13 Search Results for "Kumar, Akash"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.91
A Sublinear Time Tester for Max-Cut on Clusterable Graphs

Authors: Agastya Vibhuti Jha and Akash Kumar

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

Abstract
One natural question in the area of sublinear time algorithms asks whether we can distinguish between graphs with max-cut value at least 1-ε from graphs with max-cut value at most 1/2+ε in the adjacency list model where we can make degree queries and neighbor queries. Chiplunkar, Kapralov, Khanna, Mousavifar, and Peres (FOCS' 18) showed that in graphs of bounded degree, one cannot hope for a factor 1/2+ε approximation to the max-cut value in time n^{1/2+o(ε)}. Recently, Peng and Yoshida (SODA '23) obtained o(n) time algorithms which can distinguish expanders with max-cut value at least 1-ε from expanders with small max-cut value (their running time is n^{1/2+O(ε)}). In this paper, going beyond the results of Peng-Yoshida, we develop sublinear time algorithms for this problem on clusterable graphs (which is a graph class with a good community structure). Our algorithms run in ≈ n^{0.5001+ O(ε)} time. A natural extension of Peng-Yoshida approach does not seem to work for clusterable graphs. Indeed, their random walk based technique tracks the 𝓁₂ length of random walk vectors and they exploit the difference in the length of these vectors to tell apart expanders with large cut value from expanders with small cut-value. Such approaches fail to be reliable when graph has loosely connected clusters. Taking inspiration from [Ashish Chiplunkar et al., 2018], we exploit the more refined geometry of spectra of clusterable graphs which leads to our sublinear time implementation. We prove a novel spectral lemma which shows that in a spectral expander 2 - λ_{n-1} ≥ Ω(λ₂). This lemma is leveraged to show that there is a suitable difference between spectra of clusterable graphs with large cut value and spectra of clusterable graphs with small cut value. We use this gap to obtain our sublinear time implementation. To do this, we obtain a nuanced understanding of the eigenvector structure of clusterable graphs and in particular, we show that the eigenvectors of the normalized Laplacian of a clusterable graph, corresponding to eigenvalues which are close to 2 have a small infinity norm.
Cite as

Agastya Vibhuti Jha and Akash Kumar. A Sublinear Time Tester for Max-Cut on Clusterable Graphs. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 91:1-91:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{jha_et_al:LIPIcs.ICALP.2024.91,
  author =	{Jha, Agastya Vibhuti and Kumar, Akash},
  title =	{{A Sublinear Time Tester for Max-Cut on Clusterable Graphs}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{91:1--91:17},
  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.2024.91},
  URN =		{urn:nbn:de:0030-drops-202344},
  doi =		{10.4230/LIPIcs.ICALP.2024.91},
  annote =	{Keywords: Sublinear Algorithms, Graph Algorithms, Clusterable Graphs, Property Testung}
}
Document
DOI: 10.4230/LIPIcs.CCC.2023.20
Radical Sylvester-Gallai Theorem for Tuples of Quadratics

Authors: Abhibhav Garg, Rafael Oliveira, Shir Peleg, and Akash Kumar Sengupta

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We prove a higher codimensional radical Sylvester-Gallai type theorem for quadratic polynomials, simultaneously generalizing [Hansen, 1965; Shpilka, 2020]. Hansen’s theorem is a high-dimensional version of the classical Sylvester-Gallai theorem in which the incidence condition is given by high-dimensional flats instead of lines. We generalize Hansen’s theorem to the setting of quadratic forms in a polynomial ring, where the incidence condition is given by radical membership in a high-codimensional ideal. Our main theorem is also a generalization of the quadratic Sylvester-Gallai Theorem of [Shpilka, 2020]. Our work is the first to prove a radical Sylvester-Gallai type theorem for arbitrary codimension k ≥ 2, whereas previous works [Shpilka, 2020; Shir Peleg and Amir Shpilka, 2020; Shir Peleg and Amir Shpilka, 2021; Garg et al., 2022] considered the case of codimension 2 ideals. Our techniques combine algebraic geometric and combinatorial arguments. A key ingredient is a structural result for ideals generated by a constant number of quadratics, showing that such ideals must be radical whenever the quadratic forms are far apart. Using the wide algebras defined in [Garg et al., 2022], combined with results about integral ring extensions and dimension theory, we develop new techniques for studying such ideals generated by quadratic forms. One advantage of our approach is that it does not need the finer classification theorems for codimension 2 complete intersection of quadratics proved in [Shpilka, 2020; Garg et al., 2022].
Cite as

Abhibhav Garg, Rafael Oliveira, Shir Peleg, and Akash Kumar Sengupta. Radical Sylvester-Gallai Theorem for Tuples of Quadratics. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 20:1-20:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{garg_et_al:LIPIcs.CCC.2023.20,
  author =	{Garg, Abhibhav and Oliveira, Rafael and Peleg, Shir and Sengupta, Akash Kumar},
  title =	{{Radical Sylvester-Gallai Theorem for Tuples of Quadratics}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{20:1--20:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.20},
  URN =		{urn:nbn:de:0030-drops-182903},
  doi =		{10.4230/LIPIcs.CCC.2023.20},
  annote =	{Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.84
Exact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs

Authors: Akash Kumar, Anand Louis, and Rameesh Paul

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
The problem of finding the largest induced balanced bipartite subgraph in a given graph is NP-hard. This problem is closely related to the problem of finding the smallest Odd Cycle Transversal. In this work, we consider the following model of instances: starting with a set of vertices V, a set S ⊆ V of k vertices is chosen and an arbitrary d-regular bipartite graph is added on it; edges between pairs of vertices in S× (V⧵S) and (V⧵S) × (V⧵S) are added with probability p. Since for d = 0, the problem reduces to recovering a planted independent set, we don't expect efficient algorithms for k = o(√n). This problem is a generalization of the planted balanced biclique problem where the bipartite graph induced on S is a complete bipartite graph; [Yevgeny Levanzov, 2018] gave an algorithm for recovering S in this problem when k = Ω(√n). Our main result is an efficient algorithm that recovers (w.h.p.) the planted bipartite graph when k = Ω_p(√{n log n}) for a large range of parameters. Our results also hold for a natural semi-random model of instances, which involve the presence of a monotone adversary. Our proof shows that a natural SDP relaxation for the problem is integral by constructing an appropriate solution to it’s dual formulation. Our main technical contribution is a new approach for construction the dual solution where we calibrate the eigenvectors of the adjacency matrix to be the eigenvectors of the dual matrix. We believe that this approach may have applications to other recovery problems in semi-random models as well. When k = Ω(√n), we give an algorithm for recovering S whose running time is exponential in the number of small eigenvalues in graph induced on S; this algorithm is based on subspace enumeration techniques due to the works of [Alexandra Kolla and Madhur Tulsiani, 2007; Arora et al., 2010; Kolla, 2011].
Cite as

Akash Kumar, Anand Louis, and Rameesh Paul. Exact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 84:1-84:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kumar_et_al:LIPIcs.ICALP.2022.84,
  author =	{Kumar, Akash and Louis, Anand and Paul, Rameesh},
  title =	{{Exact Recovery Algorithm for Planted Bipartite Graph in Semi-Random Graphs}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{84:1--84:20},
  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.2022.84},
  URN =		{urn:nbn:de:0030-drops-164251},
  doi =		{10.4230/LIPIcs.ICALP.2022.84},
  annote =	{Keywords: SDP duality, Planted models, Semi-random models, Exact recovery, Threshold rank, Spectral embedding, Subspace enumeration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.42
Robust Radical Sylvester-Gallai Theorem for Quadratics

Authors: Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
We prove a robust generalization of a Sylvester-Gallai type theorem for quadratic polynomials. More precisely, given a parameter 0 < δ ≤ 1 and a finite collection ℱ of irreducible and pairwise independent polynomials of degree at most 2, we say that ℱ is a (δ, 2)-radical Sylvester-Gallai configuration if for any polynomial F_i ∈ ℱ, there exist δ(|ℱ|-1) polynomials F_j such that |rad (F_i, F_j) ∩ ℱ| ≥ 3, that is, the radical of F_i, F_j contains a third polynomial in the set. We prove that any (δ, 2)-radical Sylvester-Gallai configuration ℱ must be of low dimension: that is dim span_ℂ{ℱ} = poly(1/δ).
Cite as

Abhibhav Garg, Rafael Oliveira, and Akash Kumar Sengupta. Robust Radical Sylvester-Gallai Theorem for Quadratics. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{garg_et_al:LIPIcs.SoCG.2022.42,
  author =	{Garg, Abhibhav and Oliveira, Rafael and Sengupta, Akash Kumar},
  title =	{{Robust Radical Sylvester-Gallai Theorem for Quadratics}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.42},
  URN =		{urn:nbn:de:0030-drops-160505},
  doi =		{10.4230/LIPIcs.SoCG.2022.42},
  annote =	{Keywords: Sylvester-Gallai theorem, arrangements of hypersurfaces, locally correctable codes, algebraic complexity, polynomial identity testing, algebraic geometry, commutative algebra}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2018.40
Flipping out with Many Flips: Hardness of Testing k-Monotonicity

Authors: Elena Grigorescu, Akash Kumar, and Karl Wimmer

Published in: LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Abstract
A function f:{0,1}^n - > {0,1} is said to be k-monotone if it flips between 0 and 1 at most k times on every ascending chain. Such functions represent a natural generalization of (1-)monotone functions, and have been recently studied in circuit complexity, PAC learning, and cryptography. Our work is part of a renewed focus in understanding testability of properties characterized by freeness of arbitrary order patterns as a generalization of monotonicity. Recently, Canonne et al. (ITCS 2017) initiate the study of k-monotone functions in the area of property testing, and Newman et al. (SODA 2017) study testability of families characterized by freeness from order patterns on real-valued functions over the line [n] domain. We study k-monotone functions in the more relaxed parametrized property testing model, introduced by Parnas et al. (JCSS, 72(6), 2006). In this process we resolve a problem left open in previous work. Specifically, our results include the following. 1) Testing 2-monotonicity on the hypercube non-adaptively with one-sided error requires an exponential in sqrt{n} number of queries. This behavior shows a stark contrast with testing (1-)monotonicity, which only needs O~(sqrt{n}) queries (Khot et al. (FOCS 2015)). Furthermore, even the apparently easier task of distinguishing 2-monotone functions from functions that are far from being n^{.01}-monotone also requires an exponential number of queries. 2) On the hypercube [n]^d domain, there exists a testing algorithm that makes a constant number of queries and distinguishes functions that are k-monotone from functions that are far from being O(kd^2) -monotone. Such a dependency is likely necessary, given the lower bound above for the hypercube.
Cite as

Elena Grigorescu, Akash Kumar, and Karl Wimmer. Flipping out with Many Flips: Hardness of Testing k-Monotonicity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{grigorescu_et_al:LIPIcs.APPROX-RANDOM.2018.40,
  author =	{Grigorescu, Elena and Kumar, Akash and Wimmer, Karl},
  title =	{{Flipping out with Many Flips: Hardness of Testing k-Monotonicity}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
  pages =	{40:1--40:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-085-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{116},
  editor =	{Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.40},
  URN =		{urn:nbn:de:0030-drops-94448},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2018.40},
  annote =	{Keywords: Property Testing, Boolean Functions, k-Monotonicity, Lower Bounds}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2017.37
Finding Pseudorandom Colorings of Pseudorandom Graphs

Authors: Akash Kumar, Anand Louis, and Madhur Tulsiani

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

Abstract
We consider the problem of recovering a planted pseudorandom 3-coloring in expanding and low threshold-rank graphs. Alon and Kahale [SICOMP 1997] gave a spectral algorithm to recover the coloring for a random graph with a planted random 3-coloring. We show that their analysis can be adapted to work when coloring is pseudorandom i.e., all color classes are of equal size and the size of the intersection of the neighborhood of a random vertex with each color class has small variance. We also extend our results to partial colorings and low threshold-rank graphs to show the following: * For graphs on n vertices with threshold-rank r, for which there exists a 3-coloring that is eps-pseudorandom and properly colors the induced subgraph on (1-gamma)n vertices, we show how to recover the coloring for (1 - O(gamma + eps)) n vertices in time (rn)^{O(r)}. * For expanding graphs on n vertices, which admit a pseudorandom 3-coloring properly coloring all the vertices, we show how to recover such a coloring in polynomial time. Our results are obtained by combining the method of Alon and Kahale, with eigenspace enumeration methods used for solving constraint satisfaction problems on low threshold-rank graphs.
Cite as

Akash Kumar, Anand Louis, and Madhur Tulsiani. Finding Pseudorandom Colorings of Pseudorandom Graphs. 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. 37:1-37:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2017.37,
  author =	{Kumar, Akash and Louis, Anand and Tulsiani, Madhur},
  title =	{{Finding Pseudorandom Colorings of Pseudorandom Graphs}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{37:1--37:12},
  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.2017.37},
  URN =		{urn:nbn:de:0030-drops-83956},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.37},
  annote =	{Keywords: Graph Coloring, Expanders, Spectral algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.29
Testing k-Monotonicity

Authors: Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
A Boolean k-monotone function defined over a finite poset domain D alternates between the values 0 and 1 at most k times on any ascending chain in D. Therefore, k-monotone functions are natural generalizations of the classical monotone functions, which are the 1-monotone functions. Motivated by the recent interest in k-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of k-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are k-monotone (or are close to being k-monotone) from functions that are far from being k-monotone. Our results include the following: 1. We demonstrate a separation between testing k-monotonicity and testing monotonicity, on the hypercube domain {0,1}^d, for k >= 3; 2. We demonstrate a separation between testing and learning on {0,1}^d, for k=\omega(\log d): testing k-monotonicity can be performed with 2^{O(\sqrt d . \log d . \log{1/\eps})} queries, while learning k-monotone functions requires 2^{\Omega(k . \sqrt d .{1/\eps})} queries (Blais et al. (RANDOM 2015)). 3. We present a tolerant test for functions f\colon[n]^d\to \{0,1\}$with complexity independent of n, which makes progress on a problem left open by Berman et al. (STOC 2014). Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques. Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid [n]^d, and draw connections to distribution testing techniques.
Cite as

Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, and Karl Wimmer. Testing k-Monotonicity. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 29:1-29:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{canonne_et_al:LIPIcs.ITCS.2017.29,
  author =	{Canonne, Cl\'{e}ment L. and Grigorescu, Elena and Guo, Siyao and Kumar, Akash and Wimmer, Karl},
  title =	{{Testing k-Monotonicity}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{29:1--29:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.29},
  URN =		{urn:nbn:de:0030-drops-81583},
  doi =		{10.4230/LIPIcs.ITCS.2017.29},
  annote =	{Keywords: Boolean Functions, Learning, Monotonicity, Property Testing}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.8
Capacitated k-Center Problem with Vertex Weights

Authors: Aounon Kumar

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
We study the capacitated k-center problem with vertex weights. It is a generalization of the well known k-center problem. In this variant each vertex has a weight and a capacity. The assignment cost of a vertex to a center is given by the product of the weight of the vertex and its distance to the center. The distances are assumed to form a metric. Each center can only serve as many vertices as its capacity. We show an n^{1-epsilon}-approximation hardness for this problem, for any epsilon > 0, where n is the number of vertices in the input. Both the capacitated and the weighted versions of the k-center problem individually can be approximated within a constant factor. Yet the common extension of both the generalizations cannot be approximated efficiently within a constant factor, unless P = NP. This problem, to the best of our knowledge, is the first facility location problem with metric distances known to have a super-constant inapproximability result. The hardness result easily generalizes to versions of the problem that consider the p-norm of the assignment costs (weighted distances) as the objective function. We give n^{1- 1/p - epsilon}-approximation hardness for this problem, for p>1. We complement the hardness result by showing a simple n-approximation algorithm for this problem. We also give a bi-criteria constant factor approximation algorithm, for the case of uniform capacities, which opens at most 2k centers.
Cite as

Aounon Kumar. Capacitated k-Center Problem with Vertex Weights. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar:LIPIcs.FSTTCS.2016.8,
  author =	{Kumar, Aounon},
  title =	{{Capacitated k-Center Problem with Vertex Weights}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.8},
  URN =		{urn:nbn:de:0030-drops-68438},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.8},
  annote =	{Keywords: approximation hardness, k-center, gadget reduction}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.24
Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments

Authors: Mithilesh Kumar and Daniel Lokshtanov

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
A bipartite tournament is a directed graph T:=(A cup B, E) such that every pair of vertices (a,b), a in A, b in B are connected by an arc, and no arc connects two vertices of A or two vertices of B. A feedback vertex set is a set S of vertices in T such that T - S is acyclic. In this article we consider the Feedback Vertex Set problem in bipartite tournaments. Here the input is a bipartite tournament T on n vertices together with an integer k, and the task is to determine whether T has a feedback vertex set of size at most k. We give a new algorithm for Feedback Vertex Set in Bipartite Tournaments. The running time of our algorithm is upper-bounded by O(1.6181^k + n^{O(1)}), improving over the previously best known algorithm with running time (2^k)k^{O(1)} + n^{O(1)} [Hsiao, ISAAC 2011]. As a by-product, we also obtain the fastest currently known exact exponential-time algorithm for the problem, with running time O(1.3820^n).
Cite as

Mithilesh Kumar and Daniel Lokshtanov. Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.24,
  author =	{Kumar, Mithilesh and Lokshtanov, Daniel},
  title =	{{Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Bipartite Tournaments}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{24:1--24:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.24},
  URN =		{urn:nbn:de:0030-drops-68591},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.24},
  annote =	{Keywords: Parameterized algorithms, Exact algorithms, Feedback vertex set, Tour- naments, Bipartite tournaments}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.31
Most Likely Voronoi Diagrams in Higher Dimensions

Authors: Nirman Kumar, Benjamin Raichel, Subhash Suri, and Kevin Verbeek

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
The Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors.
Cite as

Nirman Kumar, Benjamin Raichel, Subhash Suri, and Kevin Verbeek. Most Likely Voronoi Diagrams in Higher Dimensions. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.31,
  author =	{Kumar, Nirman and Raichel, Benjamin and Suri, Subhash and Verbeek, Kevin},
  title =	{{Most Likely Voronoi Diagrams in Higher Dimensions}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{31:1--31:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.31},
  URN =		{urn:nbn:de:0030-drops-68667},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.31},
  annote =	{Keywords: Uncertainty, Lower bounds, Voronoi Diagrams, Stochastic}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.38
Finer Separations Between Shallow Arithmetic Circuits

Authors: Mrinal Kumar and Ramprasad Saptharishi

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
In this paper, we show that there is a family of polynomials P_n, where P_n is a polynomial in n variables of degree at most d = O(log^2(n)), such that * P_n can be computed by linear sized homogeneous depth-5 arithmetic circuits, * P_n can be computed by poly(n) sized non-homogeneous depth-3 arithmetic circuits. * Any homogeneous depth-4 arithmetic circuit computing P_n must have size at least n^{Omega(sqrt(d))}. This shows that the parameters for the depth reduction results of [Agrawal-Vinay 08, Koiran 12, Tavenas 13] are tight for extremely restricted classes of arithmetic circuits, for instance homogeneous depth-5 circuits and non-homogeneous depth-3 circuits, and over an appropriate range of parameters, qualitatively improve a result of [Kumar-Saraf 14], which showed that the parameters of depth reductions are optimal for algebraic branching programs. As an added advantage, our proofs are much shorter and simpler than the two known proofs of n^{Omega(sqrt(d))} lower bound for homogeneous depth-4 circuits [Kayal-Limaye-Saha-Srinivasan 14, Kumar-Saraf 14], albeit our proofs only work when d = O(log^2(n)).
Cite as

Mrinal Kumar and Ramprasad Saptharishi. Finer Separations Between Shallow Arithmetic Circuits. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 38:1-38:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kumar_et_al:LIPIcs.FSTTCS.2016.38,
  author =	{Kumar, Mrinal and Saptharishi, Ramprasad},
  title =	{{Finer Separations Between Shallow Arithmetic Circuits}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{38:1--38:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.38},
  URN =		{urn:nbn:de:0030-drops-68730},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.38},
  annote =	{Keywords: arithmetic circuits, lower bounds, separations, depth reduction}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2016.39
Sum of Products of Read-Once Formulas

Authors: Ramya C. and B. V. Raghavendra Rao

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular, 1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009]. 2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial. 3. We also show an exponential lower bound for the above model against a polynomial in VP. 4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function. Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004].
Cite as

Ramya C. and B. V. Raghavendra Rao. Sum of Products of Read-Once Formulas. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{c._et_al:LIPIcs.FSTTCS.2016.39,
  author =	{C., Ramya and Rao, B. V. Raghavendra},
  title =	{{Sum of Products of Read-Once Formulas}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.39},
  URN =		{urn:nbn:de:0030-drops-68741},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.39},
  annote =	{Keywords: Arithmetic Circuits, Permanent, Computational Complexity, Algebraic Complexity Theory}
}
Document
DOI: 10.4230/OASIcs.PPES.2011.47
An Automated Flow to Map Throughput Constrained Applications to a MPSoC

Authors: Roel Jordans, Firew Siyoum, Sander Stuijk, Akash Kumar, and Henk Corporaal

Published in: OASIcs, Volume 18, Bringing Theory to Practice: Predictability and Performance in Embedded Systems (2011)

Abstract
This paper describes a design flow to map throughput constrained applications on a Multi-processor System-on-Chip (MPSoC). It integrates several state-of-the-art mapping and synthesis tools into an automated tool flow. This flow takes as input a throughput constrained application, modeled with a synchronous dataflow graph, a C-based implementation for each actor in the graph, and a template based architecture description. Using these inputs, the tool flow generates an MPSoC platform tailored to the application requirements and it subsequently maps the application to this platform. The output of the flow is an FPGA programmable bit file. An easily extensible template based architecture is presented, this architecture allows fast and flexible generation of a predictable platform that can be synthesized using the presented tool flow. The effectiveness of the tool flow is demonstrated by mapping an MJPEG-decoder onto our MPSoC platform. This case study shows that our flow is able to provide a tight, conservative bound on the worst-case throughput of the FPGA implementation. The presented tool flow is freely available at http://www.es.ele.tue.nl/mamps.
Cite as

Roel Jordans, Firew Siyoum, Sander Stuijk, Akash Kumar, and Henk Corporaal. An Automated Flow to Map Throughput Constrained Applications to a MPSoC. In Bringing Theory to Practice: Predictability and Performance in Embedded Systems. Open Access Series in Informatics (OASIcs), Volume 18, pp. 47-58, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{jordans_et_al:OASIcs.PPES.2011.47,
  author =	{Jordans, Roel and Siyoum, Firew and Stuijk, Sander and Kumar, Akash and Corporaal, Henk},
  title =	{{An Automated Flow to Map Throughput Constrained Applications to a MPSoC}},
  booktitle =	{Bringing Theory to Practice: Predictability and Performance in Embedded Systems},
  pages =	{47--58},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-28-6},
  ISSN =	{2190-6807},
  year =	{2011},
  volume =	{18},
  editor =	{Lucas, Philipp and Wilhelm, Reinhard},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.PPES.2011.47},
  URN =		{urn:nbn:de:0030-drops-30819},
  doi =		{10.4230/OASIcs.PPES.2011.47},
  annote =	{Keywords: design flow automation, multi-processor system-on-chip, throughput constrained, synchronous data-flow graphs}
}
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