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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

We study the notion of local treewidth in sparse random graphs: the maximum treewidth over all k-vertex subgraphs of an n-vertex graph. When k is not too large, we give nearly tight bounds for this local treewidth parameter; we also derive nearly tight bounds for the local treewidth of noisy trees, trees where every non-edge is added independently with small probability. We apply our upper bounds on the local treewidth to obtain fixed parameter tractable algorithms (on random graphs and noisy trees) for edge-removal problems centered around containing a contagious process evolving over a network. In these problems, our main parameter of study is k, the number of initially "infected" vertices in the network. For the random graph models we consider and a certain range of parameters the running time of our algorithms on n-vertex graphs is 2^o(k) poly(n), improving upon the 2^Ω(k) poly(n) performance of the best-known algorithms designed for worst-case instances of these edge deletion problems.

Hermish Mehta and Daniel Reichman. Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mehta_et_al:LIPIcs.APPROX/RANDOM.2022.7, author = {Mehta, Hermish and Reichman, Daniel}, title = {{Local Treewidth of Random and Noisy Graphs with Applications to Stopping Contagion in Networks}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {7:1--7:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.7}, URN = {urn:nbn:de:0030-drops-171299}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.7}, annote = {Keywords: Graph Algorithms, Random Graphs, Data Structures and Algorithms, Discrete Mathematics} }

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

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of k ReLUs minimizing the squared error (for k > 1) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error ε. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest κ-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in 1/ε². Together with a previous work regarding improperly learning a ReLU [Surbhi Goel et al., 2017], this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on ε.

Surbhi Goel, Adam Klivans, Pasin Manurangsi, and Daniel Reichman. Tight Hardness Results for Training Depth-2 ReLU Networks. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goel_et_al:LIPIcs.ITCS.2021.22, author = {Goel, Surbhi and Klivans, Adam and Manurangsi, Pasin and Reichman, Daniel}, title = {{Tight Hardness Results for Training Depth-2 ReLU Networks}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {22:1--22:14}, 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.2021.22}, URN = {urn:nbn:de:0030-drops-135611}, doi = {10.4230/LIPIcs.ITCS.2021.22}, annote = {Keywords: ReLU, Learning Algorithm, Running Time Lower Bound} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings.
- Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol.
- Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k.
- Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem.

Alexander Golovnev, Mika Göös, Daniel Reichman, and Igor Shinkar. String Matching: Communication, Circuits, and Learning. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{golovnev_et_al:LIPIcs.APPROX-RANDOM.2019.56, author = {Golovnev, Alexander and G\"{o}\"{o}s, Mika and Reichman, Daniel and Shinkar, Igor}, title = {{String Matching: Communication, Circuits, and Learning}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {56:1--56:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.56}, URN = {urn:nbn:de:0030-drops-112717}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.56}, annote = {Keywords: string matching, communication complexity, circuit complexity, PAC learning} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Analyzing multi-dimensional data is a fundamental problem in various areas of computer science. As the amount of data is often huge, it is desirable to obtain sublinear time algorithms to understand local properties of the data.
We focus on the natural problem of testing pattern freeness: given a large d-dimensional array A and a fixed d-dimensional pattern P over a finite alphabet Gamma, we say that A is P-free if it does not contain a copy of the forbidden pattern P as a consecutive subarray. The distance of A to P-freeness is the fraction of the entries of A that need to be modified to make it P-free.
For any epsilon > 0 and any large enough pattern P over any alphabet - other than a very small set of exceptional patterns - we design a tolerant tester that distinguishes between the case that the distance is at least epsilon and the case that the distance is at most a_d epsilon, with query complexity and running time c_d epsilon^{-1}, where a_d < 1 and c_d depend only on the dimension d. These testers only need to access uniformly random blocks of samples from the input A.
To analyze the testers we establish several combinatorial results, including the following d-dimensional modification lemma, which might be of independent interest: For any large enough d-dimensional pattern P over any alphabet (excluding a small set of exceptional patterns for the binary case), and any d-dimensional array A containing a copy of P, one can delete this copy by modifying one of its locations without creating new P-copies in A.
Our results address an open question of Fischer and Newman, who asked whether there exist efficient testers for properties related to tight substructures in multi-dimensional structured data.

Omri Ben-Eliezer, Simon Korman, and Daniel Reichman. Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{beneliezer_et_al:LIPIcs.ICALP.2017.9, author = {Ben-Eliezer, Omri and Korman, Simon and Reichman, Daniel}, title = {{Deleting and Testing Forbidden Patterns in Multi-Dimensional Arrays}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.9}, URN = {urn:nbn:de:0030-drops-74427}, doi = {10.4230/LIPIcs.ICALP.2017.9}, annote = {Keywords: Property testing, Sublinear algorithms, Pattern matching} }

Document

**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational hardness of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard graph problems remain essentially as hard on percolated instances as they are in the worst-case (assuming NP !subseteq BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems, where random deletions are applied to clauses or variables.
We focus on proving the hardness of the Maximum Independent Set problem and the Graph Coloring problem on percolated instances. To show this we establish the robustness of the corresponding parameters alpha(.) and Chi(.) to percolation, which may be of independent interest. Given a graph G, let G' be the graph obtained by randomly deleting edges of G. We show that if alpha(G) is small, then alpha(G') remains small with probability at least 0.99. Similarly, we show that if Chi(G) is large, then Chi(G') remains large with probability at least 0.99.

Huck Bennett, Daniel Reichman, and Igor Shinkar. On Percolation and NP-Hardness. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 80:1-80:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bennett_et_al:LIPIcs.ICALP.2016.80, author = {Bennett, Huck and Reichman, Daniel and Shinkar, Igor}, title = {{On Percolation and NP-Hardness}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {80:1--80:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.80}, URN = {urn:nbn:de:0030-drops-62056}, doi = {10.4230/LIPIcs.ICALP.2016.80}, annote = {Keywords: percolation, NP-hardness, random subgraphs, chromatic number} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

The main paradigm of smoothed analysis on graphs suggests that for any large graph G in a certain class of graphs, perturbing slightly the edges of G at random (usually adding few random edges to G) typically results in a graph having much "nicer" properties. In this work we study smoothed analysis on trees or, equivalently, on connected graphs. Given an n-vertex connected graph G, form a random supergraph of G* of G by turning every pair of vertices of G into an edge with probability epsilon/n, where epsilon is a small positive constant. This perturbation model has been studied previously in several contexts, including smoothed analysis, small world networks, and combinatorics.
Connected graphs can be bad expanders, can have very large diameter, and possibly contain no long paths. In contrast, we show that if G is an n-vertex connected graph then typically G* has edge expansion Omega(1/(log n)), diameter O(log n), vertex expansion Omega(1/(log n)), and contains a path of length Omega(n), where for the last two properties we additionally assume that G has bounded maximum degree. Moreover, we show that if G has bounded degeneracy, then typically the mixing time of the lazy random walk on G* is O(log^2(n)). All these results are asymptotically tight.

Michael Krivelevich, Daniel Reichman, and Wojciech Samotij. Smoothed Analysis on Connected Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 810-825, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{krivelevich_et_al:LIPIcs.APPROX-RANDOM.2014.810, author = {Krivelevich, Michael and Reichman, Daniel and Samotij, Wojciech}, title = {{Smoothed Analysis on Connected Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {810--825}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.810}, URN = {urn:nbn:de:0030-drops-47407}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.810}, annote = {Keywords: Random walks and Markov chains, Random network models} }