On the pseudo-deterministic query complexity of NP search problems

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is O(√n). We further show that the quantum query complexity of the same function is O(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/log(n)) and bounded-error randomized query complexity R(g) = O(√n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is QE(g) = O(√n). These functions show that the relations D(f) = O(R1(f)2) and R0(f) = O(R(f)2) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R0, a 3/2-power separation between QE and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

In 1983, Aldous proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid \left[ {1:n} \right]^d from \Theta (n^{d - 1} ) to {\rm O}(d^{1/2} n^{d/2} ). It remains open whether randomization helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (\Omega (n))^{d - 1} on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid \left[ {1:n} \right]^d Our result can be extended to the black-box query model for Sperner's Lemma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over \left[ {1:n} \right]^d is \Theta (n^d ), the randomized query model over \left[ {1:n} \right]^d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is \Theta (n^{d - 1} ).

We analyze the efficiency of the random walk algorithm on random 3-CNF instances, and prove linear upper bounds on the running time of this algorithm for small clause density, less than 1.63. Our upper bound matches the observed running time to within a multiplicative factor. This is the first sub-exponential upper bound on the running time of a local improvement algorithm on random instances. Our proof introduces a simple, yet powerful tool for analyzing such algorithms, which may be of further use. This object, called a terminator, is a weighted satisfying assignment. We show that any CNF having a good (small weight) terminator is assured to be solved quickly by the random walk algorithm. This raises the natural question of the terminator threshold which is the maximal clause density for which such assignments exist (with high probability). We use the analysis of the pure literal heuristic presented by Broder, Frieze and Upfal and show that for small clause densities good terminators exist. Thus we show that the pure literal threshold (/spl ap/ 1.63) is a lower bound on the terminator threshold. One nice property of terminators is that they can be found efficiently, via linear programming. This makes tractable the future investigation of the terminator threshold, and also provides an efficiently computable certificate for short running time of the simple random-walk heuristic.

Longest common substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decision version of this problem, LCS with threshold \(d\) , asks whether two length- \(n\) input strings have a common substring of length \(d\) . The two extreme cases, \(d=1\) and \(d=n\) , correspond, respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case \(1\ll d\ll n\) was not fully understood. We show that the complexity of LCS with threshold \(d\) smoothly interpolates between the two extreme cases up to \(n^{o(1)}\) factors: — LCS with threshold \(d\) has a quantum algorithm in \(n^{2/3+o(1)}/d^{1/6}\) query complexity and time complexity, and requires at least \(\Omega(n^{2/3}/d^{1/6})\) quantum query complexity. Our result improves upon previous upper bounds \(\widetilde{O}(\min\{n/d^{1/2},n^{2/3}\})\) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples \(n/\tau^{1-o(1)}\) synchronizing positions in the string depending on their length- \(\Theta(\tau)\) contexts, and each synchronizing position can be reported by a quantum algorithm in \(\widetilde{O}(\tau^{1/2+o(1)})\) time. Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting. As another application of our quantum string synchronizing set, we study the \(k\) -mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most \(k\) Hamming mismatches. Using a structural result of Charalampopoulos et al. (FOCS 2020), we obtain: — \(k\) -mismatch matching has a quantum algorithm with \(k^{3/4}n^{1/2+o(1)}\) query complexity and \(\widetilde{O}(kn^{1/2})\) time complexity. We also observe a non-matching quantum query lower bound of \(\Omega(\sqrt{kn})\) .

A Note on Varying Cardinality in the Average Case Setting

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we show that the deterministic query complexity of total search problems is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13). Using our proof along with a decision-tree manipulation technique, we give a simple and self-contained proof that the $$\text{SearchCNF}$$ problem on random $$\text{k}$$ -CNF has pseudo-deterministic query complexity $$\Omega(n^{1/3})$$ ; a lower bound of $$\Omega(\sqrt{n})$$ is known, due to Goldwasser, Impagliazzo, Pitassi, and Santhanam (CCC'21), but via a significantly more complex proof. We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) We exhibit an $$\text{exp}(\widetilde{\Omega}(n^{1/4}))$$ separation for the $$\text{SearchCNF}$$ relation for random $$k$$ -CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of $$\text{SearchCNF}$$ associated with random $$k$$ -CNFs. (2) We exhibit an $${\text{exp}(\Omega(n))}$$ separation for the $$\text{ApproxHW}$$ relation. The previous best known separation for any relation was $${\text{exp}(\Omega(n^{1/2}))}$$ . We also separate pseudo-determinism from randomness in $$\text{AND}$$ and $$\text{CONJ}$$ decision trees, and determinism from pseudo-determinism in $$\text{Parity}$$ decision trees. Finally, for a hypercube colouring problem, that was introduced by Goldwasswer et al. to analyze the pseudo-deterministic complexity of a complete problem in $$\text{TFNPdt}$$ , we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is $${\Omega(n^{1/3})}$$ ; Goldwasser et al. showed an $${\Omega(n^{1/2})}$$ bound for general block-sensitivity.

Complexity of approximation with relative error criterion in worst, average, and probabilistic settings

In a sampling problem, we are given an input \(x\in\left\{ 0,1\right\} ^{n}\), and asked to sample approximately from a probability distribution \(\mathcal{D}_{x}\) over \(\operatorname*{poly}\left( n\right) \)-bit strings. In a search problem, we are given an input \(x\in\left\{ 0,1\right\} ^{n}\), and asked to find a member of a nonempty set A x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity to show that sampling and search problems are “essentially equivalent.” More precisely, for any sampling problem S, there exists a search problem R S such that, if \(\mathcal{C}\) is any “reasonable” complexity class, then R S is in the search version of \(\mathcal{C}\) if and only if S is in the sampling version. What makes this nontrivial is that the same R S works for every \(\mathcal{C}\).As an application, we prove the surprising result that SampP = SampBQP if and only if FBPP = FBQP. In other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve.KeywordsNash EquilibriumQuantum ComputerTuring MachineQuantum AlgorithmSearch ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dynamic 3-sided planar range queries with expected doubly-logarithmic time

In a landmark survey of visibility algorithms in 1974, Sutherland, Sproull, and Schumacker stated that We believe the principal untappedsource of help for hidden surface algorithms lies in frame and object coherence. Making use of such temporal coherence appears to be difficult, however, for scenes with moving objects. Almost three decades later Cohen-Or, Chrysanthou, and Silva stated that The efficient handing of dynamic scenes is an open area of research at this point. This five minute video describes a kinetic visibility algorithm formoving objects in the plane that is provably efficient in an average case setting.

Probabilistic Analysis of Simpson′s Quadrature

Termination Criteria for Linear Problems

Journal Article Complexity of deterministic and randomized methods for multivariate integration problems for the class H p Λ (I d ) Get access Fang Gensun, Fang Gensun **Email: fanggs@bnu.edu.cn Search for other works by this author on: Oxford Academic Google Scholar Ye Peixin Ye Peixin Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 25, Issue 3, July 2005, Pages 473–485, https://doi.org/10.1093/imanum/dri006 Published: 01 July 2005

We discuss the best approximation of periodic functions by trigonometric polynomials and the approximation by Fourier partial summation operators, Vallee-Poussin operators, Ces`aro operators, Abel operators, and Jackson operators, respectively, on the Sobolev space with a Gaussian measure and obtain the average error estimations. We show that, in the average case setting, the trigonometric polynomial subspaces are the asymptotically optimal subspaces in the Lq space for 1 ⩽ q < ∞, and the Fourier partial summation operators and the Vallee-Poussin operators are the asymptotically optimal linear operators and are as good as optimal nonlinear operators in the Lq space for 1 ⩽ q < ∞.

Integration and $L_2$-Approximation: Average Case Setting with Isotropic Wiener Measure for Smooth Functions