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APPROX

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

In a breakthrough work, Kawarabayashi and Thorup (J. ACM'19) gave a near-linear time deterministic algorithm to compute the weight of a minimum cut in a simple graph G = (V,E). A key component of this algorithm is finding the (1+ε)-KT partition of G, the coarsest partition {P_1, …, P_k} of V such that for every non-trivial (1+ε)-near minimum cut with sides {S, ̄{S}} it holds that P_i is contained in either S or ̄{S}, for i = 1, …, k. In this work we give a near-linear time randomized algorithm to find the (1+ε)-KT partition of a weighted graph. Our algorithm is quite different from that of Kawarabayashi and Thorup and builds on Karger’s framework of tree-respecting cuts (J. ACM'00).
We describe a number of applications of the algorithm. (i) The algorithm makes progress towards a more efficient algorithm for constructing the polygon representation of the set of near-minimum cuts in a graph. This is a generalization of the cactus representation, and was initially described by Benczúr (FOCS'95). (ii) We improve the time complexity of a recent quantum algorithm for minimum cut in a simple graph in the adjacency list model from Õ(n^{3/2}) to Õ(√{mn}), when the graph has n vertices and m edges. (iii) We describe a new type of randomized algorithm for minimum cut in simple graphs with complexity 𝒪(m + n log⁶ n). For graphs that are not too sparse, this matches the complexity of the current best 𝒪(m + n log² n) algorithm which uses a different approach based on random contractions.
The key technical contribution of our work is the following. Given a weighted graph G with m edges and a spanning tree T of G, consider the graph H whose nodes are the edges of T, and where there is an edge between two nodes of H iff the corresponding 2-respecting cut of T is a non-trivial near-minimum cut of G. We give a 𝒪(m log⁴ n) time deterministic algorithm to compute a spanning forest of H.

Simon Apers, Paweł Gawrychowski, and Troy Lee. Finding the KT Partition of a Weighted Graph in Near-Linear Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{apers_et_al:LIPIcs.APPROX/RANDOM.2022.32, author = {Apers, Simon and Gawrychowski, Pawe{\l} and Lee, Troy}, title = {{Finding the KT Partition of a Weighted Graph in Near-Linear Time}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {32:1--32:14}, 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.2022.32}, URN = {urn:nbn:de:0030-drops-171544}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.32}, annote = {Keywords: Graph theory} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Let G = (V,w) be a weighted undirected graph with m edges. The cut dimension of G is the dimension of the span of the characteristic vectors of the minimum cuts of G, viewed as vectors in {0,1}^m. For every n ≥ 2 we show that the cut dimension of an n-vertex graph is at most 2n-3, and construct graphs realizing this bound.
The cut dimension was recently defined by Graur et al. [Andrei Graur et al., 2020], who show that the maximum cut dimension of an n-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on n-vertex graphs. For every n ≥ 2, Graur et al. exhibit a graph on n vertices with cut dimension at least 3n/2 -2, giving the first lower bound larger than n on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ ℝ^{binom(n,2)} and receives the answer w^T x. Our results thus show a lower bound of 2n-3 on the number of linear queries needed by a deterministic algorithm to solve minimum cut on n-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension.
We further introduce a generalization of the cut dimension which we call the 𝓁₁-approximate cut dimension. The 𝓁₁-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on n = 3k+1 vertices with 𝓁₁-approximate cut dimension 2n-2, showing that it can be strictly larger than the cut dimension.

Troy Lee, Tongyang Li, Miklos Santha, and Shengyu Zhang. On the Cut Dimension of a Graph. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 15:1-15:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lee_et_al:LIPIcs.CCC.2021.15, author = {Lee, Troy and Li, Tongyang and Santha, Miklos and Zhang, Shengyu}, title = {{On the Cut Dimension of a Graph}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {15:1--15:35}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.15}, URN = {urn:nbn:de:0030-drops-142890}, doi = {10.4230/LIPIcs.CCC.2021.15}, annote = {Keywords: Query complexity, submodular function minimization, cut dimension} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using Õ(n^{3/2}√{τ}) queries and time. Moreover, for every integer 1 ≤ τ ≤ n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires Ω(n^{3/2}√{τ}) queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n²) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not.
In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is ̃ Θ(m). We show that the quantum query and time complexity are Õ(√{mnτ}) and Õ(√{mnτ} + n^{3/2}), respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n^{3/2}) for τ > 1 and Ω(τ n) for any 1 ≤ τ ≤ n.
Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger’s tree packing technique (STOC 1996).

Simon Apers and Troy Lee. Quantum Complexity of Minimum Cut. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 28:1-28:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{apers_et_al:LIPIcs.CCC.2021.28, author = {Apers, Simon and Lee, Troy}, title = {{Quantum Complexity of Minimum Cut}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {28:1--28:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.28}, URN = {urn:nbn:de:0030-drops-143026}, doi = {10.4230/LIPIcs.CCC.2021.28}, annote = {Keywords: Quantum algorithms, quantum query complexity, minimum cut} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a k-Fourier-sparse n-bit Boolean function from O(k^{1.5}(log k)^2) uniform quantum examples for that function. This improves over the bound of Theta~(kn) uniformly random classical examples (Haviv and Regev, CCC'15). Our main tool is an improvement of Chang’s lemma for sparse Boolean functions. Second, we show that if a concept class {C} can be exactly learned using Q quantum membership queries, then it can also be learned using O ({Q^2}/{log Q} * log|C|) classical membership queries. This improves the previous-best simulation result (Servedio-Gortler, SICOMP'04) by a log Q-factor.

Srinivasan Arunachalam, Sourav Chakraborty, Troy Lee, Manaswi Paraashar, and Ronald de Wolf. Two New Results About Quantum Exact Learning. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arunachalam_et_al:LIPIcs.ICALP.2019.16, author = {Arunachalam, Srinivasan and Chakraborty, Sourav and Lee, Troy and Paraashar, Manaswi and de Wolf, Ronald}, title = {{Two New Results About Quantum Exact Learning}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.16}, URN = {urn:nbn:de:0030-drops-105929}, doi = {10.4230/LIPIcs.ICALP.2019.16}, annote = {Keywords: quantum computing, exact learning, analysis of Boolean functions, Fourier sparse Boolean functions} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

For any relation f subseteq {0,1}^n x S and any partial Boolean function g:{0,1}^m -> {0,1,*}, we show that R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * sqrt{R_{1/3}(g)}) , where R_epsilon(*) stands for the bounded-error randomized query complexity with error at most epsilon, and f o g^n subseteq ({0,1}^m)^n x S denotes the composition of f with n instances of g.
The new composition theorem is optimal, at least, for the general case of relational problems: A relation f_0 and a partial Boolean function g_0 are constructed, such that R_{4/9}(f_0) in Theta(sqrt n), R_{1/3}(g_0)in Theta(n) and R_{1/3}(f_0 o g_0^n) in Theta(n).
The theorem is proved via introducing a new complexity measure, max-conflict complexity, denoted by bar{chi}(*). Its investigation shows that bar{chi}(g) in Omega(sqrt{R_{1/3}(g)}) for any partial Boolean function g and R_{1/3}(f o g^n) in Omega(R_{4/9}(f) * bar{chi}(g)) for any relation f, which readily implies the composition statement. It is further shown that bar{chi}(g) is always at least as large as the sabotage complexity of g.

Dmitry Gavinsky, Troy Lee, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 64:1-64:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{gavinsky_et_al:LIPIcs.ICALP.2019.64, author = {Gavinsky, Dmitry and Lee, Troy and Santha, Miklos and Sanyal, Swagato}, title = {{A Composition Theorem for Randomized Query Complexity via Max-Conflict Complexity}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {64:1--64:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.64}, URN = {urn:nbn:de:0030-drops-106407}, doi = {10.4230/LIPIcs.ICALP.2019.64}, annote = {Keywords: query complexity, lower bounds} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

Tom Bannink, Jop Briët, Harry Buhrman, Farrokh Labib, and Troy Lee. Bounding Quantum-Classical Separations for Classes of Nonlocal Games. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 12:1-12:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bannink_et_al:LIPIcs.STACS.2019.12, author = {Bannink, Tom and Bri\"{e}t, Jop and Buhrman, Harry and Labib, Farrokh and Lee, Troy}, title = {{Bounding Quantum-Classical Separations for Classes of Nonlocal Games}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {12:1--12:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.12}, URN = {urn:nbn:de:0030-drops-102512}, doi = {10.4230/LIPIcs.STACS.2019.12}, annote = {Keywords: Nonlocal games, communication complexity, bounded separations, semidefinite programming, pseudorandomness, Gowers norms} }

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

We initiate the study of quantum races, games where two or more quantum computers compete to solve a computational problem. While the problem of dueling algorithms has been studied for classical deterministic algorithms [Immorlica et al., 2011], the quantum case presents additional sources of uncertainty for the players. The foremost among these is that players do not know if they have solved the problem until they measure their quantum state. This question of "when to measure?" presents a very interesting strategic problem. We develop a game-theoretic model of a multiplayer quantum race, and find an approximate Nash equilibrium where all players play the same strategy. In the two-party case, we further show that this strategy is nearly optimal in terms of payoff among all symmetric Nash equilibria. A key role in our analysis of quantum races is played by a more tractable version of the game where there is no payout on a tie; for such races we completely characterize the Nash equilibria in the two-party case.
One application of our results is to the stability of the Bitcoin protocol when mining is done by quantum computers. Bitcoin mining is a race to solve a computational search problem, with the winner gaining the right to create a new block. Our results inform the strategies that eventual quantum miners should use, and also indicate that the collision probability - the probability that two miners find a new block at the same time - would not be too high in the case of quantum miners. Such collisions are undesirable as they lead to forking of the Bitcoin blockchain.

Troy Lee, Maharshi Ray, and Miklos Santha. Strategies for Quantum Races. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 51:1-51:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lee_et_al:LIPIcs.ITCS.2019.51, author = {Lee, Troy and Ray, Maharshi and Santha, Miklos}, title = {{Strategies for Quantum Races}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {51:1--51:21}, 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.2019.51}, URN = {urn:nbn:de:0030-drops-101446}, doi = {10.4230/LIPIcs.ITCS.2019.51}, annote = {Keywords: Game theory, Bitcoin mining, Quantum computing, Convex optimization} }

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

Let the randomized query complexity of a relation for error probability epsilon be denoted by R_epsilon(). We prove that for any relation f contained in {0,1}^n times R and Boolean function g:{0,1}^m -> {0,1}, R_{1/3}(f o g^n) = Omega(R_{4/9}(f).R_{1/2-1/n^4}(g)), where f o g^n is the relation obtained by composing f and g. We also show using an XOR lemma that R_{1/3}(f o (g^{xor}_{O(log n)})^n) = Omega(log n . R_{4/9}(f) . R_{1/3}(g))$, where g^{xor}_{O(log n)} is the function obtained by composing the XOR function on O(log n) bits and g.

Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, and Swagato Sanyal. A Composition Theorem for Randomized Query Complexity. 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. 10:1-10:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{anshu_et_al:LIPIcs.FSTTCS.2017.10, author = {Anshu, Anurag and Gavinsky, Dmitry and Jain, Rahul and Kundu, Srijita and Lee, Troy and Mukhopadhyay, Priyanka and Santha, Miklos and Sanyal, Swagato}, title = {{A Composition Theorem for Randomized Query Complexity}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {10:1--10:13}, 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.2017.10}, URN = {urn:nbn:de:0030-drops-83967}, doi = {10.4230/LIPIcs.FSTTCS.2017.10}, annote = {Keywords: Query algorithms and complexity, Decision trees, Composition theorem, XOR lemma, Hardness amplification} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma-2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank.
In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F.

Anurag Anshu, Shalev Ben-David, Ankit Garg, Rahul Jain, Robin Kothari, and Troy Lee. Separating Quantum Communication and Approximate Rank. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 24:1-24:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{anshu_et_al:LIPIcs.CCC.2017.24, author = {Anshu, Anurag and Ben-David, Shalev and Garg, Ankit and Jain, Rahul and Kothari, Robin and Lee, Troy}, title = {{Separating Quantum Communication and Approximate Rank}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {24:1--24:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.24}, URN = {urn:nbn:de:0030-drops-75303}, doi = {10.4230/LIPIcs.CCC.2017.24}, annote = {Keywords: Communication Complexity, Quantum Computing, Lower Bounds, logrank, Quantum Information} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We study how well functions over the boolean hypercube of the form f_k(x)=(|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in l_{infinity}-norm as well as in l_1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [Lee/Raghavendra/Steurer, STOC 2015] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on l_1-approximation of f_k; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [Grigoriev, Comp. Compl. 2001]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.

Troy Lee, Anupam Prakash, Ronald de Wolf, and Henry Yuen. On the Sum-of-Squares Degree of Symmetric Quadratic Functions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 17:1-17:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{lee_et_al:LIPIcs.CCC.2016.17, author = {Lee, Troy and Prakash, Anupam and de Wolf, Ronald and Yuen, Henry}, title = {{On the Sum-of-Squares Degree of Symmetric Quadratic Functions}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {17:1--17:31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.17}, URN = {urn:nbn:de:0030-drops-58383}, doi = {10.4230/LIPIcs.CCC.2016.17}, annote = {Keywords: Sum-of-squares degree, approximation theory, Positivstellensatz refutations of knapsack, quantum query complexity in expectation, extension complexity} }

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**Published in:** Dagstuhl Reports, Volume 5, Issue 2 (2015)

This report documents the program and the outcomes of Dagstuhl Seminar 15082 "Limitations of convex programming: lower bounds on extended formulations and factorization ranks" held in February 2015. Summaries of a selection of talks are given in addition to a list of open problems raised during the seminar.

Hartmut Klauck, Troy Lee, Dirk Oliver Theis, and Rekha R. Thomas. Limitations of Convex Programming: Lower Bounds on Extended Formulations and Factorization Ranks (Dagstuhl Seminar 15082). In Dagstuhl Reports, Volume 5, Issue 2, pp. 109-127, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{klauck_et_al:DagRep.5.2.109, author = {Klauck, Hartmut and Lee, Troy and Theis, Dirk Oliver and Thomas, Rekha R.}, title = {{Limitations of Convex Programming: Lower Bounds on Extended Formulations and Factorization Ranks (Dagstuhl Seminar 15082)}}, pages = {109--127}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {5}, number = {2}, editor = {Klauck, Hartmut and Lee, Troy and Theis, Dirk Oliver and Thomas, Rekha R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.2.109}, URN = {urn:nbn:de:0030-drops-50480}, doi = {10.4230/DagRep.5.2.109}, annote = {Keywords: Convex optimization, extended formulations, cone rank, positive semidefinite rank, nonnegative rank, quantum communication complexity, real algebraic geometry} }

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

Given a matrix A, we study how many epsilon-cubes are required to cover the convex hull of the columns of A. We show bounds on this cover number in terms of VC dimension and the gamma_2 norm and give algorithms for enumerating elements of a cover. This leads to algorithms for computing approximate Nash equilibria that unify and extend several previous results in the literature. Moreover, our approximation algorithms can be applied quite generally to a family of quadratic optimization problems that also includes finding the k-by-k combinatorial rectangle of a matrix. In particular, for this problem we give the first quasi-polynomial time additive approximation algorithm that works for any matrix A in [0,1]^{m x n}.

Noga Alon, Troy Lee, and Adi Shraibman. The Cover Number of a Matrix and its Algorithmic Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 34-47, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{alon_et_al:LIPIcs.APPROX-RANDOM.2014.34, author = {Alon, Noga and Lee, Troy and Shraibman, Adi}, title = {{The Cover Number of a Matrix and its Algorithmic Applications}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {34--47}, 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.2014.34}, URN = {urn:nbn:de:0030-drops-46865}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.34}, annote = {Keywords: Approximation algorithms, Approximate Nash equilibria, Cover number, VC dimension} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 2 (2013)

This report documents the program and the outcomes of Dagstuhl Seminar 13082 "Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices".

LeRoy B. Beasley, Hartmut Klauck, Troy Lee, and Dirk Oliver Theis. Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082). In Dagstuhl Reports, Volume 3, Issue 2, pp. 127-143, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{beasley_et_al:DagRep.3.2.127, author = {Beasley, LeRoy B. and Klauck, Hartmut and Lee, Troy and Theis, Dirk Oliver}, title = {{Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (Dagstuhl Seminar 13082)}}, pages = {127--143}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {3}, number = {2}, editor = {Beasley, LeRoy B. and Klauck, Hartmut and Lee, Troy and Theis, Dirk Oliver}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.2.127}, URN = {urn:nbn:de:0030-drops-40191}, doi = {10.4230/DagRep.3.2.127}, annote = {Keywords: nonnegative rank, combinatorial optimization, communication complexity, extended formulation size} }

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**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We study the communication complexity of a number of graph properties where the edges of the graph G are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are:
1. An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication.
2. A deterministic upper bound of O(n^{3/2} log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem.
3. A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for its quantum communication complexity.
4. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon [Babai et al 1986]. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if $G$ is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.

Gábor Ivanyos, Hartmut Klauck, Troy Lee, Miklos Santha, and Ronald de Wolf. New bounds on the classical and quantum communication complexity of some graph properties. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 148-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{ivanyos_et_al:LIPIcs.FSTTCS.2012.148, author = {Ivanyos, G\'{a}bor and Klauck, Hartmut and Lee, Troy and Santha, Miklos and de Wolf, Ronald}, title = {{New bounds on the classical and quantum communication complexity of some graph properties}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {148--159}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.148}, URN = {urn:nbn:de:0030-drops-38523}, doi = {10.4230/LIPIcs.FSTTCS.2012.148}, annote = {Keywords: Graph properties, communication complexity, quantum communication} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

Abstract: We will discuss a general norm based framework for showing
lower bounds on communication complexity. An advantage of this approach is that one can use duality theory to obtain a lower bound quantity phrased as a
maximization problem, which can be more convenient to work with in showing lower bounds.
We discuss two applications of this approach.
1. The approximation rank of a matrix A is the minimum rank of a
matrix close to A in ell_infty norm. The logarithm of approximation rank lower bounds quantum communication complexity and is one of the most powerful techniques available, albeit difficult to compute in practice. We
show that an approximation norm known as gamma_2 is polynomially
related to approximation rank.
This results in a polynomial time algorithm to approximate
approximation rank, and also shows that the logarithm of approximation rank lower bounds quantum communication complexity even with entanglement which was previously not known.
2. By means of an approximation norm which lower bounds multiparty
number-on-the-forehead complexity, we show non-trivial lower bounds on the complexity of the disjointness function for up to c log log n players, c <1.

Troy Lee and Adi Shraibman. Approximation norms and duality for communication complexity lower bounds. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{lee_et_al:DagSemProc.08381.3, author = {Lee, Troy and Shraibman, Adi}, title = {{Approximation norms and duality for communication complexity lower bounds}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--9}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.3}, URN = {urn:nbn:de:0030-drops-17768}, doi = {10.4230/DagSemProc.08381.3}, annote = {Keywords: Communication complexity, lower bounds} }