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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

A graph is 2-degenerate if every subgraph contains a vertex of degree at most 2. We show that every 2-degenerate graph can be drawn with straight lines such that the drawing decomposes into 4 plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of 2-degenerate graphs is at most 4. On the other hand, we show that there are 2-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into 2 plane graphs. That is, there are 2-degenerate graphs with geometric thickness, and hence geometric arboricity, at least 3. This answers two questions posed by Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004].

Rahul Jain, Marco Ricci, Jonathan Rollin, and André Schulz. On the Geometric Thickness of 2-Degenerate Graphs. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 44:1-44:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{jain_et_al:LIPIcs.SoCG.2023.44, author = {Jain, Rahul and Ricci, Marco and Rollin, Jonathan and Schulz, Andr\'{e}}, title = {{On the Geometric Thickness of 2-Degenerate Graphs}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {44:1--44:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.44}, URN = {urn:nbn:de:0030-drops-178946}, doi = {10.4230/LIPIcs.SoCG.2023.44}, annote = {Keywords: Degeneracy, geometric thickness, geometric arboricity} }

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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 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

The problem of graph Reachability is to decide whether there is a path from one vertex to another in a given graph. In this paper, we study the Reachability problem on three distinct graph families - intersection graphs of Jordan regions, unit contact disk graphs (penny graphs), and chordal graphs. For each of these graph families, we present space-efficient algorithms for the Reachability problem.
For intersection graphs of Jordan regions, we show how to obtain a "good" vertex separator in a space-efficient manner and use it to solve the Reachability in polynomial time and O(m^{1/2} log n) space, where n is the number of Jordan regions, and m is the total number of crossings among the regions. We use a similar approach for chordal graphs and obtain a polynomial time and O(m^{1/2} log n) space algorithm, where n and m are the number of vertices and edges, respectively. However, for unit contact disk graphs (penny graphs), we use a more involved technique and obtain a better algorithm. We show that for every ε > 0, there exists a polynomial time algorithm that can solve Reachability in an n vertex directed penny graph, using O(n^{1/4+ε}) space. We note that the method used to solve penny graphs does not extend naturally to the class of geometric intersection graphs that include arbitrary size cliques.

Sujoy Bhore and Rahul Jain. Space-Efficient Algorithms for Reachability in Directed Geometric Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 63:1-63:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhore_et_al:LIPIcs.ISAAC.2021.63, author = {Bhore, Sujoy and Jain, Rahul}, title = {{Space-Efficient Algorithms for Reachability in Directed Geometric Graphs}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {63:1--63:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.63}, URN = {urn:nbn:de:0030-drops-154961}, doi = {10.4230/LIPIcs.ISAAC.2021.63}, annote = {Keywords: Reachablity, Geometric intersection graphs, Space-efficient algorithms} }

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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 200, 36th Computational Complexity Conference (CCC 2021)

We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f ⊆ 𝒳×𝒴×𝒵. For any 0 < ε < δ < 1/2 and any k≥1, we show that
Q¹_{1-(1-ε)^{Ω(k/log|𝒵|)}}(f^k) = Ω(k⋅Q¹_{δ}(f)),
where Q¹_{ε}(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and f^k denotes k parallel instances of f.
As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [Rahul Jain et al., 2014], and under anchored distributions due to Bavarian, Vidick and Yuen [Bavarian et al., 2017], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [Rahul Jain et al., 2005]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on 𝒳×𝒴 that is anchored on one side, i.e., there exists a y^* such that q(y^*) is constant and q(x|y^*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation f̃ which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for f̃ under q can be used to give a protocol that fails with probability at most ε + ζ for f under p.
Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y^* as previously specified, or there exists an x^* such that q(x^*) is constant and q(y|x^*) = q(y) for all y. In particular, we show that for any game G = (q, 𝒳×𝒴, 𝒜×ℬ, 𝖵) where q is a distribution on 𝒳×𝒴 anchored on any one side with constant anchoring probability, then
ω^*(G^k) = (1 - (1-ω^*(G))⁵) ^{Ω(k/(log(|𝒜|⋅|ℬ|)))}
where ω^*(G) represents the entangled value of the game G. This is a generalization of the result of [Bavarian et al., 2017], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x^* and a special y^* exist, and potentially a simplification of their proof.

Rahul Jain and Srijita Kundu. A Direct Product Theorem for One-Way Quantum Communication. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 27:1-27:28, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jain_et_al:LIPIcs.CCC.2021.27, author = {Jain, Rahul and Kundu, Srijita}, title = {{A Direct Product Theorem for One-Way Quantum Communication}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {27:1--27:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.27}, URN = {urn:nbn:de:0030-drops-143017}, doi = {10.4230/LIPIcs.CCC.2021.27}, annote = {Keywords: Direct product theorem, parallel repetition theorem, quantum communication, one-way protocols, communication complexity} }

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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 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Let the randomized query complexity of a relation for error probability epsilon be denoted by R_epsilon(). We prove that for any relation f contained in {0,1}^n times R and Boolean function g:{0,1}^m -> {0,1}, R_{1/3}(f o g^n) = Omega(R_{4/9}(f).R_{1/2-1/n^4}(g)), where f o g^n is the relation obtained by composing f and g. We also show using an XOR lemma that R_{1/3}(f o (g^{xor}_{O(log n)})^n) = Omega(log n . R_{4/9}(f) . R_{1/3}(g))$, where g^{xor}_{O(log n)} is the function obtained by composing the XOR function on O(log n) bits and g.

Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 10:1-10:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{anshu_et_al:LIPIcs.FSTTCS.2017.10, author = {Anshu, Anurag and Gavinsky, Dmitry and Jain, Rahul and Kundu, Srijita and Lee, Troy and Mukhopadhyay, Priyanka and Santha, Miklos and Sanyal, Swagato}, title = {{A Composition Theorem for Randomized Query Complexity}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {10:1--10:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.10}, URN = {urn:nbn:de:0030-drops-83967}, doi = {10.4230/LIPIcs.FSTTCS.2017.10}, annote = {Keywords: Query algorithms and complexity, Decision trees, Composition theorem, XOR lemma, Hardness amplification} }

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

One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma-2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank.
In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F.

Anurag Anshu, Shalev Ben-David, Ankit Garg, Rahul Jain, Robin Kothari, and Troy Lee. Separating Quantum Communication and Approximate Rank. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 24:1-24:33, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{anshu_et_al:LIPIcs.CCC.2017.24, author = {Anshu, Anurag and Ben-David, Shalev and Garg, Ankit and Jain, Rahul and Kothari, Robin and Lee, Troy}, title = {{Separating Quantum Communication and Approximate Rank}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {24:1--24:33}, 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.2017.24}, URN = {urn:nbn:de:0030-drops-75303}, doi = {10.4230/LIPIcs.CCC.2017.24}, annote = {Keywords: Communication Complexity, Quantum Computing, Lower Bounds, logrank, Quantum Information} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Let f: {0,1}^n*{0,1}^n -> {0,1} be a 2-party function. For every product distribution mu on {0,1}^n*{0,1}^n, we show that
CC^{mu}_{0.49}(f) = O(log(prt_{1/8}(f))*log(log(prt_{1/8}(f)))^2),
where CC^{mu}_{epsilon}(f) is the distributional communication complexity of f with error at most epsilon under the distribution mu and prt_{1/8}(f) is the partition bound of f, as defined by Jain and Klauck [Proc. 25th CCC, 2010]. We also prove a similar bound in terms of IC_{1/8}(f), the information complexity of f, namely,
CC^{mu}_{0.49}(f) = O((IC_{1/8}(f)*log(IC_{1/8}(f)))^2).
The latter bound was recently and independently established by Kol [Proc. 48th STOC, 2016] using a different technique.
We show a similar result for query complexity under product distributions. Let g: {0,1}^n -> {0,1} be a function. For every bit-wise product distribution mu on {0,1}^n, we show that
QC^{mu}_{0.49}(g) = O((log(qprt_{1/8}(g))*log(log(qprt_{1/8}(g))))^2),
where QC^{mu}_{epsilon}(g) is the distributional query complexity of f with error at most epsilon under the distribution mu and qprt_{1/8}(g) is the query partition bound of the function g.
Partition bounds were introduced (in both communication complexity and query complexity models) to provide LP-based lower bounds for randomized communication complexity and randomized query complexity. Our results demonstrate that these lower bounds are polynomially tight for product distributions.

Prahladh Harsha, Rahul Jain, and Jaikumar Radhakrishnan. Partition Bound Is Quadratically Tight for Product Distributions. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 135:1-135:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{harsha_et_al:LIPIcs.ICALP.2016.135, author = {Harsha, Prahladh and Jain, Rahul and Radhakrishnan, Jaikumar}, title = {{Partition Bound Is Quadratically Tight for Product Distributions}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {135:1--135:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.135}, URN = {urn:nbn:de:0030-drops-62708}, doi = {10.4230/LIPIcs.ICALP.2016.135}, annote = {Keywords: partition bound, product distribution, communication complexity, query complexity} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We consider unidirectional data streams with restricted access, such as read-only and write-only streams. For read-write streams, we also
introduce a new complexity measure called expansion, the ratio between
the space used on the stream and the input size. We give tight bounds for the complexity of reversing a stream of length n in several of the possible models. In the read-only and write-only model, we show that p-pass algorithms need memory space Theta(n/p). But if either the output stream or the input stream is read-write, then the complexity falls to Theta(n/p^2). It becomes polylog(n) if p = O(log n) and both streams are read-write. We also study the complexity of sorting a stream and give two algorithms with small expansion. Our main sorting algorithm is randomized and has O(1) expansion, O(log n) passes and O(log n) memory.

Nathanaël François, Rahul Jain, and Frédéric Magniez. Unidirectional Input/Output Streaming Complexity of Reversal and Sorting. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 654-668, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{francois_et_al:LIPIcs.APPROX-RANDOM.2014.654, author = {Fran\c{c}ois, Nathana\"{e}l and Jain, Rahul and Magniez, Fr\'{e}d\'{e}ric}, title = {{Unidirectional Input/Output Streaming Complexity of Reversal and Sorting}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {654--668}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.654}, URN = {urn:nbn:de:0030-drops-47298}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.654}, annote = {Keywords: Streaming Algorithms, Multiple Streams, Reversal, Sorting} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

The main result of this paper is an optimal strong direct
product result for the two-party public-coin randomized communication
complexity of the Tribes function. This is proved by providing an
alternate proof of the optimal lower bound of Omega(n) for the randomised communication complexity of the Tribes function using the so-called smooth-rectangle bound, introduced by Jain and Klauck [CCC/2010]. The optimal Omega(n) lower bound for Tribes was originally proved by Jayram, Kumar and Sivakumar [STOC/2003], using a more powerful lower bound technique, namely the information complexity bound. The information complexity bound is known to be at least as strong a lower bound method as the smooth-rectangle bound [Kerenidis et al, 2012]. On the other hand, we are not aware of any function or relation for which the smooth-rectangle bound is (asymptotically) smaller than its public-coin randomized communication complexity. The optimal direct product for Tribes is obtained by combining our smooth-rectangle bound for tribes with the strong direct product result of Jain and Yao (2012) in terms of smooth-rectangle bound.

Prahladh Harsha and Rahul Jain. A Strong Direct Product Theorem for the Tribes Function via the Smooth-Rectangle Bound. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 141-152, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{harsha_et_al:LIPIcs.FSTTCS.2013.141, author = {Harsha, Prahladh and Jain, Rahul}, title = {{A Strong Direct Product Theorem for the Tribes Function via the Smooth-Rectangle Bound}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {141--152}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.141}, URN = {urn:nbn:de:0030-drops-43670}, doi = {10.4230/LIPIcs.FSTTCS.2013.141}, annote = {Keywords: Rectangle bound, Tribes function, Strong direct product} }

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