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

We define a novel notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it.
We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and n^{k/2} / ε² constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+O_k(ε) fraction of constraints.
Previously, this was known for even k, but for odd k one needed n^{k/2} (log n)^{O(1)} / ε² random constraints to achieve the same conclusion.
Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck’s inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n^⌈k/2⌉) and the third one cannot work when the number of constraints is o(n^{k/2} √{log n}).
We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with n^{k/2} / ε² constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.

Tommaso d'Orsi and Luca Trevisan. A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dorsi_et_al:LIPIcs.CCC.2023.27, author = {d'Orsi, Tommaso and Trevisan, Luca}, title = {{A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {27:1--27:16}, 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.2023.27}, URN = {urn:nbn:de:0030-drops-182979}, doi = {10.4230/LIPIcs.CCC.2023.27}, annote = {Keywords: CSP, k-XOR, strong refutation, sum-of-squares, tensor, graph, hypergraph, non-backtracking walk} }

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**Published in:** LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

Full-bond percolation with parameter p is the process in which, given a graph, for every edge independently, we keep the edge with probability p and delete it with probability 1-p. Bond percolation is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Moreover, the full-bond percolation is equivalent to the Reed-Frost process, a network version of SIR epidemic spreading.
We consider one-dimensional power-law small-world graphs with parameter α obtained as the union of a cycle with additional long-range random edges: each pair of nodes {u,v} at distance L on the cycle is connected by a long-range edge {u,v}, with probability proportional to 1/L^α. Our analysis determines three phases for the percolation subgraph G_p of the small-world graph, depending on the value of α.
- If α < 1, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in 𝒪(log n) hops;
- If 1 < α < 2, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in log^{𝒪(1)}(n) hops;
- If α > 2, for every p < 1, with high probability all connected components of G_p have size 𝒪(log n).

Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Bond Percolation in Small-World Graphs with Power-Law Distribution. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{becchetti_et_al:LIPIcs.SAND.2023.3, author = {Becchetti, Luca and Clementi, Andrea and Pasquale, Francesco and Trevisan, Luca and Ziccardi, Isabella}, title = {{Bond Percolation in Small-World Graphs with Power-Law Distribution}}, booktitle = {2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-275-4}, ISSN = {1868-8969}, year = {2023}, volume = {257}, editor = {Doty, David and Spirakis, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.3}, URN = {urn:nbn:de:0030-drops-179392}, doi = {10.4230/LIPIcs.SAND.2023.3}, annote = {Keywords: Information spreading, gossiping, epidemics, fault-tolerance, network self-organization and formation, complex systems, social and transportation networks} }

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Extended Abstract

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Consensus and Broadcast are two fundamental problems in distributed computing, whose solutions have several applications. Intuitively, Consensus should be no harder than Broadcast, and this can be rigorously established in several models. Can Consensus be easier than Broadcast?
In models that allow noiseless communication, we prove a reduction of (a suitable variant of) Broadcast to binary Consensus, that preserves the communication model and all complexity parameters such as randomness, number of rounds, communication per round, etc., while there is a loss in the success probability of the protocol. Using this reduction, we get, among other applications, the first logarithmic lower bound on the number of rounds needed to achieve Consensus in the uniform GOSSIP model on the complete graph. The lower bound is tight and, in this model, Consensus and Broadcast are equivalent.
We then turn to distributed models with noisy communication channels that have been studied in the context of some bio-inspired systems. In such models, only one noisy bit is exchanged when a communication channel is established between two nodes, and so one cannot easily simulate a noiseless protocol by using error-correcting codes. An Ω(ε^{-2} n) lower bound is proved by Boczkowski et al. [PLOS Comp. Bio. 2018] on the convergence time of binary Broadcast in one such model (noisy uniform PULL), where ε is a parameter that measures the amount of noise).
We prove an O(ε^{-2} log n) upper bound on the convergence time of binary Consensus in such model, thus establishing an exponential complexity gap between Consensus versus Broadcast. We also prove our upper bound above is tight and this implies, for binary Consensus, a further strong complexity gap between noisy uniform PULL and noisy uniform PUSH. Finally, we show a Θ(ε^{-2} n log n) bound for Broadcast in the noisy uniform PULL.

Andrea Clementi, Luciano Gualà, Emanuele Natale, Francesco Pasquale, Giacomo Scornavacca, and Luca Trevisan. Consensus vs Broadcast, with and Without Noise (Extended Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{clementi_et_al:LIPIcs.ITCS.2020.42, author = {Clementi, Andrea and Gual\`{a}, Luciano and Natale, Emanuele and Pasquale, Francesco and Scornavacca, Giacomo and Trevisan, Luca}, title = {{Consensus vs Broadcast, with and Without Noise}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {42:1--42:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.42}, URN = {urn:nbn:de:0030-drops-117277}, doi = {10.4230/LIPIcs.ITCS.2020.42}, annote = {Keywords: Distributed Computing, Consensus, Broadcast, Gossip Models, Noisy Communication Channels} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say +/- 1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average.
We provide a rigorous analysis of the above process showing that, if G exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process. Our analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest.
We then exploit our analysis to design the first opportunistic protocols that approximately recover community structure using only logarithmic (or polylogarithmic, depending on the sparsity of the cut) work per node.

Luca Becchetti, Andrea Clementi, Pasin Manurangsi, Emanuele Natale, Francesco Pasquale, Prasad Raghavendra, and Luca Trevisan. Average Whenever You Meet: Opportunistic Protocols for Community Detection. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{becchetti_et_al:LIPIcs.ESA.2018.7, author = {Becchetti, Luca and Clementi, Andrea and Manurangsi, Pasin and Natale, Emanuele and Pasquale, Francesco and Raghavendra, Prasad and Trevisan, Luca}, title = {{Average Whenever You Meet: Opportunistic Protocols for Community Detection}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.7}, URN = {urn:nbn:de:0030-drops-94705}, doi = {10.4230/LIPIcs.ESA.2018.7}, annote = {Keywords: Community Detection, Random Processes, Spectral Analysis} }

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

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n).
- A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time.
- An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time.
Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010].

Pasin Manurangsi and Luca Trevisan. Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{manurangsi_et_al:LIPIcs.APPROX-RANDOM.2018.20, author = {Manurangsi, Pasin and Trevisan, Luca}, title = {{Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {20:1--20:17}, 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.2018.20}, URN = {urn:nbn:de:0030-drops-94241}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.20}, annote = {Keywords: Approximation algorithms, Exponential-time algorithms, Vertex Cover, Sparsest Cut, Balanced Separator} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

In this paper, we prove an almost-optimal hardness for Max k-CSP_R based on Khot's Unique Games Conjecture (UGC). In Max k-CSP_R, we are given a set of predicates each of which depends on exactly k variables. Each variable can take any value from 1, 2, ..., R. The goal is to find an assignment to variables that maximizes the number of satisfied predicates.
Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max k-CSP_R to within factor 2^{O(k log k)}(log R)^{k/2}/R^{k - 1} for any k, R.
To the best of our knowledge, this result improves on all the known hardness of approximation results when 3 <= k = o(log R/log log R). In this case, the previous best hardness result was NP-hardness of approximating within a factor O(k/R^{k-2}) by Chan. When k = 2, our result matches the best known UGC-hardness result of Khot, Kindler, Mossel and O'Donnell.
In addition, by extending an algorithm for Max 2-CSP_R by Kindler, Kolla and Trevisan, we provide an Omega(log R/R^{k - 1})-approximation algorithm for Max k-CSP_R. This algorithm implies that our inapproximability result is tight up to a factor of 2^{O(k \log k)}(\log R)^{k/2 - 1}. In comparison, when 3 <= k is a constant, the previously known gap was $O(R)$, which is significantly larger than our gap of O(polylog R).
Finally, we show that we can replace the Unique Games Conjecture assumption with Khot's d-to-1 Conjecture and still get asymptotically the same hardness of approximation.

Pasin Manurangsi, Preetum Nakkiran, and Luca Trevisan. Near-Optimal UGC-hardness of Approximating Max k-CSP_R. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 15:1-15:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{manurangsi_et_al:LIPIcs.APPROX-RANDOM.2016.15, author = {Manurangsi, Pasin and Nakkiran, Preetum and Trevisan, Luca}, title = {{Near-Optimal UGC-hardness of Approximating Max k-CSP\underlineR}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {15:1--15:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.15}, URN = {urn:nbn:de:0030-drops-66388}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.15}, annote = {Keywords: inapproximability, unique games conjecture, constraint satisfaction problem, invariance principle} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We show that for any odd k and any instance I of the max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a 1/2 + Omega(1/sqrt(D)) fraction of I's constraints, where D is a bound on the number of constraints that each variable occurs in.
This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a quantum algorithm to find an assignment satisfying a 1/2 Omega(D^{-3/4}) fraction of the equations.
For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a mu + Omega(1/sqrt(degree)) fraction of constraints, where mu is the fraction that would be satisfied by a uniformly random assignment.

Boaz Barak, Ankur Moitra, Ryan O’Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright. Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 110-123, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{barak_et_al:LIPIcs.APPROX-RANDOM.2015.110, author = {Barak, Boaz and Moitra, Ankur and O’Donnell, Ryan and Raghavendra, Prasad and Regev, Oded and Steurer, David and Trevisan, Luca and Vijayaraghavan, Aravindan and Witmer, David and Wright, John}, title = {{Beating the Random Assignment on Constraint Satisfaction Problems of Bounded Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {110--123}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.110}, URN = {urn:nbn:de:0030-drops-52981}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.110}, annote = {Keywords: constraint satisfaction problems, bounded degree, advantage over random} }

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