The class QMA(k), introduced by , consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
First, in their original paper on @math , @cite_20 proved that a positive answer to the second question implies a positive answer to the third. That is, if @math protocols can be amplified, then @math for all @math .
@cite_20
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Given a linear code @math , one can define the @math -th power of @math as the span of all componentwise products of @math elements of @math . A power of @math may quickly fill the whole space. Our purpose is to answer the following question: does the square of a code "typically" fill the whole space? We give a positive answer, for codes of dimension @math and length roughly @math or smaller. Moreover, the convergence speed is exponential if the difference @math is at least linear in @math . The proof uses random coding and combinatorial arguments, together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank, and the number of their zeros. Let @math be an arbitrary polynomial of degree @math with @math non-zero integer coefficients of absolute value less than @math . In this paper, we answer the open question whether the real roots of @math can be computed with a number of arithmetic operations over the rational numbers that is polynomial in the input size of the sparse representation of @math . More precisely, we give a deterministic, complete, and certified algorithm that determines isolating intervals for all real roots of @math with @math many exact arithmetic operations over the rational numbers. When using approximate but certified arithmetic, the bit complexity of our algorithm is bounded by @math , where @math means that we ignore logarithmic. Hence, for sufficiently sparse polynomials (i.e. @math for a positive constant @math ), the bit complexity is @math . We also prove that the latter bound is optimal up to logarithmic factors. We study a one-parameter family of discrete dynamical systems called the @math -color firefly cellular automata (FCAs), which were introduced recently by the author. At each discrete time @math , each vertex in a graph has a state in @math , and a special state @math is designated as the blinking' state. At step @math , simultaneously for all vertices, the state of a vertex increments from @math to @math unless @math and at least one of its neighbors is in the state @math . A central question about this system is that on what class of network topologies synchrony is guaranteed to emerge. In a previous work, we have shown that for @math , every @math -coloring on a finite tree synchronizes iff the maximum degree is less than @math , and asked whether this behavior holds for all @math . In this paper, we answer the question positively for @math and negatively for all @math by constructing counterexamples on trees with maximum degree at most @math . What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Toth a number of years ago. The answer is trivial for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai abgsMiJoaakaaabaGaaG4maaWcbeaakiaac+cacaaIYaaaaa!3AF6! @math and, surprisingly, it is not difficult to prove [M2] for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai aa-1dacaWGUbWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaikdaaaa!3AF2! @math , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertesz [K] solved the case MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83DaG qaaiaa+bcacaGF8aWaaOaaaeaacaaIYaaaleqaaaaa!392C! @math . Here we fill the first gap, i.e., the maximal density is determined for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaie aacaWFZaaaleqaaOGaai4laiaaikdacqGH8aapcaWG3bGaeyizIm6a aOaaaeaacaaIZaaaleqaaaaa!3CCB! @math . A family of sets is said to be if its automorphism group is transitive, and if any two sets in the family have nonempty intersection. Our purpose here is to study the following question: for @math with @math , how large can a symmetric intersecting family of @math -element subsets of @math be? As a first step towards a complete answer, we prove that such a family has size at most [ (- c(n-2k) n k( n - k) ) n k , ] where @math is a universal constant. We also describe various combinatorial and algebraic approaches to constructing such families. Let @math be an induced subgraph of the hypercube @math for some @math . We show that if @math is a power of @math then, for sufficiciently large @math , the vertex set of @math can be partitioned into induced copies of @math . This answers a question of Offner. In fact, we prove a stronger statement: if @math is a subset of @math for some @math and if @math is a power of @math , then, for sufficiently large @math , @math can be partitioned into isometric copies of @math . For an @math -variate degree– @math real polynomial @math , we prove that @math is determined up to an additive @math as long as @math is a @math -wise independent distribution over @math for @math . This gives a broad class of explicit pseudorandom generators against degree- @math boolean threshold functions, and answers an open question of (FOCS 2009). Adaptivity is an important feature of data analysis---the choice of questions to ask about a dataset often depends on previous interactions with the same dataset. However, statistical validity is typically studied in a nonadaptive model, where all questions are specified before the dataset is drawn. Recent work by (STOC, 2015) and Hardt and Ullman (FOCS, 2014) initiated the formal study of this problem, and gave the first upper and lower bounds on the achievable generalization error for adaptive data analysis. Specifically, suppose there is an unknown distribution @math and a set of @math independent samples @math is drawn from @math . We seek an algorithm that, given @math as input, accurately answers a sequence of adaptively chosen queries about the unknown distribution @math . How many samples @math must we draw from the distribution, as a function of the type of queries, the number of queries, and the desired level of accuracy? In this work we make two new contributions: (i) We give upper bounds on the number of samples @math that are needed to answer statistical queries. The bounds improve and simplify the work of (STOC, 2015), and have been applied in subsequent work by those authors (Science, 2015, NIPS, 2015). (ii) We prove the first upper bounds on the number of samples required to answer more general families of queries. These include arbitrary low-sensitivity queries and an important class of optimization queries. As in , our algorithms are based on a connection with algorithmic stability in the form of differential privacy. We extend their work by giving a quantitatively optimal, more general, and simpler proof of their main theorem that stability implies low generalization error. We also study weaker stability guarantees such as bounded KL divergence and total variation distance. Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on @math with i.i.d. exponential weights on the vertices. Fix two points @math and @math for some @math , and consider the maximal paths @math and @math starting at @math and @math respectively to the point @math for @math . Our object of study is the point of coalescence, i.e., the point @math with smallest @math . We establish that the distance to coalescence @math scales as @math , by showing the upper tail bound @math for some @math . We also consider the problem of coalescence for semi-infinite geodesics. For the almost surely unique semi-infinite geodesics in the direction @math starting from @math and @math , we establish the optimal tail estimate @math , for the point of coalescence @math . This answers a question left open by Pimentel (Ann. Probab., 2016) who proved the corresponding lower bound. One of the fundamental questions of Algorithmic Mechanism Design is whether there exists an inherent clash between truthfulness and computational tractability: in particular, whether polynomial-time truthful mechanisms for combinatorial auctions are provably weaker in terms of approximation ratio than non-truthful ones. This question was very recently answered for universally truthful mechanisms for combinatorial auctions D11 , and even for truthful-in-expectation mechanisms DughmiV11 . However, both of these results are based on information-theoretic arguments for valuations given by a value oracle, and leave open the possibility of polynomial-time truthful mechanisms for succinctly described classes of valuations. This paper is the first to prove computational hardness results for truthful mechanisms for combinatorial auctions with succinctly described valuations. We prove that there is a class of succinctly represented submodular valuations for which no deterministic truthful mechanism provides an @math -approximation for a constant @math , unless @math ( @math denotes the number of items). Furthermore, we prove that even truthful-in-expectation mechanisms cannot approximate combinatorial auctions with certain succinctly described submodular valuations better than within @math , where @math is the number of bidders and @math some absolute constant, unless @math . In addition, we prove computational hardness results for two related problems. We prove that the determinantal complexity of a hypersurface of degree @math d>2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the @math 3×3 permanent is 7. We also prove that for @math n>3, there is no nonsingular hypersurface in @math Pn of degree d that has an expression as a determinant of a @math d×d matrix of linear forms, while on the other hand for @math n≤3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. Given a static array of @math totally ordered objects, the range minimum query problem is to build a data structure that allows us to answer efficiently subsequent on-line queries of the form “what is the position of a minimum element in the subarray ranging from @math to @math ?”. We focus on two settings, where (1) the input array is available at query time, and (2) the input array is available only at construction time. In setting (1), we show new data structures (a) of size @math bits and query time @math for any positive integer function @math for an arbitrary constant @math , or (b) with @math bits and @math query time, where @math denotes the empirical entropy of @math th order of the input array. In setting (2), we give a data structure of size @math bits and query time @math . All data structures can be constructed in linear time and almost in-place. The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a computational resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a natural supervised learning problem --- we consider agnostic PAC learning of halfspaces over @math -sparse vectors in @math . This class is inefficiently learnable using @math examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random @math formulas is hard, it is impossible to efficiently learn this class using only @math examples. We further show that under stronger hardness assumptions, even @math examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using @math examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem. The distributed single-source shortest paths problem is one of the most fundamental and central problems in the message-passing distributed computing. Classical Bellman-Ford algorithm solves it in @math time, where @math is the number of vertices in the input graph @math . Peleg and Rubinovich (FOCS'99) showed a lower bound of @math for this problem, where @math is the hop-diameter of @math . Whether or not this problem can be solved in @math time when @math is relatively small is a major notorious open question. Despite intensive research LP13,N14,HKN15,EN16,BKKL16 that yielded near-optimal algorithms for the approximate variant of this problem, no progress was reported for the original problem. In this paper we answer this question in the affirmative. We devise an algorithm that requires @math time, for @math , and @math time, for larger @math . This running time is sublinear in @math in almost the entire range of parameters, specifically, for @math . For the all-pairs shortest paths problem, our algorithm requires @math time, regardless of the value of @math . We also devise the first algorithm with non-trivial complexity guarantees for computing exact shortest paths in the multipass semi-streaming model of computation. From the technical viewpoint, our algorithm computes a hopset @math of a skeleton graph @math of @math without first computing @math itself. We then conduct a Bellman-Ford exploration in @math , while computing the required edges of @math on the fly. As a result, our algorithm computes exactly those edges of @math that it really needs, rather than computing approximately the entire @math . We prove new lower bounds for locally decodable codes and private information retrieval. We show that a 2-query LDC encoding n-bit strings over an l-bit alphabet, where the decoder only uses b bits of each queried position, needs code length @math Similarly, a 2-server PIR scheme with an n-bit database and t-bit queries, where the user only needs b bits from each of the two l-bit answers, unknown to the servers, satisfies @math This implies that several known PIR schemes are close to optimal. Our results generalize those of [8], who proved roughly the same bounds for linear LDCs and PIRs. Like earlier work by Kerenidis and de Wolf [12], our classical bounds are proved using quantum computational techniques. In particular, we give a tight analysis of how well a 2-input function can be computed from a quantum superposition of both inputs. We show a parallel repetition theorem for the entangled value @math of any two-player one-round game @math where the questions @math to Alice and Bob are drawn from a product distribution on @math . We show that for the @math -fold product @math of the game @math (which represents the game @math played in parallel @math times independently), @math , where @math and @math represent the sets from which the answers of Alice and Bob are drawn. The Gap-Hamming-Distance problem arose in the context of proving space lower bounds for a number of key problems in the data stream model. In this problem, Alice and Bob have to decide whether the Hamming distance between their @math -bit input strings is large (i.e., at least @math ) or small (i.e., at most @math ); they do not care if it is neither large nor small. This @math gap in the problem specification is crucial for capturing the approximation allowed to a data stream algorithm. Thus far, for randomized communication, an @math lower bound on this problem was known only in the one-way setting. We prove an @math lower bound for randomized protocols that use any constant number of rounds. As a consequence we conclude, for instance, that @math -approximately counting the number of distinct elements in a data stream requires @math space, even with multiple (a constant number of) passes over the input stream. This extends earlier one-pass lower bounds, answering a long-standing open question. We obtain similar results for approximating the frequency moments and for approximating the empirical entropy of a data stream. In the process, we also obtain tight @math lower and upper bounds on the one-way deterministic communication complexity of the problem. Finally, we give a simple combinatorial proof of an @math lower bound on the one-way randomized communication complexity. We give efficient algorithms for volume sampling, i.e., for picking @math -subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). In other words, we can efficiently sample @math -subsets of @math with probabilities proportional to the corresponding @math by @math principal minors of any given @math by @math positive semi definite matrix. This solves an open problem from the monograph on spectral algorithms by Kannan and Vempala (see Section @math of KV , also implicit in BDM, DRVW ). Our first algorithm for volume sampling @math -subsets of rows from an @math -by- @math matrix runs in @math arithmetic operations (where @math is the exponent of matrix multiplication) and a second variant of it for @math -approximate volume sampling runs in @math arithmetic operations, which is almost linear in the size of the input (i.e., the number of entries) for small @math . Our efficient volume sampling algorithms imply the following results for low-rank matrix approximation: (1) Given @math , in @math arithmetic operations we can find @math of its rows such that projecting onto their span gives a @math -approximation to the matrix of rank @math closest to @math under the Frobenius norm. This improves the @math -approximation of Boutsidis, Drineas and Mahoney BDM and matches the lower bound shown in DRVW . The method of conditional expectations gives a algorithm with the same complexity. The running time can be improved to @math at the cost of losing an extra @math in the approximation factor. (2) The same rows and projection as in the previous point give a @math -approximation to the matrix of rank @math closest to @math under the spectral norm. In this paper, we show an almost matching lower bound of @math , even for @math . We study the aggregate group nearest neighbor searching for the MAX operator in the plane. For a set @math of @math points and a query set @math of @math points, the query asks for a point of @math whose maximum distance to the points in @math is minimized. We present data structures for answering such queries for both @math and @math distance measures. Previously, only heuristic and approximation algorithms were given for both versions. For the @math version, we build a data structure of O(n) size in @math time, such that each query can be answered in @math time. For the @math version, we build a data structure in @math time and @math space, such that each query can be answered in @math time, and alternatively, we build a data structure in @math time and space for any @math , such that each query can be answered in @math time. Further, we extend our result for the @math version to the top- @math queries where each query asks for the @math points of @math whose maximum distances to @math are the smallest for any @math with @math : We build a data structure of O(n) size in @math time, such that each top- @math query can be answered in @math time.