Document

**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Planar Embedding is a drawing of a graph on the plane such that the edges do not intersect each other except at the vertices. We know that testing the planarity of a graph and computing its embedding (if it exists), can efficiently be computed, both sequentially [John E. Hopcroft and Robert Endre Tarjan, 1974] and in parallel [Vijaya Ramachandran and John H. Reif, 1994], when the entire graph is presented as input.
In the dynamic setting, the input graph changes one edge at a time through insertion and deletions and planarity testing/embedding has to be updated after every change. By storing auxilliary information we can improve the complexity of dynamic planarity testing/embedding over the obvious recomputation from scratch. In the sequential dynamic setting, there has been a series of works [David Eppstein et al., 1996; Giuseppe F. Italiano et al., 1993; Jacob Holm et al., 2018; Jacob Holm and Eva Rotenberg, 2020], culminating in the breakthrough result of polylog(n) sequential time (amortized) planarity testing algorithm of Holm and Rotenberg [Jacob Holm and Eva Rotenberg, 2020].
In this paper we study planar embedding through the lens of DynFO, a parallel dynamic complexity class introduced by Patnaik et al [Sushant Patnaik and Neil Immerman, 1997] (also [Guozhu Dong et al., 1995]). We show that it is possible to dynamically maintain whether an edge can be inserted to a planar graph without causing non-planarity in DynFO. We extend this to show how to maintain an embedding of a planar graph under both edge insertions and deletions, while rejecting edge insertions that violate planarity.
Our main idea is to maintain embeddings of only the triconnected components and a special two-colouring of separating pairs that enables us to side-step cascading flips when embedding of a biconnected planar graph changes, a major issue for sequential dynamic algorithms [Jacob Holm and Eva Rotenberg, 2020; Jacob Holm and Eva Rotenberg, 2020].

Samir Datta, Asif Khan, and Anish Mukherjee. Dynamic Planar Embedding Is in DynFO. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 39:1-39:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{datta_et_al:LIPIcs.MFCS.2023.39, author = {Datta, Samir and Khan, Asif and Mukherjee, Anish}, title = {{Dynamic Planar Embedding Is in DynFO}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {39:1--39:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.39}, URN = {urn:nbn:de:0030-drops-185736}, doi = {10.4230/LIPIcs.MFCS.2023.39}, annote = {Keywords: Dynamic Complexity, Planar graphs, Planar embedding} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Reachability, distance, and matching are some of the most fundamental graph problems that have been of particular interest in dynamic complexity theory in recent years [Samir Datta et al., 2018; Samir Datta et al., 2018; Samir Datta et al., 2020]. Reachability can be maintained with first-order update formulas, or equivalently in DynFO in general graphs with n nodes [Samir Datta et al., 2018], even under O(log(n)/log log(n)) changes per step [Samir Datta et al., 2018]. In the context of how large the number of changes can be handled, it has recently been shown [Samir Datta et al., 2020] that under a polylogarithmic number of changes, reachability is in DynFOpar in planar, bounded treewidth, and related graph classes - in fact in any graph where small non-zero circulation weights can be computed in NC.
We continue this line of investigation and extend the meta-theorem for reachability to distance and bipartite maximum matching with the same bounds. These are amongst the most general classes of graphs known where we can maintain these problems deterministically without using a majority quantifier and even maintain witnesses. For the bipartite matching result, modifying the approach from [Stephen A. Fenner et al., 2016], we convert the static non-zero circulation weights to dynamic matching-isolating weights.
While reachability is in DynFOar under O(log(n)/log log(n)) changes, no such bound is known for either distance or matching in any non-trivial class of graphs under non-constant changes. We show that, in the same classes of graphs as before, bipartite maximum matching is in DynFOar under O(log(n)/log log(n)) changes per step. En route to showing this we prove that the rank of a matrix can be maintained in DynFOar, also under O(log(n)/log log(n)) entry changes, improving upon the previous O(1) bound [Samir Datta et al., 2018]. This implies a similar extension for the non-uniform DynFO bound for maximum matching in general graphs and an alternate algorithm for maintaining reachability under O(log(n)/log log(n)) changes [Samir Datta et al., 2018].

Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari. Dynamic Meta-Theorems for Distance and Matching. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 118:1-118:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{datta_et_al:LIPIcs.ICALP.2022.118, author = {Datta, Samir and Gupta, Chetan and Jain, Rahul and Mukherjee, Anish and Sharma, Vimal Raj and Tewari, Raghunath}, title = {{Dynamic Meta-Theorems for Distance and Matching}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {118:1--118:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.118}, URN = {urn:nbn:de:0030-drops-164598}, doi = {10.4230/LIPIcs.ICALP.2022.118}, annote = {Keywords: Dynamic Complexity, Distance, Matching, Derandomization, Isolation, Matrix Rank} }

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

We show that for each single crossing graph H, a polynomially bounded weight function for all H-minor free graphs G can be constructed in logspace such that it gives nonzero weights to all the cycles in G. This class of graphs subsumes almost all classes of graphs for which such a weight function is known to be constructed in logspace. As a consequence, we obtain that for the class of H-minor free graphs where H is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani [David Eppstein and Vijay V. Vazirani, 2021], where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.

Samir Datta, Chetan Gupta, Rahul Jain, Anish Mukherjee, Vimal Raj Sharma, and Raghunath Tewari. Reachability and Matching in Single Crossing Minor Free Graphs. 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. 16:1-16:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2021.16, author = {Datta, Samir and Gupta, Chetan and Jain, Rahul and Mukherjee, Anish and Sharma, Vimal Raj and Tewari, Raghunath}, title = {{Reachability and Matching in Single Crossing Minor Free Graphs}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {16:1--16:16}, 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.2021.16}, URN = {urn:nbn:de:0030-drops-155277}, doi = {10.4230/LIPIcs.FSTTCS.2021.16}, annote = {Keywords: Reachability, Matching, Logspace, Single-crossing minor free graphs} }

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Track B: Automata, Logic, Semantics, and Theory of Programming

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges. Later, in 2018, this was extended for changes of size (log n)/(log log n), where n is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used.
In this paper we extend these results by showing that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining (1) undirected reachability, and (2) directed reachability under insertions. For classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes of polylogarithmic size. Examples for these classes include the class of planar graphs and graphs with bounded treewidth. The latter is shown here.
As the logics we consider cannot maintain reachability under changes of larger sizes, our results are optimal with respect to the size of the changes.

Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, and Thomas Zeume. Dynamic Complexity of Reachability: How Many Changes Can We Handle?. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 122:1-122:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{datta_et_al:LIPIcs.ICALP.2020.122, author = {Datta, Samir and Kumar, Pankaj and Mukherjee, Anish and Tawari, Anuj and Vortmeier, Nils and Zeume, Thomas}, title = {{Dynamic Complexity of Reachability: How Many Changes Can We Handle?}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {122:1--122:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.122}, URN = {urn:nbn:de:0030-drops-125291}, doi = {10.4230/LIPIcs.ICALP.2020.122}, annote = {Keywords: Dynamic complexity, reachability, complex changes} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Perfect matchings in planar graphs have been extensively studied and understood in the context of parallel complexity [P W Kastelyn, 1967; Vijay Vazirani, 1988; Meena Mahajan and Kasturi R. Varadarajan, 2000; Datta et al., 2010; Nima Anari and Vijay V. Vazirani, 2017]. However, corresponding results for maximum matchings have been elusive. We partly bridge this gap by proving:
1) An SPL upper bound for planar bipartite maximum matching search.
2) Planar maximum matching search reduces to planar maximum matching decision.
3) Planar maximum matching count reduces to planar bipartite maximum matching count and planar maximum matching decision.
The first bound improves on the known [Thanh Minh Hoang, 2010] bound of L^{C_=L} and is adaptable to any special bipartite graph class with non-zero circulation such as bounded genus graphs, K_{3,3}-free graphs and K_5-free graphs. Our bounds and reductions non-trivially combine techniques like the Gallai-Edmonds decomposition [L. Lovász and M.D. Plummer, 1986], deterministic isolation [Datta et al., 2010; Samir Datta et al., 2012; Rahul Arora et al., 2016], and the recent breakthroughs in the parallel search for planar perfect matchings [Nima Anari and Vijay V. Vazirani, 2017; Piotr Sankowski, 2018].

Samir Datta, Raghav Kulkarni, Ashish Kumar, and Anish Mukherjee. Planar Maximum Matching: Towards a Parallel Algorithm. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 21:1-21:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{datta_et_al:LIPIcs.ISAAC.2018.21, author = {Datta, Samir and Kulkarni, Raghav and Kumar, Ashish and Mukherjee, Anish}, title = {{Planar Maximum Matching: Towards a Parallel Algorithm}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.21}, URN = {urn:nbn:de:0030-drops-99695}, doi = {10.4230/LIPIcs.ISAAC.2018.21}, annote = {Keywords: maximum matching, planar graphs, parallel complexity, reductions} }

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**Published in:** LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

The well-known k-disjoint path problem (k-DPP) asks for pairwise vertex-disjoint paths between k specified pairs of vertices (s_i, t_i) in a given graph, if they exist. The decision version of the shortest k-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly, the search and counting versions ask for one such and the number of such shortest set of paths, respectively.
We restrict attention to the shortest k-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdière and Schrijver [Éric Colin de Verdière and Alexander Schrijver, 2011] for the general one-face problem. We do so by providing a randomised NC^2 algorithm along with an O(n^{omega/2}) time randomised sequential algorithm, for any fixed k. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals.
The algorithms are based on a bijection between a shortest k-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "pure" cycle covers.

Samir Datta, Siddharth Iyer, Raghav Kulkarni, and Anish Mukherjee. Shortest k-Disjoint Paths via Determinants. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 19:1-19:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{datta_et_al:LIPIcs.FSTTCS.2018.19, author = {Datta, Samir and Iyer, Siddharth and Kulkarni, Raghav and Mukherjee, Anish}, title = {{Shortest k-Disjoint Paths via Determinants}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {19:1--19:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-093-4}, ISSN = {1868-8969}, year = {2018}, volume = {122}, editor = {Ganguly, Sumit and Pandya, Paritosh}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.19}, URN = {urn:nbn:de:0030-drops-99183}, doi = {10.4230/LIPIcs.FSTTCS.2018.19}, annote = {Keywords: disjoint paths, planar graph, parallel algorithm, cycle cover, determinant, inclusion-exclusion} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Read-only memory (ROM) model is a classical model of computation to study time-space tradeoffs of algorithms. A classical result on the ROM model is that any algorithm to sort n numbers using O(s) words of extra space requires Omega (n^2/s) comparisons for lg n <= s <= n/lg n and the bound has also been recently matched by an algorithm. However, if we relax the model, we do have sorting algorithms (say Heapsort) that can sort using O(n lg n) comparisons using O(lg n) bits of extra space, even keeping a permutation of the given input sequence at anytime during the algorithm.
We address similar relaxations for graph algorithms. We show that a simple natural relaxation of ROM model allows us to implement fundamental graph search methods like BFS and DFS more space efficiently than in ROM. By simply allowing elements in the adjacency list of a vertex to be permuted, we show that, on an undirected or directed connected graph G having n vertices and m edges, the vertices of G can be output in a DFS or BFS order using O(lg n) bits of extra space and O(n^3 lg n) time. Thus we obtain similar bounds for reachability and shortest path distance (both for undirected and directed graphs). With a little more (but still polynomial) time, we can also output vertices in the lex-DFS order. As reachability in directed graphs (even in DAGs) and shortest path distance (even in undirected graphs) are NL-complete, and lex-DFS is P-complete, our results show that our model is more powerful than ROM if L != P.
En route, we also introduce and develop algorithms for another relaxation of ROM where the adjacency lists of the vertices are circular lists and we can modify only the heads of the lists. Here we first show a linear time DFS implementation using n + O(lg n) bits of extra space. Improving the extra space exponentially to only O(lg n) bits, we also obtain BFS and DFS albeit with a slightly slower running time. Both the models we propose maintain the graph structure throughout the algorithm, only the order of vertices in the adjacency list changes. In sharp contrast, for BFS and DFS, to the best of our knowledge, there are no algorithms in ROM that use even O(n^{1-epsilon}) bits of extra space; in fact, implementing DFS using cn bits for c<1 has been mentioned as an open problem. Furthermore, DFS (BFS, respectively) algorithms using n+o(n) (o(n), respectively) bits of extra use Reingold's [JACM, 2008] or Barnes et al's reachability algorithm [SICOMP, 1998] and hence have high runtime. Our results can be contrasted with the recent result of Buhrman et al. [STOC, 2014] which gives an algorithm for directed st-reachability on catalytic Turing machines using O(lg n) bits with catalytic space O(n^2 lg n) and time O(n^9).

Sankardeep Chakraborty, Anish Mukherjee, Venkatesh Raman, and Srinivasa Rao Satti. A Framework for In-place Graph Algorithms. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 13:1-13:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chakraborty_et_al:LIPIcs.ESA.2018.13, author = {Chakraborty, Sankardeep and Mukherjee, Anish and Raman, Venkatesh and Satti, Srinivasa Rao}, title = {{A Framework for In-place Graph Algorithms}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {13:1--13:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.13}, URN = {urn:nbn:de:0030-drops-94760}, doi = {10.4230/LIPIcs.ESA.2018.13}, annote = {Keywords: DFS, BFS, in-place algorithm, space-efficient graph algorithms, logspace} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Recently it was shown that the transitive closure of a directed graph can be updated using first-order formulas after insertions and deletions of single edges in the dynamic descriptive complexity framework by Dong, Su, and Topor, and Patnaik and Immerman. In other words, Reachability is in DynFO.
In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes. We show that the former problem can be maintained in DynFO(+, x) under changes affecting O({log n}/{log log n}) nodes, for graphs with n nodes. If the update formulas may use a majority quantifier then both Reachability and Distance can be maintained under changes that affect O(log^c n) nodes, for fixed c in N. Some preliminary results towards showing that distances are in DynFO are discussed.

Samir Datta, Anish Mukherjee, Nils Vortmeier, and Thomas Zeume. Reachability and Distances under Multiple Changes. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 120:1-120:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{datta_et_al:LIPIcs.ICALP.2018.120, author = {Datta, Samir and Mukherjee, Anish and Vortmeier, Nils and Zeume, Thomas}, title = {{Reachability and Distances under Multiple Changes}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {120:1--120:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.120}, URN = {urn:nbn:de:0030-drops-91245}, doi = {10.4230/LIPIcs.ICALP.2018.120}, annote = {Keywords: dynamic complexity, reachability, distances, complex changes} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

A strategy for constructing dynamic programs is introduced that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. It is established that if some program can maintain a query for log n change steps after an AC^1-computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in DynFO. As an application, it is shown that decision and optimisation problems defined by monadic second-order (MSO) and guarded second-order logic (GSO) formulas are in DynFO, if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result, Feferman-Vaught-type composition theorems for MSO and GSO are established that might be useful in their own right.

Samir Datta, Anish Mukherjee, Thomas Schwentick, Nils Vortmeier, and Thomas Zeume. A Strategy for Dynamic Programs: Start over and Muddle Through. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 98:1-98:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{datta_et_al:LIPIcs.ICALP.2017.98, author = {Datta, Samir and Mukherjee, Anish and Schwentick, Thomas and Vortmeier, Nils and Zeume, Thomas}, title = {{A Strategy for Dynamic Programs: Start over and Muddle Through}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {98:1--98:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.98}, URN = {urn:nbn:de:0030-drops-74470}, doi = {10.4230/LIPIcs.ICALP.2017.98}, annote = {Keywords: dynamic complexity, treewidth, monadic second order logic} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, using only logarithmic space. This deviates from the well known Baker's approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any "recursively sparse" graph class which contains a linear size matching and also for certain other classes like bounded genus graphs.
The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker's method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. The bounded degree assumption allows us to obtain a Logspace algorithm.

Samir Datta, Raghav Kulkarni, and Anish Mukherjee. Space-Efficient Approximation Scheme for Maximum Matching in Sparse Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 28:1-28:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{datta_et_al:LIPIcs.MFCS.2016.28, author = {Datta, Samir and Kulkarni, Raghav and Mukherjee, Anish}, title = {{Space-Efficient Approximation Scheme for Maximum Matching in Sparse Graphs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {28:1--28:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.28}, URN = {urn:nbn:de:0030-drops-64436}, doi = {10.4230/LIPIcs.MFCS.2016.28}, annote = {Keywords: maximum matching, approximation scheme, logspace, planar graphs} }

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