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

In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n^{2-o(1)} time to solve, for all d=omega(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph.
We prove that OVC is true in several computational models of interest:
- For all sufficiently large n and d, OV for n vectors in {0,1}^d has branching program complexity Theta~(n * min(n,2^d)). In particular, the lower and upper bounds match up to polylog factors.
- OV has Boolean formula complexity Theta~(n * min(n,2^d)), over all complete bases of O(1) fan-in.
- OV requires Theta~(n * min(n,2^d)) wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in.
Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV.
The proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the "average case" even for AC^0 formulas:
For all p in (0,1) there is a delta_p > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of O(n^{2-delta_p}) size, on all but a o_n(1) fraction of instances. Moreover, lim_{p - > 1}delta_p = 1.

Daniel M. Kane and Richard Ryan Williams. The Orthogonal Vectors Conjecture for Branching Programs and Formulas. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 48:1-48:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kane_et_al:LIPIcs.ITCS.2019.48, author = {Kane, Daniel M. and Williams, Richard Ryan}, title = {{The Orthogonal Vectors Conjecture for Branching Programs and Formulas}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {48:1--48:15}, 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.2019.48}, URN = {urn:nbn:de:0030-drops-101418}, doi = {10.4230/LIPIcs.ITCS.2019.48}, annote = {Keywords: fine-grained complexity, orthogonal vectors, branching programs, symmetric functions, Boolean formulas} }

Document

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

We define a model of size-S R-way branching programs with oracles that can make up to S distinct oracle queries over all of their possible inputs, and generalize a lower bound proof strategy of Beame [SICOMP 1991] to apply in the case of random oracles. Through a series of succinct reductions, we prove that the following problems require randomized algorithms where the product of running time and space usage must be Omega(n^2/poly(log n)) to obtain correct answers with constant nonzero probability, even for algorithms with constant-time access to a uniform random oracle (i.e., a uniform random hash function):
- Given an unordered list L of n elements from [n] (possibly with repeated elements), output [n]-L.
- Counting satisfying assignments to a given 2CNF, and printing any satisfying assignment to a given 3CNF. Note it is a major open problem to prove a time-space product lower bound of n^{2-o(1)} for the decision version of SAT, or even for the decision problem Majority-SAT.
- Printing the truth table of a given CNF formula F with k inputs and n=O(2^k) clauses, with values printed in lexicographical order (i.e., F(0^k), F(0^{k-1}1), ..., F(1^k)). Thus we have a 4^k/poly(k) lower bound in this case.
- Evaluating a circuit with n inputs and O(n) outputs.
As our lower bounds are based on R-way branching programs, they hold for any reasonable model of computation (e.g. log-word RAMs and multitape Turing machines).

Dylan M. McKay and Richard Ryan Williams. Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mckay_et_al:LIPIcs.ITCS.2019.56, author = {McKay, Dylan M. and Williams, Richard Ryan}, title = {{Quadratic Time-Space Lower Bounds for Computing Natural Functions with a Random Oracle}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {56:1--56:20}, 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.2019.56}, URN = {urn:nbn:de:0030-drops-101493}, doi = {10.4230/LIPIcs.ITCS.2019.56}, annote = {Keywords: branching programs, random oracles, time-space tradeoffs, lower bounds, SAT, counting complexity} }

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Invited Paper

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

In 2010, the author proposed a program for proving lower bounds in circuit complexity, via faster algorithms for circuit satisfiability and related problems. This talk will give an overview of how the program works, report on the successes of this program so far, and outline open frontiers that have yet to be resolved.

Richard Ryan Williams. Lower Bounds by Algorithm Design: A Progress Report (Invited Paper). In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{williams:LIPIcs.ICALP.2018.4, author = {Williams, Richard Ryan}, title = {{Lower Bounds by Algorithm Design: A Progress Report}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {4:1--4:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.4}, URN = {urn:nbn:de:0030-drops-90088}, doi = {10.4230/LIPIcs.ICALP.2018.4}, annote = {Keywords: circuit complexity, satisfiability, derandomization} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over R) of functions from some "simple" class C. In particular, given C we are interested in finding low-complexity functions lacking sparse representations. When C forms a basis for the space of Boolean functions (e.g., the set of PARITY functions or the set of conjunctions) this sort of problem has a well-understood answer; the problem becomes interesting when C is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where C is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts.
We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. Let alpha(n) be an unbounded function such that n^{alpha(n)} is time constructible (e.g. alpha(n) = log^*(n)). We show:
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many linear threshold functions to represent (depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds).
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many ReLU gates to represent (depth-two neural networks with ReLU activation function).
- Functions in NTIME[n^{alpha(n)}] that require super-polynomially many O(1)-degree F_p-polynomials to represent exactly, for every prime p (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in E^{NP} requiring 2^{Omega(n)} linear combinations.
- Functions in NTIME[n^{poly(log n)}] that require super-polynomially many ACC ° THR circuits to represent exactly (further generalizing the recent lower bounds of Murray and the author). We also obtain "fixed-polynomial" lower bounds for functions in NP, for the first three representation classes. All our lower bounds are obtained via algorithms for analyzing linear combinations of simple functions in the above scenarios, in ways which substantially beat exhaustive search.

Richard Ryan Williams. Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 6:1-6:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{williams:LIPIcs.CCC.2018.6, author = {Williams, Richard Ryan}, title = {{Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {6:1--6:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.6}, URN = {urn:nbn:de:0030-drops-88846}, doi = {10.4230/LIPIcs.CCC.2018.6}, annote = {Keywords: linear threshold functions, lower bounds, neural networks, low-degree polynomials} }

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

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit C(x_1,...,x_n) of size s and degree d over a field F, and any inputs a_1,...,a_K in F}^n,
- the Prover sends the Verifier the values C(a_1), ..., C(a_K) in F and a proof of ~O(K * d) length, and
- the Verifier tosses poly(log(dK|F|epsilon)) coins and can check the proof in about ~O}(K * (n + d) + s) time, with probability of error less than epsilon.
For small degree d, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in c^{n} time (for various c < 2) for the Permanent, #Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of 0-1 linear programs. In general, the value of any polynomial in Valiant's class VP can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas running in 2^{2n/3+o(n)} time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in n^{k/2+O(1)}-time for counting k-cliques in graphs.
We point to some potential future directions for refuting the Nondeterministic Strong ETH.

Richard Ryan Williams. Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{williams:LIPIcs.CCC.2016.2, author = {Williams, Richard Ryan}, title = {{Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {2:1--2:17}, 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.2016.2}, URN = {urn:nbn:de:0030-drops-58307}, doi = {10.4230/LIPIcs.CCC.2016.2}, annote = {Keywords: counting complexity, exponential-time hypothesis, interactive proofs, Merlin-Arthur games} }

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Invited Talk

**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

In circuit complexity, the polynomial method is a general approach to proving circuit lower bounds in restricted settings. One shows that functions computed by sufficiently restricted circuits are "correlated" in some way with a low-complexity polynomial, where complexity may be measured by the degree of the polynomial or the number of monomials. Then, results limiting the capabilities of low-complexity polynomials are extended to the restricted circuits.
Old theorems proved by this method have recently found interesting applications to the design of algorithms for basic problems in the theory of computing. This paper surveys some of these applications, and gives a few new ones.

Richard Ryan Williams. The Polynomial Method in Circuit Complexity Applied to Algorithm Design (Invited Talk). In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 47-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{williams:LIPIcs.FSTTCS.2014.47, author = {Williams, Richard Ryan}, title = {{The Polynomial Method in Circuit Complexity Applied to Algorithm Design}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {47--60}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.47}, URN = {urn:nbn:de:0030-drops-48328}, doi = {10.4230/LIPIcs.FSTTCS.2014.47}, annote = {Keywords: algorithm design, circuit complexity, polynomial method} }

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