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Track A: Algorithms, Complexity and Games

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

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.

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} }

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Track A: Algorithms, Complexity and Games

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

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].

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} }

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

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.

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} }

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

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.

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} }

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

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.

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} }

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

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.

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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