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Documents authored by Kulkarni, Shubhang

Document
DOI: 10.4230/LIPIcs.ESA.2024.22
Hypergraph Connectivity Augmentation in Strongly Polynomial Time

Authors: Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We consider hypergraph network design problems where the goal is to construct a hypergraph that satisfies certain connectivity requirements. For graph network design problems where the goal is to construct a graph that satisfies certain connectivity requirements, the number of edges in every feasible solution is at most quadratic in the number of vertices. In contrast, for hypergraph network design problems, we might have feasible solutions in which the number of hyperedges is exponential in the number of vertices. This presents an additional technical challenge in hypergraph network design problems compared to graph network design problems: in order to solve the problem in polynomial time, we first need to show that there exists a feasible solution in which the number of hyperedges is polynomial in the input size. The central theme of this work is to overcome this additional technical challenge for certain hypergraph network design problems. We show that these hypergraph network design problems admit solutions in which the number of hyperedges is polynomial in the number of vertices and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges and (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges.
Cite as

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni. Hypergraph Connectivity Augmentation in Strongly Polynomial Time. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berczi_et_al:LIPIcs.ESA.2024.22,
  author =	{B\'{e}rczi, Krist\'{o}f and Chandrasekaran, Karthekeyan and Kir\'{a}ly, Tam\'{a}s and Kulkarni, Shubhang},
  title =	{{Hypergraph Connectivity Augmentation in Strongly Polynomial Time}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{22:1--22:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.22},
  URN =		{urn:nbn:de:0030-drops-210938},
  doi =		{10.4230/LIPIcs.ESA.2024.22},
  annote =	{Keywords: Hypergraphs, Hypergraph Connectivity, Submodular Functions, Combinatorial Optimization}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.42
From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

Authors: Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. - We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. - We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k).
Cite as

Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chekuri_et_al:LIPIcs.ESA.2024.42,
  author =	{Chekuri, Chandra and Jain, Rhea and Kulkarni, Shubhang and Zheng, Da Wei and Zhu, Weihao},
  title =	{{From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{42:1--42:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.42},
  URN =		{urn:nbn:de:0030-drops-211134},
  doi =		{10.4230/LIPIcs.ESA.2024.42},
  annote =	{Keywords: Directed Planar Graphs, Submodular Functions, Steiner Tree, Network Design}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.23
Splitting-Off in Hypergraphs

Authors: Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lovász [Lov{á}sz, 1974; Lov{á}sz, 1993] and Mader [Mader, 1978] showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature [Lovász, 1976; Mader, 1978; Frank, 1993; Frank and Király, 2002; Király and Lau, 2008; Frank, 1992; Goemans and Bertsimas, 1993; Frank, 1994; Bang-Jensen et al., 1995; Frank, 2011; Nagamochi and Ibaraki, 2008; Nagamochi et al., 1997; Henzinger and Williamson, 1996; Goemans, 2001; Jordán, 2003; Kriesell, 2003; Jain et al., 2003; Chan et al., 2011; Bhalgat et al., 2008; Lau, 2007; Chekuri and Shepherd, 2008; Nägele and Zenklusen, 2020; Blauth and Nägele, 2023]. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of k-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Király and Lau [Király and Lau, 2008]). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.
Cite as

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, and Shubhang Kulkarni. Splitting-Off in Hypergraphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berczi_et_al:LIPIcs.ICALP.2024.23,
  author =	{B\'{e}rczi, Krist\'{o}f and Chandrasekaran, Karthekeyan and Kir\'{a}ly, Tam\'{a}s and Kulkarni, Shubhang},
  title =	{{Splitting-Off in Hypergraphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.23},
  URN =		{urn:nbn:de:0030-drops-201660},
  doi =		{10.4230/LIPIcs.ICALP.2024.23},
  annote =	{Keywords: Hypergraphs, Hypergraph Connectivity, Splitting-off, Constructive Characterizations, Hypergraph Orientations, Submodular Functions, Combinatorial Optimization}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2020.7
Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree

Authors: Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Abstract
A heapable sequence is a sequence of numbers that can be arranged in a min-heap data structure. Finding a longest heapable subsequence of a given sequence was proposed by Byers, Heeringa, Mitzenmacher, and Zervas (ANALCO 2011) as a generalization of the well-studied longest increasing subsequence problem and its complexity still remains open. An equivalent formulation of the longest heapable subsequence problem is that of finding a maximum-sized binary tree in a given permutation directed acyclic graph (permutation DAG). In this work, we study parameterized algorithms for both longest heapable subsequence and maximum-sized binary tree. We introduce alphabet size as a new parameter in the study of computational problems in permutation DAGs and show that this parameter with respect to a fixed topological ordering admits a complete characterization and a polynomial time algorithm. We believe that this parameter is likely to be useful in the context of optimization problems defined over permutation DAGs.
Cite as

Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu. Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chandrasekaran_et_al:LIPIcs.IPEC.2020.7,
  author =	{Chandrasekaran, Karthekeyan and Grigorescu, Elena and Istrate, Gabriel and Kulkarni, Shubhang and Lin, Young-San and Zhu, Minshen},
  title =	{{Fixed-Parameter Algorithms for Longest Heapable Subsequence and Maximum Binary Tree}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.7},
  URN =		{urn:nbn:de:0030-drops-133102},
  doi =		{10.4230/LIPIcs.IPEC.2020.7},
  annote =	{Keywords: maximum binary tree, heapability, permutation directed acyclic graphs}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2020.16
Locally Decodable/Correctable Codes for Insertions and Deletions

Authors: Alexander R. Block, Jeremiah Blocki, Elena Grigorescu, Shubhang Kulkarni, and Minshen Zhu

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

Abstract
Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and decoding algorithms. In many applications, polynomial running time may still be prohibitively expensive, which has motivated the study of codes with super-efficient decoding algorithms. These have led to the well-studied notions of Locally Decodable Codes (LDCs) and Locally Correctable Codes (LCCs). Inspired by these notions, Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015) generalized Hamming LDCs to insertions and deletions. To the best of our knowledge, these are the only known results that study the analogues of Hamming LDCs in channels performing insertions and deletions. Here we continue the study of insdel codes that admit local algorithms. Specifically, we reprove the results of Ostrovsky and Paskin-Cherniavsky for insdel LDCs using a different set of techniques. We also observe that the techniques extend to constructions of LCCs. Specifically, we obtain insdel LDCs and LCCs from their Hamming LDCs and LCCs analogues, respectively. The rate and error-correction capability blow up only by a constant factor, while the query complexity blows up by a poly log factor in the block length. Since insdel locally decodable/correctble codes are scarcely studied in the literature, we believe our results and techniques may lead to further research. In particular, we conjecture that constant-query insdel LDCs/LCCs do not exist.
Cite as

Alexander R. Block, Jeremiah Blocki, Elena Grigorescu, Shubhang Kulkarni, and Minshen Zhu. Locally Decodable/Correctable Codes for Insertions and Deletions. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{block_et_al:LIPIcs.FSTTCS.2020.16,
  author =	{Block, Alexander R. and Blocki, Jeremiah and Grigorescu, Elena and Kulkarni, Shubhang and Zhu, Minshen},
  title =	{{Locally Decodable/Correctable Codes for Insertions and Deletions}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{16:1--16:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.16},
  URN =		{urn:nbn:de:0030-drops-132577},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.16},
  annote =	{Keywords: Locally decodable/correctable codes, insert-delete channel}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.30
The Maximum Binary Tree Problem

Authors: Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ≠ NP. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.
Cite as

Karthekeyan Chandrasekaran, Elena Grigorescu, Gabriel Istrate, Shubhang Kulkarni, Young-San Lin, and Minshen Zhu. The Maximum Binary Tree Problem. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 30:1-30:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chandrasekaran_et_al:LIPIcs.ESA.2020.30,
  author =	{Chandrasekaran, Karthekeyan and Grigorescu, Elena and Istrate, Gabriel and Kulkarni, Shubhang and Lin, Young-San and Zhu, Minshen},
  title =	{{The Maximum Binary Tree Problem}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{30:1--30:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.30},
  URN =		{urn:nbn:de:0030-drops-128967},
  doi =		{10.4230/LIPIcs.ESA.2020.30},
  annote =	{Keywords: maximum binary tree, heapability, inapproximability, fixed-parameter tractability}
}
Document
DOI: 10.4230/LIPIcs.ITC.2020.16
On Locally Decodable Codes in Resource Bounded Channels

Authors: Jeremiah Blocki, Shubhang Kulkarni, and Samson Zhou

Published in: LIPIcs, Volume 163, 1st Conference on Information-Theoretic Cryptography (ITC 2020)

Abstract
Constructions of locally decodable codes (LDCs) have one of two undesirable properties: low rate or high locality (polynomial in the length of the message). In settings where the encoder/decoder have already exchanged cryptographic keys and the channel is a probabilistic polynomial time (PPT) algorithm, it is possible to circumvent these barriers and design LDCs with constant rate and small locality. However, the assumption that the encoder/decoder have exchanged cryptographic keys is often prohibitive. We thus consider the problem of designing explicit and efficient LDCs in settings where the channel is slightly more constrained than the encoder/decoder with respect to some resource e.g., space or (sequential) time. Given an explicit function f that the channel cannot compute, we show how the encoder can transmit a random secret key to the local decoder using f(⋅) and a random oracle 𝖧(⋅). We then bootstrap the private key LDC construction of Ostrovsky, Pandey and Sahai (ICALP, 2007), thereby answering an open question posed by Guruswami and Smith (FOCS 2010) of whether such bootstrapping techniques are applicable to LDCs in channel models weaker than just PPT algorithms. Specifically, in the random oracle model we show how to construct explicit constant rate LDCs with locality of polylog in the security parameter against various resource constrained channels.
Cite as

Jeremiah Blocki, Shubhang Kulkarni, and Samson Zhou. On Locally Decodable Codes in Resource Bounded Channels. In 1st Conference on Information-Theoretic Cryptography (ITC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 163, pp. 16:1-16:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blocki_et_al:LIPIcs.ITC.2020.16,
  author =	{Blocki, Jeremiah and Kulkarni, Shubhang and Zhou, Samson},
  title =	{{On Locally Decodable Codes in Resource Bounded Channels}},
  booktitle =	{1st Conference on Information-Theoretic Cryptography (ITC 2020)},
  pages =	{16:1--16:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-151-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{163},
  editor =	{Tauman Kalai, Yael and Smith, Adam D. and Wichs, Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2020.16},
  URN =		{urn:nbn:de:0030-drops-121216},
  doi =		{10.4230/LIPIcs.ITC.2020.16},
  annote =	{Keywords: Locally Decodable Codes, Resource Bounded Channels}
}
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