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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value.
Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.

Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6, author = {Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos}, title = {{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.6}, URN = {urn:nbn:de:0030-drops-169444}, doi = {10.4230/LIPIcs.ESA.2022.6}, annote = {Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum} }

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**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

The famous least squares Monte Carlo (LSM) algorithm combines linear least square regression with Monte Carlo simulation to approximately solve problems in stochastic optimal stopping theory. In this work, we propose a quantum LSM based on quantum access to a stochastic process, on quantum circuits for computing the optimal stopping times, and on quantum techniques for Monte Carlo. For this algorithm, we elucidate the intricate interplay of function approximation and quantum algorithms for Monte Carlo. Our algorithm achieves a nearly quadratic speedup in the runtime compared to the LSM algorithm under some mild assumptions. Specifically, our quantum algorithm can be applied to American option pricing and we analyze a case study for the common situation of Brownian motion and geometric Brownian motion processes.

João F. Doriguello, Alessandro Luongo, Jinge Bao, Patrick Rebentrost, and Miklos Santha. Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 2:1-2:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{doriguello_et_al:LIPIcs.TQC.2022.2, author = {Doriguello, Jo\~{a}o F. and Luongo, Alessandro and Bao, Jinge and Rebentrost, Patrick and Santha, Miklos}, title = {{Quantum Algorithm for Stochastic Optimal Stopping Problems with Applications in Finance}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {2:1--2:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.2}, URN = {urn:nbn:de:0030-drops-165091}, doi = {10.4230/LIPIcs.TQC.2022.2}, annote = {Keywords: Quantum computation complexity, optimal stopping time, stochastic processes, American options, quantum finance} }

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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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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} }

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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 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.

Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gharibian_et_al:LIPIcs.MFCS.2018.58, author = {Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin}, title = {{Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.58}, URN = {urn:nbn:de:0030-drops-96409}, doi = {10.4230/LIPIcs.MFCS.2018.58}, annote = {Keywords: Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity} }

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

Let F_{q} be the finite field of size q and let l: F_{q}^{n} -> F_{q} be a linear function. We introduce the Learning From Subset problem LFS(q,n,d) of learning l, given samples u in F_{q}^{n} from a special distribution depending on l: the probability of sampling u is a function of l(u) and is non zero for at most d values of l(u). We provide a randomized algorithm for LFS(q,n,d) with sample complexity (n+d)^{O(d)} and running time polynomial in log q and (n+d)^{O(d)}. Our algorithm generalizes and improves upon previous results [Friedl et al., 2014; Gábor Ivanyos, 2008] that had provided algorithms for LFS(q,n,q-1) with running time (n+q)^{O(q)}. We further present applications of our result to the Hidden Multiple Shift problem HMS(q,n,r) in quantum computation where the goal is to determine the hidden shift s given oracle access to r shifted copies of an injective function f: Z_{q}^{n} -> {0, 1}^{l}, that is we can make queries of the form f_{s}(x,h) = f(x-hs) where h can assume r possible values. We reduce HMS(q,n,r) to LFS(q,n, q-r+1) to obtain a polynomial time algorithm for HMS(q,n,r) when q=n^{O(1)} is prime and q-r=O(1). The best known algorithms [Andrew M. Childs and Wim van Dam, 2007; Friedl et al., 2014] for HMS(q,n,r) with these parameters require exponential time.

Gábor Ivanyos, Anupam Prakash, and Miklos Santha. On Learning Linear Functions from Subset and Its Applications in Quantum Computing. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ivanyos_et_al:LIPIcs.ESA.2018.66, author = {Ivanyos, G\'{a}bor and Prakash, Anupam and Santha, Miklos}, title = {{On Learning Linear Functions from Subset and Its Applications in Quantum Computing}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {66:1--66:14}, 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.2018.66}, URN = {urn:nbn:de:0030-drops-95299}, doi = {10.4230/LIPIcs.ESA.2018.66}, annote = {Keywords: Learning from subset, hidden shift problem, quantum algorithms, linearization} }

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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)

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems.
Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.

Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 30:1-30:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{belovs_et_al:LIPIcs.CCC.2017.30, author = {Belovs, Aleksandrs and Ivanyos, G\'{a}bor and Qiao, Youming and Santha, Miklos and Yang, Siyi}, title = {{On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {30:1--30:24}, 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.2017.30}, URN = {urn:nbn:de:0030-drops-75260}, doi = {10.4230/LIPIcs.CCC.2017.30}, annote = {Keywords: Chevalley-Warning Theorem, Combinatorail Nullstellensatz, Polynomial Parity Argument, arithmetic circuit} }

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

A canonical result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Q_{ij} on a system of n qubits, and the task is to decide whether the Hamiltonian H = sum Q_{ij} has a 0-eigenvalue, or it is larger than 1/n^c for some c = O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding the ground state of 2-local frustration-free Hamiltonians of spin 1/2, a well-studied model believed to capture certain key properties in modern condensed matter physics. While Bravyi has shown that the quantum 2-SAT problem has a classical polynomial-time algorithm, the running time of his algorithm is O(n^4). In this paper we give a classical algorithm with linear running time in the number of local projectors, therefore achieving the best possible complexity.

Itai Arad, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. Linear Time Algorithm for Quantum 2SAT. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 15:1-15:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.ICALP.2016.15, author = {Arad, Itai and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu}, title = {{Linear Time Algorithm for Quantum 2SAT}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {15:1--15:14}, 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.2016.15}, URN = {urn:nbn:de:0030-drops-62795}, doi = {10.4230/LIPIcs.ICALP.2016.15}, annote = {Keywords: Quantum SAT, Davis-Putnam Procedure, Linear Time Algorithm} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC'13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.

Itai Arad, Adam Bouland, Daniel Grier, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.MFCS.2016.12, author = {Arad, Itai and Bouland, Adam and Grier, Daniel and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu}, title = {{On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.12}, URN = {urn:nbn:de:0030-drops-64284}, doi = {10.4230/LIPIcs.MFCS.2016.12}, annote = {Keywords: computational complexity, satisfiability problems, trial and error, quantum computing, learning theory} }

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

The subcube partition model of computation is at least as powerful as decision trees but no separation between these models was known. We show that there exists a function whose deterministic subcube partition complexity is asymptotically smaller than its randomized decision tree complexity, resolving an open problem of Friedgut, Kahn, and Wigderson (2002). Our lower bound is based on the information-theoretic techniques first introduced to lower bound the randomized decision tree complexity of the recursive majority function.
We also show that the public-coin partition bound, the best known lower bound method for randomized decision tree complexity subsuming other general techniques such as block sensitivity, approximate degree, randomized certificate complexity, and the classical adversary bound, also lower bounds randomized subcube partition complexity. This shows that all these lower bound techniques cannot prove optimal lower bounds for randomized decision tree complexity, which answers an open question of Jain and Klauck (2010) and Jain, Lee, and Vishnoi (2014).

Robin Kothari, David Racicot-Desloges, and Miklos Santha. Separating Decision Tree Complexity from Subcube Partition Complexity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 915-930, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kothari_et_al:LIPIcs.APPROX-RANDOM.2015.915, author = {Kothari, Robin and Racicot-Desloges, David and Santha, Miklos}, title = {{Separating Decision Tree Complexity from Subcube Partition Complexity}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {915--930}, 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.2015.915}, URN = {urn:nbn:de:0030-drops-53445}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.915}, annote = {Keywords: Decision tree complexity, query complexity, randomized algorithms, subcube partition complexity} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the nxn matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, while symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one.
Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independent of the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.

Gábor Ivanyos, Marek Karpinski, Youming Qiao, and Miklos Santha. Generalized Wong sequences and their applications to Edmonds' problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 397-408, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ivanyos_et_al:LIPIcs.STACS.2014.397, author = {Ivanyos, G\'{a}bor and Karpinski, Marek and Qiao, Youming and Santha, Miklos}, title = {{Generalized Wong sequences and their applications to Edmonds' problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {397--408}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.397}, URN = {urn:nbn:de:0030-drops-44741}, doi = {10.4230/LIPIcs.STACS.2014.397}, annote = {Keywords: symbolic determinantal identity testing, Edmonds' problem, maximum rank matrix completion, derandomization, Wong sequences} }

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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} }