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Documents authored by Uniyal, Sumedha

Document
DOI: 10.4230/LIPIcs.WADS.2025.16
An Improved Guillotine Cut for Squares

Authors: Parinya Chalermsook, Axel Kugelmann, Ly Orgo, Sumedha Uniyal, and Minoo Zarsav

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Given a set of n non-overlapping geometric objects, can we separate a constant fraction of them using straight-line cuts that extend from edge to edge? In 1996, Urrutia posed this question for compact convex objects. Pach and Tardos later refuted it for general line segments by constructing a family where any separable subfamily has size at most O (n^{log₃ 2}). However, for axis-parallel rectangles, they provided positive evidence, showing that an Ω(1/log n)-fraction can be separated. This problem naturally arises in geometric approximation algorithms. In particular, when restricting cuts to only orthogonal straight lines, known as a guillotine cut sequence, any bound on the separability ratio directly translates into a clean and simple dynamic programming for computing a maximum independent set of geometric objects. This paper focuses on the case when the objects are squares. For squares of arbitrary sizes, an Ω(1)-fraction can be separated (Abed et al., APPROX 2015), recently improved to 1/40 (and 1/160 ≈ 0.62% for the weighted case) (Khan and Pittu, APPROX 2020). We further improve this bound, showing that a 9/256 ≈ 3.51% can be separated for the weighted case. This result significantly narrows the possible range for squares to [3.51%, 50%]. The key to our improvement is a refined analysis of the existing framework.
Cite as

Parinya Chalermsook, Axel Kugelmann, Ly Orgo, Sumedha Uniyal, and Minoo Zarsav. An Improved Guillotine Cut for Squares. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 16:1-16:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chalermsook_et_al:LIPIcs.WADS.2025.16,
  author =	{Chalermsook, Parinya and Kugelmann, Axel and Orgo, Ly and Uniyal, Sumedha and Zarsav, Minoo},
  title =	{{An Improved Guillotine Cut for Squares}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{16:1--16:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.16},
  URN =		{urn:nbn:de:0030-drops-242472},
  doi =		{10.4230/LIPIcs.WADS.2025.16},
  annote =	{Keywords: Guillotine cuts, Geometric Approximation Algorithms, Rectangles, Squares}
}
Document
DOI: 10.4230/LIPIcs.ESA.2024.38
Local Optimization Algorithms for Maximum Planar Subgraph

Authors: Gruia Călinescu and Sumedha Uniyal

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

Abstract
Consider the NP-hard problem of, given a simple graph G, to find a planar subgraph of G with the maximum number of edges. This is called the Maximum Planar Subgraph problem and the best known approximation is 4/9 and is obtained by sophisticated Graphic Matroid Parity algorithms. Here we show that applying a local optimization phase to the output of this known algorithm improves this approximation ratio by a small {ε} = 1/747 > 0. This is the first improvement in approximation ratio in more than a quarter century. The analysis relies on a more refined extremal bound on the Lovász cactus number in planar graphs, compared to the earlier (tight) bound of [Gruia Călinescu et al., 1998; Chalermsook et al., 2019]. A second local optimization algorithm achieves a tight ratio of 5/12 for Maximum Planar Subgraph without using Graphic Matroid Parity. We also show that applying a greedy algorithm before this second optimization algorithm improves its ratio to at least 91/216 < 4/9. The motivation for not using Graphic Matroid Parity is that it requires sophisticated algorithms that are not considered practical by previous work. The best previously published [Chalermsook and Schmid, 2017] approximation ratio without Graphic Matroid Parity is 13/33 < 5/12.
Cite as

Gruia Călinescu and Sumedha Uniyal. Local Optimization Algorithms for Maximum Planar Subgraph. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{calinescu_et_al:LIPIcs.ESA.2024.38,
  author =	{C\u{a}linescu, Gruia and Uniyal, Sumedha},
  title =	{{Local Optimization Algorithms for Maximum Planar Subgraph}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{38:1--38:18},
  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.2024.38},
  URN =		{urn:nbn:de:0030-drops-211090},
  doi =		{10.4230/LIPIcs.ESA.2024.38},
  annote =	{Keywords: planar graph, maximum subgraph, approximation algorithm, matroid parity, local optimization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.85
A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints

Authors: Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant epsilon >0 achieves a guarantee of 1-(1/e)-epsilon while violating only the covering constraints by a multiplicative factor of 1-epsilon. Our algorithm is based on a novel enumeration method, which unlike previously known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraint as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach. While this approach does not give tight bounds it yields deterministic and in some special cases also considerably faster algorithms. For example, for the well-studied special case of only packing constraints (Kulik et al. [Math. Oper. Res. `13] and Chekuri et al. [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.
Cite as

Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, and Sumedha Uniyal. A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 85:1-85:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mizrachi_et_al:LIPIcs.ICALP.2019.85,
  author =	{Mizrachi, Eyal and Schwartz, Roy and Spoerhase, Joachim and Uniyal, Sumedha},
  title =	{{A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{85:1--85:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.85},
  URN =		{urn:nbn:de:0030-drops-106610},
  doi =		{10.4230/LIPIcs.ICALP.2019.85},
  annote =	{Keywords: submodular function, approximation algorithm, covering, packing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2019.19
A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

Authors: Parinya Chalermsook, Andreas Schmid, and Sumedha Uniyal

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

Abstract
A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least a 1/6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing "dense planar structures" inside any graph: (i) A 1/6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous 1/11-approximation; (ii) An alternate (and arguably more illustrative) proof of the 4/9 approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind.
Cite as

Parinya Chalermsook, Andreas Schmid, and Sumedha Uniyal. A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chalermsook_et_al:LIPIcs.STACS.2019.19,
  author =	{Chalermsook, Parinya and Schmid, Andreas and Uniyal, Sumedha},
  title =	{{A Tight Extremal Bound on the Lov\'{a}sz Cactus Number in Planar Graphs}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{19:1--19:14},
  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.2019.19},
  URN =		{urn:nbn:de:0030-drops-102583},
  doi =		{10.4230/LIPIcs.STACS.2019.19},
  annote =	{Keywords: Graph Drawing, Matroid Matching, Maximum Planar Subgraph, Local Search Algorithms}
}
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