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Documents authored by Upasana, Anannya

Document
DOI: 10.4230/LIPIcs.ISAAC.2025.42
Parameterized Reunion with Achromatic Number

Authors: Satyabrata Jana, Souvik Saha, Saket Saurabh, and Anannya Upasana

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract
In this paper, we study the Achromatic Number problem. Given a graph G and an integer k, the task is to determine whether there exists a proper coloring of G, using at least k colors, in which every pair of distinct colors appears on the endpoints of some edge. It was established early on that the problem is fixed-parameter tractable (FPT)- even before the formal development of parameterized complexity. In fact, Farber, Hahn, Hell, and Miller [JCTB, 1986] devised an algorithm with a running time of 𝒪(f(k) ⋅ |E(G)|). Although the exact form of f(k) was not specified, it appears to be at least doubly exponential in k. In our work, we first present an algorithm with an explicit dependence on k, and then introduce another algorithm that is parameterized by the vertex cover number of the graph. More formally, we show the following. - Achromatic Number is solvable in time 2^𝒪(k⁵)+𝒪(|E(G)|). - Achromatic Number admits a polynomial kernel when the input is restricted to a d-degenerate graph and a more efficient kernel on trees. - We also study the parameterized complexity of the problem with respect to Vertex Cover and show that it admits an FPT algorithm running in time 2^𝒪(𝓁²) ⋅ n^𝒪(1), where 𝓁 is the size of a vertex cover.
Cite as

Satyabrata Jana, Souvik Saha, Saket Saurabh, and Anannya Upasana. Parameterized Reunion with Achromatic Number. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jana_et_al:LIPIcs.ISAAC.2025.42,
  author =	{Jana, Satyabrata and Saha, Souvik and Saurabh, Saket and Upasana, Anannya},
  title =	{{Parameterized Reunion with Achromatic Number}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{42:1--42:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.42},
  URN =		{urn:nbn:de:0030-drops-249502},
  doi =		{10.4230/LIPIcs.ISAAC.2025.42},
  annote =	{Keywords: Achromatic number, Coloring, Fixed-parameter tractability, Kernelization, Lower bound, W-hardness}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.88
Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints

Authors: Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket Saurabh, and Anannya Upasana

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

Abstract
In the MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula Φ, and a positive integer k, and the goal is to find an assignment β with at most k variables set to true (also called a weight k-assignment) such that the number of clauses satisfied by β is maximized. Maximum Coverage can be seen as a special case of CC-MaxSat, where the formula Φ is monotone, i.e., does not contain any negative literals. CC-MaxSat and Maximum Coverage are extremely well-studied problems in the approximation algorithms as well as the parameterized complexity literature. Our first conceptual contribution is that CC-MaxSat and Maximum Coverage are equivalent to each other in the context of FPT-Approximation parameterized by k (here, the approximation is in terms of the number of clauses satisfied/elements covered). In particular, we give a randomized reduction from CC-MaxSat to Maximum Coverage running in time 𝒪(1/ε)^{k} ⋅ (m+n)^{𝒪(1)} that preserves the approximation guarantee up to a factor of (1-ε). Furthermore, this reduction also works in the presence of "fairness" constraints on the satisfied clauses, as well as matroid constraints on the set of variables that are assigned true. Here, the "fairness" constraints are modeled by partitioning the clauses of the formula Φ into r different colors, and the goal is to find an assignment that satisfies at least t_j clauses of each color 1 ≤ j ≤ r. Armed with this reduction, we focus on designing FPT-Approximation schemes (FPT-ASes) for Maximum Coverage and its generalizations. Our algorithms are based on a novel combination of a variety of ideas, including a carefully designed probability distribution that exploits sparse coverage functions. These algorithms substantially generalize the results in Jain et al. [SODA 2023] for CC-MaxSat and Maximum Coverage for K_{d,d}-free set systems (i.e., no d sets share d elements), as well as a recent FPT-AS for Matroid Constrained Maximum Coverage by Sellier [ESA 2023] for frequency-d set systems.
Cite as

Tanmay Inamdar, Pallavi Jain, Daniel Lokshtanov, Abhishek Sahu, Saket Saurabh, and Anannya Upasana. Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 88:1-88:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{inamdar_et_al:LIPIcs.ICALP.2024.88,
  author =	{Inamdar, Tanmay and Jain, Pallavi and Lokshtanov, Daniel and Sahu, Abhishek and Saurabh, Saket and Upasana, Anannya},
  title =	{{Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{88:1--88:18},
  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.2024.88},
  URN =		{urn:nbn:de:0030-drops-202318},
  doi =		{10.4230/LIPIcs.ICALP.2024.88},
  annote =	{Keywords: Partial Vertex Cover, Max SAT, FPT Approximation, Matroids}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.15
Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!

Authors: Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study a graph coloring problem that is otherwise easy in the RAM model but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our main work is on estimating the number of conflicting or monochromatic edges given a coloring function that is streaming along with the graph; we call the problem Conflict-Est. The coloring function on a vertex can be read or accessed only when the vertex is revealed in the stream. If we need the color on a vertex that has streamed past, then that color, along with its vertex, has to be stored explicitly. We provide algorithms for a graph that is streaming in different variants of the vertex arrival in one-pass streaming model, viz. the Vertex Arrival (VA), Vertex Arrival With Degree Oracle (VAdeg), Vertex Arrival in Random Order (VArand) models, with special focus on the random order model. We also provide matching lower bounds for most of the cases. The mainstay of our work is in showing that the properties of a random order stream can be exploited to design efficient streaming algorithms for estimating the number of monochromatic edges. We have also obtained a lower bound, though not matching the upper bound, for the random order model. Among all the three models vis-a-vis this problem, we can show a clear separation of power in favor of the VArand model.
Cite as

Anup Bhattacharya, Arijit Bishnu, Gopinath Mishra, and Anannya Upasana. Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bhattacharya_et_al:LIPIcs.ITCS.2021.15,
  author =	{Bhattacharya, Anup and Bishnu, Arijit and Mishra, Gopinath and Upasana, Anannya},
  title =	{{Even the Easiest(?) Graph Coloring Problem Is Not Easy in Streaming!}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{15:1--15:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.15},
  URN =		{urn:nbn:de:0030-drops-135544},
  doi =		{10.4230/LIPIcs.ITCS.2021.15},
  annote =	{Keywords: Streaming, random ordering, graph coloring, estimation, lower bounds}
}
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