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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.

Matthew Coulson, Ewan Davies, Alexandra Kolla, Viresh Patel, and Guus Regts. Statistical Physics Approaches to Unique Games. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 13:1-13:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{coulson_et_al:LIPIcs.CCC.2020.13, author = {Coulson, Matthew and Davies, Ewan and Kolla, Alexandra and Patel, Viresh and Regts, Guus}, title = {{Statistical Physics Approaches to Unique Games}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {13:1--13:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.13}, URN = {urn:nbn:de:0030-drops-125650}, doi = {10.4230/LIPIcs.CCC.2020.13}, annote = {Keywords: Unique Games Conjecture, approximation algorithm, Potts model, cluster expansion, polynomial interpolation} }

Document

**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of finding symmetric signings of matrices with natural spectral properties. Our results are the following:
1) We characterize matrices that have an invertible signing: a symmetric matrix has an invertible symmetric signing if and only if the support graph of the matrix contains a perfect 2-matching. Further, we present an efficient algorithm to search for an invertible symmetric signing.
2) We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing.
3) We show NP-completeness of the following problems: verifying whether a given matrix has a symmetric signing that is singular or has bounded eigenvalues. However, we also illustrate that the complexity could differ substantially for input matrices that are adjacency matrices of graphs.
We use combinatorial techniques in addition to classic results from matching theory.

Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, and Alexandra Kolla. Spectral Aspects of Symmetric Matrix Signings. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{carlson_et_al:LIPIcs.MFCS.2019.81, author = {Carlson, Charles and Chandrasekaran, Karthekeyan and Chang, Hsien-Chih and Kakimura, Naonori and Kolla, Alexandra}, title = {{Spectral Aspects of Symmetric Matrix Signings}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {81:1--81:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.81}, URN = {urn:nbn:de:0030-drops-110258}, doi = {10.4230/LIPIcs.MFCS.2019.81}, annote = {Keywords: Spectral Graph Theory, Matrix Signing, Matchings} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We initiate the study of spectral generalizations of the graph isomorphism problem.
b) The Spectral Graph Dominance (SGD) problem: On input of two graphs G and H does there exist a permutation pi such that G preceq pi(H)?
c) The Spectrally Robust Graph Isomorphism (kappa-SRGI) problem: On input of two graphs G and H, find the smallest number kappa over all permutations pi such that pi(H) preceq G preceq kappa c pi(H) for some c. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology.
G preceq c H means that for all vectors x we have x^T L_G x <= c x^T L_H x, where L_G is the Laplacian G.
We prove NP-hardness for SGD. We also present a kappa^3-approximation algorithm for SRGI for the case when both G and H are bounded-degree trees. The algorithm runs in polynomial time when kappa is a constant.

Alexandra Kolla, Ioannis Koutis, Vivek Madan, and Ali Kemal Sinop. Spectrally Robust Graph Isomorphism. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kolla_et_al:LIPIcs.ICALP.2018.84, author = {Kolla, Alexandra and Koutis, Ioannis and Madan, Vivek and Sinop, Ali Kemal}, title = {{Spectrally Robust Graph Isomorphism}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {84:1--84:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.84}, URN = {urn:nbn:de:0030-drops-90887}, doi = {10.4230/LIPIcs.ICALP.2018.84}, annote = {Keywords: Network Alignment, Graph Isomorphism, Graph Similarity} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

A k-lift of an n-vertex base graph G is a graph H on nxk vertices, where each vertex v of G is replaced by k vertices v_1,...,v_k and each edge uv in G is replaced by a matching representing a bijection pi_{uv} so that the edges of H are of the form (u_i,v_{pi_{uv}(i)}). Lifts have been investigated as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are:
1. A uniform random lift by a cyclic group of order k of any n-vertex d-regular base graph G, with the nontrivial eigenvalues of the adjacency matrix of G bounded by lambda in magnitude, has the new nontrivial eigenvalues bounded by lambda+O(sqrt{d}) in magnitude with probability 1-ke^{-Omega(n/d^2)}. The probability bounds as well as the dependency on lambda are almost optimal. As a special case, we obtain that there is a constant c_1 such that for every k<=2^{c_1n/d^2}, there exists a lift H of every Ramanujan graph by a cyclic group of order k such that H is almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most O(sqrt{d}) in magnitude). We also show how this result leads to a quasi-polynomial time deterministic algorithm to construct almost Ramanujan expanders.
2. There is a constant c_2 such that for every k>=2^{c_2nd}, there does not exist an abelian k-lift H of any n-vertex d-regular base graph such that H is almost Ramanujan. This can be viewed as an analogue of the well-known no-expansion result for constant degree abelian Cayley graphs.
Suppose k_0 is the order of the largest abelian group that produces expanding lifts. Our two results highlight lower and upper bounds on k_0 that are tight upto a factor of d^3 in the exponent, thus suggesting a threshold phenomenon.

Naman Agarwal, Karthekeyan Chandrasekaran, Alexandra Kolla, and Vivek Madan. On the Expansion of Group-Based Lifts. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{agarwal_et_al:LIPIcs.APPROX-RANDOM.2017.24, author = {Agarwal, Naman and Chandrasekaran, Karthekeyan and Kolla, Alexandra and Madan, Vivek}, title = {{On the Expansion of Group-Based Lifts}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.24}, URN = {urn:nbn:de:0030-drops-75739}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.24}, annote = {Keywords: Expanders, Lifts, Spectral Graph Theory} }

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