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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-dev.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-dev.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 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P.
Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n.
We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado.

Nikhil Balaji, Samir Datta, Raghav Kulkarni, and Supartha Podder. Graph Properties in Node-Query Setting: Effect of Breaking Symmetry. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{balaji_et_al:LIPIcs.MFCS.2016.17, author = {Balaji, Nikhil and Datta, Samir and Kulkarni, Raghav and Podder, Supartha}, title = {{Graph Properties in Node-Query Setting: Effect of Breaking Symmetry}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {17:1--17:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.17}, URN = {urn:nbn:de:0030-drops-64329}, doi = {10.4230/LIPIcs.MFCS.2016.17}, annote = {Keywords: query complexity, graph properties, symmetry and computation, forbidden subgraph} }

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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-dev.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} }

Document

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

Let H be a (non-empty) graph on n vertices, possibly containing isolated vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let alpha_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Omega( sqrt{alpha_H * n}), where Q(f_H) denotes the quantum query complexity of f_H.
As a consequence we obtain lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the critical probability.
We also use the above bound to show that Q(f_H) = Omega(n^{3/4}) for any H, improving on the previously best known bound of Omega(n^{2/3}) [M. Santha/A. Chi-Chih Yao, unpublished manuscript]. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Omega(n^{3/4}) bound for Q(f_H) matches the square root of the current best known bound for the randomized query complexity of f_H, which is Omega(n^{3/2}) due to Groger. Interestingly, the randomized bound of Omega(alpha_H * n) for f_H still remains open.
We also study the Subgraph Homomorphism Problem, denoted by f_{[H]}, and show that Q(f_{[H]}) = Omega(n).
Finally we extend our results to the 3-uniform hypergraphs. In particular, we show an Omega(n^{4/5}) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Omega(n^{3/4}) bound. For the Subgraph Homomorphism, we obtain an Omega(n^{3/2}) bound for the same.

Raghav Kulkarni and Supartha Podder. Quantum Query Complexity of Subgraph Isomorphism and Homomorphism. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kulkarni_et_al:LIPIcs.STACS.2016.48, author = {Kulkarni, Raghav and Podder, Supartha}, title = {{Quantum Query Complexity of Subgraph Isomorphism and Homomorphism}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {48:1--48:13}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.48}, URN = {urn:nbn:de:0030-drops-57495}, doi = {10.4230/LIPIcs.STACS.2016.48}, annote = {Keywords: quantum query complexity, subgraph isomorphism, monotone graph properties} }

Document

**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-dev.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-dev.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} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

A Boolean function on $N$ variables is called \emph{evasive} if its decision-tree complexity is $N$. A sequence $B_n$ of Boolean functions is \emph{eventually evasive} if $B_n$ is evasive for all sufficiently large $n$.
We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$, ``forbidden subgraph $H$'' is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.)
While Chowla's conjecture is not known to follow from the Extended Riemann Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's $L$ functions), we show (b) with the bound $O(n^{5/4-\epsilon})$ under ERH.
We also prove unconditional results: (a$'$) for any graph $H$, the query complexity of ``forbidden subgraph $H$'' is $\binom{n}{2} - O(1)$; (b$'$) for some constant $c>0$, all nontrivial monotone properties of graphs with $\le cn\log n+O(1)$ edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number theory such as Vinogradov's theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti, Khot, and Shi (2002), with a deeper analysis of the orbital structure of permutation groups and their connection to the distribution of prime numbers. Our unconditional results include stronger versions and generalizations of some result of Chakrabarti et al.

László Babai, Anandam Banerjee, Raghav Kulkarni, and Vipul Naik. Evasiveness and the Distribution of Prime Numbers. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 71-82, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{babai_et_al:LIPIcs.STACS.2010.2445, author = {Babai, L\'{a}szl\'{o} and Banerjee, Anandam and Kulkarni, Raghav and Naik, Vipul}, title = {{Evasiveness and the Distribution of Prime Numbers}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {71--82}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2445}, URN = {urn:nbn:de:0030-drops-24451}, doi = {10.4230/LIPIcs.STACS.2010.2445}, annote = {Keywords: Decision tree complexity, evasiveness, graph property, group action, Dirichlet primes, Extended Riemann Hypothesis} }

Document

**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

We present a deterministic way of assigning small (log bit) weights
to the edges of a bipartite planar graph so that the minimum weight
perfect matching becomes unique. The isolation lemma as described
in (Mulmuley et al. 1987) achieves the same for general graphs
using a randomized weighting scheme, whereas we can do it
deterministically when restricted to bipartite planar graphs. As a
consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under
the promise that its determinant is $0$ or $1$, thus obtaining a
highly parallel SPL algorithm for bipartite planar graphs. This
improves the earlier known bounds of non-uniform SPL by (Allender
et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and
Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect
matching in non-bipartite planar graphs, which has been open for a
long time. Our techniques are elementary and simple.

Samir Datta, Raghav Kulkarni, and Sambuddha Roy. Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 229-240, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{datta_et_al:LIPIcs.STACS.2008.1346, author = {Datta, Samir and Kulkarni, Raghav and Roy, Sambuddha}, title = {{Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {229--240}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1346}, URN = {urn:nbn:de:0030-drops-13465}, doi = {10.4230/LIPIcs.STACS.2008.1346}, annote = {Keywords: } }

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