problem}}, booktitle = {13th International Symposium on Parameterized and Exact Computation (IPEC 2018)}, pages = {25:1--25:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-084-2}, ISSN = {1868-8969}, year = {2019}, volume = {115}, editor = {Paul, Christophe and Pilipczuk, Michal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.25}, URN = {urn:nbn:de:0030-drops-102265}, doi = {10.4230/LIPIcs.IPEC.2018.25}, annote = {Keywords: Embedded Planar Diameter Improvement, Constructive Algorithms, Nooses} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence F, a positive integer t, and a connected graph G of maximum degree at most D, and determines, in time f(|F|,t)*2^O(D*t)*|G|^O(t), whether G has a supergraph G' of treewidth at most t such that G' satisfies F.
The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)*2^O(D*d)*|G|^O(d), whether a connected graph of maximum degree D has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has a k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d.
This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.

Mateus de Oliveira Oliveira. On Supergraphs Satisfying CMSO Properties. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deoliveiraoliveira:LIPIcs.CSL.2017.33, author = {de Oliveira Oliveira, Mateus}, title = {{On Supergraphs Satisfying CMSO Properties}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.33}, URN = {urn:nbn:de:0030-drops-77058}, doi = {10.4230/LIPIcs.CSL.2017.33}, annote = {Keywords: On Supergraphs Satisfying CMSO Properties} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

We introduce the notion of monotone linear-programming circuits (MLP circuits), a model of computation for partial Boolean functions. Using this model, we prove the following results.
1. MLP circuits are superpolynomially stronger than monotone Boolean circuits.
2. MLP circuits are exponentially stronger than monotone span programs.
3. MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovasz-Schrijver proof systems, and for mixed Lovasz-Schrijver proof systems.
4. The Lovasz-Schrijver proof system cannot be polynomially simulated by the cutting planes proof system. This is the first result showing a separation between these two proof systems.
Finally, we discuss connections between the problem of proving lower bounds on the size of MLPs and the problem of proving lower bounds on extended formulations of polytopes.

Mateus de Oliveira Oliveira and Pavel Pudlák. Representations of Monotone Boolean Functions by Linear Programs. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deoliveiraoliveira_et_al:LIPIcs.CCC.2017.3, author = {de Oliveira Oliveira, Mateus and Pudl\'{a}k, Pavel}, title = {{Representations of Monotone Boolean Functions by Linear Programs}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {3:1--3:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.3}, URN = {urn:nbn:de:0030-drops-75200}, doi = {10.4230/LIPIcs.CCC.2017.3}, annote = {Keywords: Monotone Linear Programming Circuits, Lov\'{a}sz-Schrijver Proof System, Cutting-Planes Proof System, Feasible Interpolation, Lower Bounds} }

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**Published in:** LIPIcs, Volume 78, 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)

In the Maximum-Duo Preservation String Mapping (Max-Duo PSM) problem, the input consists of two related strings A and B of length n and a nonnegative integer k. The objective is to determine whether there exists a mapping m from the set of positions of A to the set of positions of B that maps only to positions with the same character and preserves at least k duos, which are pairs of adjacent positions. We develop a randomized algorithm that solves Max-Duo PSM in time 4^k * n^{O(1)}, and a deterministic algorithm that solves this problem in time 6.855^k * n^{O(1)}. The previous best known (deterministic) algorithm for this problem has running time (8e)^{2k+o(k)} * n^{O(1)} [Beretta et al., Theor. Comput. Sci. 2016]. We also show that Max-Duo PSM admits a problem kernel of size O(k^3), improving upon the previous best known problem kernel of size O(k^6).

Christian Komusiewicz, Mateus de Oliveira Oliveira, and Meirav Zehavi. Revisiting the Parameterized Complexity of Maximum-Duo Preservation String Mapping. In 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 78, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{komusiewicz_et_al:LIPIcs.CPM.2017.11, author = {Komusiewicz, Christian and de Oliveira Oliveira, Mateus and Zehavi, Meirav}, title = {{Revisiting the Parameterized Complexity of Maximum-Duo Preservation String Mapping}}, booktitle = {28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017)}, pages = {11:1--11:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-039-2}, ISSN = {1868-8969}, year = {2017}, volume = {78}, editor = {K\"{a}rkk\"{a}inen, Juha and Radoszewski, Jakub and Rytter, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2017.11}, URN = {urn:nbn:de:0030-drops-73436}, doi = {10.4230/LIPIcs.CPM.2017.11}, annote = {Keywords: comparative genomics, parameterized complexity, kernelization} }

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**Published in:** LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016)

The ground term reachability problem consists in determining whether a given variable-free term t can be transformed into a given variable-free term t' by the application of rules from a term rewriting system R. The joinability problem, on the other hand, consists in determining whether there exists a variable-free term t'' which is reachable both from t and from t'. Both problems have proven to be of fundamental importance for several subfields of computer science. Nevertheless, these problems are undecidable even when restricted to linear term rewriting systems. In this work, we approach reachability and joinability in linear term rewriting systems from the perspective of parameterized complexity theory, and show that these problems are fixed parameter tractable with respect to the depth of derivations. More precisely, we consider a notion of parallel rewriting, in which an unbounded number of rules can be applied simultaneously to a term as long as these rules do not interfere with each other. A term t_1 can reach a term t_2 in depth d if t_2 can be obtained from t_1 by the application of d parallel rewriting steps. Our main result states that for some function f(R,d), and for any linear term rewriting system R, one can determine in time f(R,d)*|t_1|*|t_2| whether a ground term t_2 can be reached from a ground term t_1 in depth at most d by the application of rules from R. Additionally, one can determine in time f(R,d)^2*|t_1|*|t_2| whether there exists a ground term u, such that u can be reached from both t_1 and t_2 in depth at most d. Our algorithms improve exponentially on exhaustive search, which terminates in time 2^{|t_1|*2^{O(d)}}*|t_2|, and can be applied with regard to any linear term rewriting system, irrespective of whether the rewriting system in question is terminating or confluent.

Mateus de Oliveira Oliveira. Ground Reachability and Joinability in Linear Term Rewriting Systems are Fixed Parameter Tractable with Respect to Depth. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 63, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{deoliveiraoliveira:LIPIcs.IPEC.2016.25, author = {de Oliveira Oliveira, Mateus}, title = {{Ground Reachability and Joinability in Linear Term Rewriting Systems are Fixed Parameter Tractable with Respect to Depth}}, booktitle = {11th International Symposium on Parameterized and Exact Computation (IPEC 2016)}, pages = {25:1--25:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-023-1}, ISSN = {1868-8969}, year = {2017}, volume = {63}, editor = {Guo, Jiong and Hermelin, Danny}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.25}, URN = {urn:nbn:de:0030-drops-69256}, doi = {10.4230/LIPIcs.IPEC.2016.25}, annote = {Keywords: Linear Term Rewriting Systems, Ground Reachability, Ground Joinability, Fixed Parameter Tractability} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

In this work we study the relationship between size and treewidth of circuits computing variants of the element distinctness function. First, we show that for each n, any circuit of treewidth t computing the element distinctness function delta_n:{0,1}^n -> {0,1} must have size at least Omega((n^2)/(2^{O(t)}*log(n))). This result provides a non-trivial generalization of a super-linear lower bound for the size of Boolean formulas (treewidth 1) due to Neciporuk. Subsequently, we turn our attention to read-once circuits, which are circuits where each variable labels at most one input vertex. For each n, we show that any read-once circuit of treewidth t and size s computing a variant tau_n:{0,1}^n -> {0,1} of the element distinctness function must satisfy the inequality t * log(s) >= Omega(n/log(n)). Using this inequality in conjunction with known results in structural graph theory, we show that for each fixed graph H, read-once circuits computing tau_n which exclude H as a minor must have size at least Omega(n^2/log^{4}(n)). For certain well studied functions, such as the triangle-freeness function, this last lower bound can be improved to Omega(n^2/log^2(n)).

Mateus de Oliveira Oliveira. Size-Treewidth Tradeoffs for Circuits Computing the Element Distinctness Function. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{deoliveiraoliveira:LIPIcs.STACS.2016.56, author = {de Oliveira Oliveira, Mateus}, title = {{Size-Treewidth Tradeoffs for Circuits Computing the Element Distinctness Function}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {56:1--56:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.56}, URN = {urn:nbn:de:0030-drops-57571}, doi = {10.4230/LIPIcs.STACS.2016.56}, annote = {Keywords: non-linear lower bounds, treewidth, element distinctness} }