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

A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present a polynomial-time algorithm that uses O(g^{1/2} n^{1/2} log n)-space to find an O(g^{1/2} n^{1/2})-sized separator of a graph having n vertices and embedded on an orientable surface of genus g.

Chetan Gupta, Rahul Jain, and Raghunath Tewari. Time Space Optimal Algorithm for Computing Separators in Bounded Genus 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. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2021.23, author = {Gupta, Chetan and Jain, Rahul and Tewari, Raghunath}, title = {{Time Space Optimal Algorithm for Computing Separators in Bounded Genus Graphs}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {23:1--23:15}, 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.2021.23}, URN = {urn:nbn:de:0030-drops-155344}, doi = {10.4230/LIPIcs.FSTTCS.2021.23}, annote = {Keywords: Graph algorithms, space-bounded algorithms, surface embedded graphs, reachability, Euler genus, algorithmic graph theory, computational complexity theory} }

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

We show that given an embedding of an O(log n) genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists.
As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for O(log n) genus bipartite graphs.

Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gupta_et_al:LIPIcs.MFCS.2020.43, author = {Gupta, Chetan and Sharma, Vimal Raj and Tewari, Raghunath}, title = {{Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {43:1--43:13}, 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.2020.43}, URN = {urn:nbn:de:0030-drops-127099}, doi = {10.4230/LIPIcs.MFCS.2020.43}, annote = {Keywords: Logspace computation, High genus, Matching isolation} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The catalytic Turing machine is a model of computation defined by Buhrman, Cleve, Koucký, Loff, and Speelman (STOC 2014). Compared to the classical space-bounded Turing machine, this model has an extra space which is filled with arbitrary content in addition to the clean space. In such a model we study if this additional filled space can be used to increase the power of computation or not, with the condition that the initial content of this extra filled space must be restored at the end of the computation.
In this paper, we define the notion of unambiguous catalytic Turing machine and prove that under a standard derandomization assumption, the class of problems solved by an unambiguous catalytic Turing machine is same as the class of problems solved by a general nondeterministic catalytic Turing machine in the logspace setting.

Chetan Gupta, Rahul Jain, Vimal Raj Sharma, and Raghunath Tewari. Unambiguous Catalytic Computation. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 16:1-16:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gupta_et_al:LIPIcs.FSTTCS.2019.16, author = {Gupta, Chetan and Jain, Rahul and Sharma, Vimal Raj and Tewari, Raghunath}, title = {{Unambiguous Catalytic Computation}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {16:1--16:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.16}, URN = {urn:nbn:de:0030-drops-115782}, doi = {10.4230/LIPIcs.FSTTCS.2019.16}, annote = {Keywords: Catalytic computation, Logspace, Reinhardt-Allender} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

The reachability problem is to determine if there exists a path from one vertex to another in a graph. Grid graphs are the class of graphs where vertices are present on the lattice points of a two-dimensional grid, and an edge can occur between a vertex and its immediate horizontal or vertical neighbor only.
Asano et al. presented the first simultaneous time space bound for reachability in grid graphs by presenting an algorithm that solves the problem in polynomial time and O(n^(1/2 + epsilon)) space. In 2018, the space bound was improved to O~(n^(1/3)) by Ashida and Nakagawa.
In this paper, we show that reachability in an n vertex grid graph can be decided by an algorithm using O(n^(1/4 + epsilon)) space and polynomial time simultaneously.

Rahul Jain and Raghunath Tewari. An O(n^(1/4 +epsilon)) Space and Polynomial Algorithm for Grid Graph Reachability. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jain_et_al:LIPIcs.FSTTCS.2019.19, author = {Jain, Rahul and Tewari, Raghunath}, title = {{An O(n^(1/4 +epsilon)) Space and Polynomial Algorithm for Grid Graph Reachability}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.19}, URN = {urn:nbn:de:0030-drops-115813}, doi = {10.4230/LIPIcs.FSTTCS.2019.19}, annote = {Keywords: graph reachability, grid graph, graph algorithm, sublinear space algorithm} }

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**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability; however, their space complexity is also linear. On the other hand, Savitch’s algorithm takes quasipolynomial time although the space bound is O(log^2 n). Here, we study space efficient algorithms for deciding reachability that run in polynomial time.
In this paper, we show that given an n vertex directed graph of treewidth w along with its tree decomposition, there exists an algorithm running in polynomial time and O(w log n) space that solves the reachability problem.

Rahul Jain and Raghunath Tewari. Reachability in High Treewidth Graphs. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{jain_et_al:LIPIcs.ISAAC.2019.12, author = {Jain, Rahul and Tewari, Raghunath}, title = {{Reachability in High Treewidth Graphs}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.12}, URN = {urn:nbn:de:0030-drops-115087}, doi = {10.4230/LIPIcs.ISAAC.2019.12}, annote = {Keywords: graph reachability, simultaneous time-space upper bound, tree decomposition} }

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**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We show that given an embedding of an O(log n) genus graph G and two vertices s and t in G, deciding if there is a path from s to t in G is in unambiguous logarithmic space.
Unambiguous computation is a restriction of nondeterministic computation where the nondeterministic machine has at most one accepting computation path on each input. An important fundamental question in computational complexity theory is whether this is an actual restriction or are unambiguous computations as powerful as general nondeterminism. We investigate this problem in the domain of logarithmic space bounded computations, where the corresponding unambiguous and general nondeterministic classes are UL and NL respectively.
In 1997 Reinhardt and Allender showed that NL and UL are equal in a non-uniform model. More specifically they showed that if one can efficiently construct an O(log n)-bit min-unique weight function for a graph, then these classes are equal unconditionally as well. In other words, they gave a UL algorithm to solve reachability in graphs with a min-unique weight assignment. Using this approach reachability in various classes of graphs such as planar graphs, constant genus graphs, minor free graphs, etc., have been shown to be in UL by devising min-unique weight functions for those classes.
In this paper we improve these results by constructing a min-unique weight function for O(log n) genus graphs. We define signature of a path in a graph as the parity of the number of crossings of that path with respect to each handle of the surface on which the graph is embedded. We construct our weight function in two steps. First we ensure that between any pair of vertices, amongst all paths having the same signature, the minimum weight path is unique. Now since in a genus g graph there are 2^{2g} many possible signatures, we use the hashing scheme of Fredman, Komlós and Szemerédi to isolate a unique minimum weight path among these 2^{2g} many paths isolated in the first step.

Chetan Gupta, Vimal Raj Sharma, and Raghunath Tewari. Reachability in O(log n) Genus Graphs is in Unambiguous Logspace. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gupta_et_al:LIPIcs.STACS.2019.34, author = {Gupta, Chetan and Sharma, Vimal Raj and Tewari, Raghunath}, title = {{Reachability in O(log n) Genus Graphs is in Unambiguous Logspace}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {34:1--34:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.34}, URN = {urn:nbn:de:0030-drops-102730}, doi = {10.4230/LIPIcs.STACS.2019.34}, annote = {Keywords: logspace unambiguity, high genus, path isolation} }

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

Savitch showed in 1970 that nondeterministic logspace (NL) is contained in deterministic O(log^2(n)) space but his algorithm requires quasipolynomial time. The question whether we can have a deterministic algorithm for every problem in NL that requires polylogarithmic space and simultaneously runs in polynomial time was left open.
In this paper we give a partial solution to this problem and show that for every language in NL there exists an unambiguous nondeterministic algorithm that requires O(log^2(n)) space and simultaneously runs in polynomial time.

Vivek Anand T Kallampally and Raghunath Tewari. Trading Determinism for Time in Space Bounded Computations. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kallampally_et_al:LIPIcs.MFCS.2016.10, author = {Kallampally, Vivek Anand T and Tewari, Raghunath}, title = {{Trading Determinism for Time in Space Bounded Computations}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {10:1--10:13}, 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.2016.10}, URN = {urn:nbn:de:0030-drops-64268}, doi = {10.4230/LIPIcs.MFCS.2016.10}, annote = {Keywords: space complexity, unambiguous computations, Savitch's Theorem} }

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

The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for K3,3-free and K5-free bipartite graphs. That is, we give a deterministic log-space construction of such a weight assignment for these graphs. Such a construction was known previously for planar bipartite graphs. Our result implies that the perfect matching problem for K3,3-free and K5-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result – reachability for K3,3-free and K5-free graphs is in UL.

Rahul Arora, Ashu Gupta, Rohit Gurjar, and Raghunath Tewari. Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arora_et_al:LIPIcs.STACS.2016.10, author = {Arora, Rahul and Gupta, Ashu and Gurjar, Rohit and Tewari, Raghunath}, title = {{Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {10:1--10:15}, 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.2016.10}, URN = {urn:nbn:de:0030-drops-57116}, doi = {10.4230/LIPIcs.STACS.2016.10}, annote = {Keywords: bipartite matching, derandomization, isolation lemma, SPL, minor-free graph} }

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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We obtain the following new simultaneous time-space upper bounds for the directed reachability problem. (1) A polynomial-time, O(n^{2/3} * g^{1/3})-space algorithm for directed graphs embedded on orientable surfaces of genus g. (2) A polynomial-time, O(n^{2/3})-space algorithm for all H-minor-free graphs given the tree decomposition, and (3) for K_{3,3}-free and K_5-free graphs, a polynomial-time, O(n^{1/2 + epsilon})-space algorithm, for every epsilon > 0.
For the general directed reachability problem, the best known simultaneous time-space upper bound is the BBRS bound, due to Barnes, Buss, Ruzzo, and Schieber, which achieves a space bound of O(n/2^{k * sqrt(log(n))}) with polynomial running time, for any constant k. It is a significant open question to improve this bound for reachability over general directed graphs. Our algorithms beat the BBRS bound for graphs embedded on surfaces of genus n/2^{omega(sqrt(log(n))}, and for all H-minor-free graphs. This significantly broadens the class of directed graphs for which the BBRS bound can be improved.

Diptarka Chakraborty, A. Pavan, Raghunath Tewari, N. V. Vinodchandran, and Lin Forrest Yang. New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 585-595, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2014.585, author = {Chakraborty, Diptarka and Pavan, A. and Tewari, Raghunath and Vinodchandran, N. V. and Yang, Lin Forrest}, title = {{New Time-Space Upperbounds for Directed Reachability in High-genus and H-minor-free Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {585--595}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.585}, URN = {urn:nbn:de:0030-drops-48730}, doi = {10.4230/LIPIcs.FSTTCS.2014.585}, annote = {Keywords: Reachability, Space complexity, Time-Space Efficient Algorithms, Graphs on Surfaces, Minor Free Graphs, Savitch's Algorithm, BBRS Bound} }

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

We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus:
(1) BPM in planar graphs is in UL (improves upon the SPL bound from Datta, Kulkarni, and Roy;
(2) BPM in constant genus graphs is in NL (orthogonal to the SPL bound from Datta, Kulkarni, Tewari, and Vinodchandran.;
(3) BPM in poly-logarithmic genus graphs is in NC; (extends the NC bound for O(log n) genus graphs from Mahajan and Varadarajan, and Kulkarni, Mahajan, and Varadarajan.
For Part (1) we combine the flow technique of Miller and Naor with the double counting technique of Reinhardt and Allender . For Part (2) and (3) we extend Miller and Naor's result to higher genus surfaces in the spirit of Chambers, Erickson and Nayyeri.

Samir Datta, Arjun Gopalan, Raghav Kulkarni, and Raghunath Tewari. Improved Bounds for Bipartite Matching on Surfaces. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 254-265, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{datta_et_al:LIPIcs.STACS.2012.254, author = {Datta, Samir and Gopalan, Arjun and Kulkarni, Raghav and Tewari, Raghunath}, title = {{Improved Bounds for Bipartite Matching on Surfaces}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {254--265}, 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.2012.254}, URN = {urn:nbn:de:0030-drops-34141}, doi = {10.4230/LIPIcs.STACS.2012.254}, annote = {Keywords: Perfect Matching, Graphs on Surfaces, Space Complexity, NC, UL} }

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

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class SPL. Since SPL is contained in the logspace counting classes oplus L (in fact in mod_k for all k >= 2), C=L, and PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in FL^SPL. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier.
As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.

Samir Datta, Raghav Kulkarni, Raghunath Tewari, and N. Variyam Vinodchandran. Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 579-590, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{datta_et_al:LIPIcs.STACS.2011.579, author = {Datta, Samir and Kulkarni, Raghav and Tewari, Raghunath and Vinodchandran, N. Variyam}, title = {{Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {579--590}, 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.2011.579}, URN = {urn:nbn:de:0030-drops-30450}, doi = {10.4230/LIPIcs.STACS.2011.579}, annote = {Keywords: perfect matching, bounded genus graphs, isolation problem} }