Document

Track A: Algorithms, Complexity and Games

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

We consider a very wide class of models for sparse random Boolean 2CSPs; equivalently, degree-2 optimization problems over {±1}ⁿ. For each model ℳ, we identify the "high-probability value" s^*_ℳ of the natural SDP relaxation (equivalently, the quantum value). That is, for all ε > 0 we show that the SDP optimum of a random n-variable instance is (when normalized by n) in the range (s^*_ℳ-ε, s^*_ℳ+ε) with high probability. Our class of models includes non-regular CSPs, and ones where the SDP relaxation value is strictly smaller than the spectral relaxation value.

Amulya Musipatla, Ryan O'Donnell, Tselil Schramm, and Xinyu Wu. The SDP Value of Random 2CSPs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 97:1-97:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{musipatla_et_al:LIPIcs.ICALP.2022.97, author = {Musipatla, Amulya and O'Donnell, Ryan and Schramm, Tselil and Wu, Xinyu}, title = {{The SDP Value of Random 2CSPs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {97:1--97:19}, 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.2022.97}, URN = {urn:nbn:de:0030-drops-164381}, doi = {10.4230/LIPIcs.ICALP.2022.97}, annote = {Keywords: Random constraint satisfaction problems} }

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

We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^Ω(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent).
We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k.
Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis.

Pasin Manurangsi, Aviad Rubinstein, and Tselil Schramm. The Strongish Planted Clique Hypothesis and Its Consequences. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 10:1-10:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{manurangsi_et_al:LIPIcs.ITCS.2021.10, author = {Manurangsi, Pasin and Rubinstein, Aviad and Schramm, Tselil}, title = {{The Strongish Planted Clique Hypothesis and Its Consequences}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {10:1--10:21}, 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.2021.10}, URN = {urn:nbn:de:0030-drops-135491}, doi = {10.4230/LIPIcs.ITCS.2021.10}, annote = {Keywords: Planted Clique, Densest k-Subgraph, Hardness of Approximation} }

Document

**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

Let G be any n-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by 1/sqrt{Delta} (for example, a random graph G of average degree Theta(Delta) typically has this property). We show that the exp(c (log n)/(log Delta))-round Sherali - Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1 % (in fact, at most 1/2 + 2^{-Omega(c)}). For example, in random graphs with n^{1.01} edges, O(1) rounds suffice; in random graphs with n * polylog(n) edges, n^{O(1/log log n)} = n^{o(1)} rounds suffice.
Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for max-cut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali - Adams can strongly refute random Boolean k-CSP instances with n^{ceil[k/2] + delta} constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.

Ryan O'Donnell and Tselil Schramm. Sherali - Adams Strikes Back. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 8:1-8:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{odonnell_et_al:LIPIcs.CCC.2019.8, author = {O'Donnell, Ryan and Schramm, Tselil}, title = {{Sherali - Adams Strikes Back}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {8:1--8:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.8}, URN = {urn:nbn:de:0030-drops-108309}, doi = {10.4230/LIPIcs.CCC.2019.8}, annote = {Keywords: Linear programming, Sherali, Adams, max-cut, graph eigenvalues, Sum-of-Squares} }

Document

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value beta, it did not necessarily actually "satisfy" the constraint "objective = beta". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating global constraints into local constraints. Using these ideas, we also show that degree-Omega(sqrt{n}) SOS does not provide a (4/3 - epsilon)-approximation for Min-Bisection, and degree-Omega(n) SOS does not provide a (11/12 + epsilon)-approximation for Max-Bisection or a (5/4 - epsilon)-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.

Pravesh K. Kothari, Ryan O'Donnell, and Tselil Schramm. SOS Lower Bounds with Hard Constraints: Think Global, Act Local. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 49:1-49:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kothari_et_al:LIPIcs.ITCS.2019.49, author = {Kothari, Pravesh K. and O'Donnell, Ryan and Schramm, Tselil}, title = {{SOS Lower Bounds with Hard Constraints: Think Global, Act Local}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {49:1--49:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.49}, URN = {urn:nbn:de:0030-drops-101420}, doi = {10.4230/LIPIcs.ITCS.2019.49}, annote = {Keywords: sum-of-squares hierarchy, random constraint satisfaction problems} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem:
on query S \subset V, the oracle returns the size of the cut between S and V \ S.
We provide algorithms computing an exact minimum $s$-$t$ cut in $G$ with ~{O}(n^{5/3}) queries, and computing an exact global minimum cut of G with only ~{O}(n) queries (while learning the graph requires ~{\Theta}(n^2) queries).

Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing Exact Minimum Cuts Without Knowing the Graph. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{rubinstein_et_al:LIPIcs.ITCS.2018.39, author = {Rubinstein, Aviad and Schramm, Tselil and Weinberg, S. Matthew}, title = {{Computing Exact Minimum Cuts Without Knowing the Graph}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {39:1--39:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.39}, URN = {urn:nbn:de:0030-drops-83168}, doi = {10.4230/LIPIcs.ITCS.2018.39}, annote = {Keywords: query complexity, minimum cut} }

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

In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness epsilon and soundness 1/2 is at least as difficult as Small-Set Expansion with completeness epsilon and soundness f(epsilon), for any function f(epsilon) which grows faster than (epsilon)^(1/2). We achieve this amplification via random walks--the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same.

Prasad Raghavendra and Tselil Schramm. Gap Amplification for Small-Set Expansion via Random Walks. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 381-391, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{raghavendra_et_al:LIPIcs.APPROX-RANDOM.2014.381, author = {Raghavendra, Prasad and Schramm, Tselil}, title = {{Gap Amplification for Small-Set Expansion via Random Walks}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {381--391}, 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.2014.381}, URN = {urn:nbn:de:0030-drops-47108}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.381}, annote = {Keywords: Gap amplification, Small-Set Expansion, random walks, graph products, Unique Games} }

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

The stochastic block model is a classical cluster-exhibiting random graph model that has been widely studied in statistics, physics and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intra-cluster edge probability p, and inter-cluster edge probability q. We focus on the sparse case, i.e. p, q = O(1/n), which is practically more relevant and also mathematically more challenging. A conjecture of Decelle, Krzakala, Moore and Zdeborova, based on ideas from statistical physics, predicted a specific threshold for clustering. The negative direction
of the conjecture was proved by Mossel, Neeman and Sly (2012), and more recently the positive direction was proven independently by Massoulie and Mossel, Neeman, and Sly.
In many real network clustering problems, nodes contain information as well. We study the interplay between node and network information in clustering by studying a labeled block model, where in addition to the edge information, the true cluster labels of a small fraction of the nodes are revealed. In the case of two clusters, we show that below the threshold, a small amount of node information does not affect recovery. On the other hand, we show that for any small amount of information efficient local clustering is achievable as long as the number of clusters is sufficiently large (as a function of the amount of revealed information).

Varun Kanade, Elchanan Mossel, and Tselil Schramm. Global and Local Information in Clustering Labeled Block Models. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 779-792, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{kanade_et_al:LIPIcs.APPROX-RANDOM.2014.779, author = {Kanade, Varun and Mossel, Elchanan and Schramm, Tselil}, title = {{Global and Local Information in Clustering Labeled Block Models}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {779--792}, 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.2014.779}, URN = {urn:nbn:de:0030-drops-47384}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.779}, annote = {Keywords: stochastic block models, information flow on trees} }